matrix regular orders on operator spaces by walter...
TRANSCRIPT
MATRIX REGULAR ORDERS ON OPERATOR SPACES
BY
WALTER JAMES SCHREINER
B.A., College of St. Thomas, 1968M.A., College of St. Thomas, 1969
M.S., University of Notre Dame, 1973
THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 1995
Urbana, Illinois
Abstract
In this thesis, the concept of the regular (or Riesz) norm on ordered real Banach spaces is
generalized to matrix ordered complex operator spaces in a way that respects the matricial
structure of the operator space. A norm on an ordered real Banach space E is regular if:
(1) −x ≤ y ≤ x implies that ‖y‖ ≤ ‖x‖; and (2) ‖y‖ < 1 implies the existence of x ∈ E
such that ‖x‖ < 1 and −x ≤ y ≤ x. A matrix ordered operator space is called matrix
regular if, at each matrix level, the restriction of the norm to the self-adjoint elements is
a regular norm. In such a space, elements at each matrix level can be written as linear
combinations of four positive elements.
After providing the necessary background material on operator spaces, especially with
respect to the Haagerup (⊗h), operator projective (∧⊗), and operator injective (
∨⊗) tensor
products, the concept of the matrix ordered operator space is made specific in such a
way as to be a natural generalization of ordered real and complex Banach spaces. For
the case where V is a matrix ordered operator space, a natural cone is defined on the
operator space X∗⊗h V ⊗h X so as to make it a matrix ordered operator space. Exploiting
the advantages gained by taking X to be the column Hilbert space Hc, an equivalence is
established between the matrix regularity of a space and that of its operator dual.
Beginning with the fact that all C∗-algebras, and in fact all operator systems, are matrix
regular, it is shown that all operator spaces of the form X∗ ⊗h V ⊗h X and CB(V,B(H))
are matrix regular whenever V is. Reverse implications are also shown in some cases. The
replacement of B(H) by an injective von Neumann algebra R is explored, leading to more
general results and some extra results regarding R′-module projective tensor products.
Complex interpolation is used to define operator space structures on the Schatten class
spaces Sp and the commutative Lp-spaces. These spaces are then shown to be matrix
regular. Some generalized Schatten class spaces are also shown to be matrix regular.
Finally, as an application, an alternative proof is presented for the Christensen-Sinclair
Multilinear Representation Theorem that depends on matrix regularity rather than on
Wittstock’s complicated concept of matricial sublinearity.
iii
Dedication
I dedicate this thesis to my parents.
iv
Acknowledgement
Though I have written this thesis, I have hardly been alone throughout the process of
its development. I wish to thank the following for their role in making this thesis possible:
z Professor Zhong-Jin Ruan, my advisor, for all his help, understanding, patience,
and insights into Chinese history and culture.
z My parents Walter and Helen for their constant love, encouragement, and support.
I am only sorry that my mother did not live to see its completion, but I’m sure she
knows from her place in heaven.
z The St. Paul-Minneapolis Province of the Christian Brothers, the Catholic religious
order to which I belong, for giving me the opportunity, time, encouragement, and
support needed to pursue graduate studies in Mathematics. I especially wish to
thank Brothers Dominic Ehrmantraut, Frank Walsh, Thomas Sullivan, and Frank
Carr for all their support and help, and the De La Salle communities of Minneapolis
and Chicago for being my homes away from home.
z Professor Wilson Zaring, retired Director of Graduate Studies, for a most encourag-
ing and helpful letter when I inquired about the possibility of beginning a graduate
program at the age of 44.
z Professor Richard Jerrard, current Director of Graduate Studies, for his help in
navigating the practicalities of graduate student life.
z Adam Lewenberg for preparing the TEX thesis style that was used to prepare this
manuscript, and for responding to several special requests.
z The United States Department of Education and the University of Illinois for two
National Need Fellowships and one University Fellowship, respectively.
z The Research Board of the University of Illinois for a Research Assistantship funded
by an Arnold O. Beckman Research Grant to my advisor, Professor Zhong-Jin
Ruan.
z The Fields Institute for Research in the Mathematical Sciences for their invitation
to participate in their year on operator algebras.
z My special friends and officemates that kept me going throughout the past six
years, especially Kevin Fitzgerald, Rick Faber, Jan Figa, and Chris Oliver.
v
Table of Contents
1. Operator Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Operator Space Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Operator Space Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Matrix Ordered Operator Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1. Ordered Real Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2. Ordered Complex Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3. Matrix Ordered Operator Spaces . . . . . . . . . . . . . . . . . . . . . . . . 22
3. Matrix Regularity and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1. Matrix Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2. Matrix Regularity and Duality . . . . . . . . . . . . . . . . . . . . . . . . . 48
4. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1. Expanding the Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2. Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5. Matrix Regularity of Schatten Class and Lp-spaces . . . . . . . . . . . 74
5.1. Generalized Schatten Class Spaces . . . . . . . . . . . . . . . . . . . . . . . 74
5.2. Commutative Lp-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6. An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Appendix: Ordered Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 106
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
vi
1. Operator Spaces
1.1. Introduction
Given a Hilbert space H, we let B(H) denote the bounded operators on H . A (concrete)
operator space V on H is a linear subspace of B(H). In this case, we can then assign norms
to each of the linear spaces M n(V ) by means of the inclusion M n(V ) ⊆ M n(B(H)) =
B(Hn).
An abstract operator space is a vector space V together with a norm ‖·‖ defined on each
of the matrix spaces M n(V ) such that
(1) ‖v ⊕ w‖ = max‖v‖ , ‖w‖, and
(2) ‖αvβ‖ ≤ ‖α‖ ‖v‖ ‖β‖
for all v ∈ M n(V ), w ∈ M m(V ), and α, β ∈ M n, where M n denotes M n(C ) = B(C n). We
let Mn(V ) denote M n(V ) with the given norm. It is clear that each concrete operator
space is an abstract operator space.
Given abstract operator spaces V and W and a linear map ϕ : V → W , there are
corresponding linear maps ϕn : Mn(V ) → Mn(W ) defined by ϕn(v) = [ϕ(vij)] for all
v = [vij ] ∈ Mn(V ). The completely bounded norm of ϕ is defined by
‖ϕ‖cb = sup‖ϕn‖ : n ∈ N
(which may be infinite). We say ϕ is completely bounded (respectively, completely con-
tractive) if ‖ϕ‖cb < ∞ (respectively, ‖ϕ‖cb ≤ 1). We note that the norms ‖ϕn‖ form an
increasing sequence
‖ϕ‖ ≤ ‖ϕ2‖ ≤ · · · ≤ ‖ϕn‖ ≤ · · · ≤ ‖ϕ‖cb ,
and that ‖ϕ‖cb is a norm on the linear space CB(V,W ) of completely bounded maps. We
define ϕ to be a complete isometry if each map ϕn : Mn(V ) → Mn(W ) is an isometry.
Ruan [47, 28] has shown that each abstract operator space V is completely isometric to
a concrete operator space W ⊆ B(H), so we will no longer distinguish between concrete
and abstract operator spaces. Also, if V is an operator space, so is its closure V since an
embedding V → B(H) can be extended to V .
1
From [29], if W is a closed subspace of an operator space V , then Mn(W ) is closed in
Mn(V ) for each n ∈ N , and furthermore, an operator space V is complete if and only if
any one of the normed spaces Mn(V ) is complete.
Assumption. Throughout this thesis we shall assume, without further statement, that all
given operator spaces are complete.
In this thesis we consider order structures, in particular matrix order structures, on
operator spaces. The only C∗-algebras with a Banach lattice structure for the self-adjoint
elements are those of the form C(X) where X is a compact Hausdorff space. In general,
unital C∗-algebras (including B(H)) have an order unit and a (proper) cone which is
Archimedean. However, an operator space may not contain a unit and may possess an
order structure unrelated to that of the B(H) in which it is completely isometrically
embedded.
We will consider operator spaces where we can define a set of self-adjoint elements at
each matrix level by means of an involution which is an isometry, and a partial ordering
of those self-adjoint elements such that the the intersection of those self-adjoint elements
with the open unit ball is absolutely order convex and absolutely dominated. A convex
subset S is absolutely order convex if x ∈ S and −x ≤ y ≤ x implies y ∈ S, and absolutely
dominated if y ∈ S implies the existence of x ∈ X with −x ≤ y ≤ x. Primary examples
of such operator spaces are all C∗-algebras along with what we shall later refer to as their
operator duals. We shall call such operator spaces matrix regular. In these spaces, there
is a relationship between norm and order, and there are enough positive elements so that
each element can be written as a linear combination of positive elements.
In the remainder of this chapter, we will gather the majority of the results that we will
need from the theory of operator spaces. We will give special focus to three operator space
tensor products that are crucial to our work.
In Chapter 2, we show how there is a natural progression from ordered real Banach
spaces to ordered complex Banach spaces, and then to matrix ordered operator spaces. We
develop the matrix order properties that we will need for our main work, concentrating
especially on operator spaces of the form X∗ ⊗h V ⊗h X where V is a matrix ordered
operator space and X is an operator space.
2
The main part of our work is found in Chapter 3. We define the concept of the matrix
regular operator space and show that an operator space is matrix regular if and only if
its operator dual is. To accomplish this, we exploit the use of the column Hilbert space
Hc and row Hilbert space Hr in operator spaces of the form Hr ⊗h V ⊗h Hc where V is a
matrix regular operator space.
In Chapter 4, we provide some general extensions to the class of operator spaces having
the property of matrix regularity. We first generalize on the spaces Hr ⊗h V ⊗h Hc. Then
we get further expansion of our results by replacing instances of use of B(H) by injective
von Neumann algebras.
We look at two important classes of examples in Chapter 5. These are the generalized
Schatten class spaces and the commutative Lp(µ) spaces. We use the method of complex
interpolation to define “natural” operator space structures on these spaces, and then prove
that all these spaces are matrix regular.
We develop a representation theorem for a class of completely bounded self-adjoint
maps in Chapter 6. This leads to a new proof of the Christensen-Sinclair Multilinear
Representation Theorem that depends on matrix regularity rather than on the rather
complicated concept of matricial sublinearity as developed by Wittstock.
Finally, we provide an Appendix which contains a development of a key theorem from
the theory of ordered real Banach spaces.
1.2. Operator Space Properties
In this section we will consider general properties of operator spaces that we will need to
refer to later in the context of ordered operator spaces. We first introduce some notation.
Notation. For an operator space V ⊆ B(H), we let M I(V ) denote the set of square
matrices [vij ] with vij ∈ V , i, j ∈ I, I an arbitrary index set. We let M I = M I(C ). For
I finite, we identify M I(V ) and M n(V ) where I ↔ 1, . . . , n is a bijection. We have
M n = M n(C ). For I countable, we identify M I(V ) and M ∞(V ), and let M ∞ = M ∞(C ).
For a finite subset ∆ ⊆ I and v ∈ M I(V ), we have the truncated matrix v∆ = [vij]i,j∈∆ ∈
M ∆(V ). Regarding v∆ as a net in M I(V ) by ordering the finite sets ∆ by inclusion,
‖v‖ = sup∆
∥∥v∆∥∥ = lim
∆
∥∥v∆∥∥ ,
3
and we define MI(V ) to be the linear space of all such matrices with ‖v‖ < ∞.
We define FI(V ) =⋃
M∆(V ) and FI =⋃
M∆. We let KI(V ) denote the norm closure
of FI(V ) in MI(V ).
Letting l2(I) denote the Hilbert space of square summable generalized sequences (αi)i∈I ,
αi ∈ C , we have that for any index set I , MI(V ) is an operator space since it may be
identified with the linear space of operators v = [vij ] ∈ B(H ⊗ l2(I)) for which vij ∈ V .
From [24], MI(V ) is norm complete if V is. With E∆ the projection on the subspace
H ⊗ l2(∆) of H ⊗ l2(I), we have that v∆ = E∆vE∆. We denote E∆ as En when ∆ =
1, . . . , n. We have (also from [24]) that KI(V ) is the set of v ∈ MI(V ) for which
‖v − E∆vE∆‖ → 0. We identify KI = KI(C ) with the C∗-algebra of compact operators
on l2(I). KI(V ) is norm complete if V is.
In most cases, a Banach space V can be given multiple matrix norm structures, each of
which satisfies the operator space conditions. Then any two such norms must be equivalent
on Mn(V ) for all n ∈ N . To see how this follows directly from the definition of an operator
space, we let Ei = [ 0 . . . 1i . . . 0 ] ∈ M1,n. Then ‖Ei‖ = 1 and, for v ∈ Mn(V ),
‖vij‖ =∥∥EivE∗j
∥∥ ≤ ‖v‖ and
‖v‖ ≤∥∥∥∑
i,j
EivijE∗j
∥∥∥ ≤∑
i,j
‖vij‖ .
Further, we can also see that a sequence (vk) in Mn(V ) converges if and only if the entries
vkij converge, and a linear map F = [Fij ] : V → Mn is continuous if and only if each
Fij : V → C is continuous. We now look at two instances of multiple matrix norms that
will be needed later.
In the first instance, given m, n ∈ N , we consider two additional norms on M m,n(C ).
The Hilbert-Schmidt norm is
‖α‖2 = [trace(α∗α)]1/2
=[∑
|αij|2]1/2
and the trace class norm is
‖α‖1 = trace(|α|),
where |α| = (α∗α)1/2.
4
We write HSm,n and Tm,n for the vector space M m,n with the norms ‖·‖2 and ‖·‖1,
respectively, and HSn = HSn,n and Tn = Tn,n.
Also, for I an infinite set, we let TI indicate (KI)′, the trace class operators on l2(I)
with the trace class norm (the “prime” indicates Banach space duality). For a countable
index set I, we may use T∞.
Definition 1.2.1. Given an operator space V , for v ∈ M n(V ), we define
‖v‖1 = inf‖α‖2 ‖v‖‖β‖2 : v = αvβ where v ∈ Mn(V ), α ∈ HSn,m, β ∈ HSm,n.
Proposition 1.2.2. Given an operator space V , ‖·‖1 is a norm on M n(V ).
We let Tn(V ) denote M n(V ) with the norm ‖·‖1.
Next we consider two natural operator space structures on any Hilbert space H ([5,
27]). The first is the column operator space structure, which arises by identifying H with
B(C ,H). We refer to H with this operator space structure as the column Hilbert space
Hc. We have h ∈ Hc corresponding to ϕh ∈ B(C ,H) where ϕh(1) = h.
The second operator space structure results from identifying H with B(H, C ), where
H = H ′ is the dual Hilbert space. We refer to H with this operator space structure as the
row Hilbert space Hr. In this case, h ∈ Hr corresponds to 〈· | h〉 : H → C . We recall that
we may consider H as having the same elements as H (h corresponds to 〈· | h〉), but with
αh = αh and 〈h | k〉 = 〈k | h〉, with h denoting that we are considering h as an element of
H rather than H .
By definition, we have
Mm,n(Hc) = B(C n, Hm) and
Mm,n(Hr) = B(Hn, C m).
In general, given ξ1, . . . , ξn ∈ H , we let ξr denote the row matrix [ ξ1 . . . ξn ] and ξc
the column matrix
ξ1...
ξn
. Then ξr ∈ M1,n(Hr) has norm
‖ξr‖ =(∑
‖ξj‖2)1/2
,
5
and ξc ∈ Mn,1(Hc) has norm
‖ξc‖ =(∑
‖ξj‖2)1/2
.
If ξ1, . . . , ξn is a set of orthogonal elements in H , then ξr ∈ M1,n(Hc) has norm
‖ξr‖ = max‖ξj‖,
and ξc ∈ Mn,1(Hr) has norm
‖ξc‖ = max‖ξj‖.
Given a closed subspace W of an operator space V , Mn(W ) is closed in Mn(V ) for all
n ∈ N . Therefore Mn(V )/Mn(W ) is a Banach space , and we use the identification
Mn(V/W ) = Mn(V )/Mn(W )
to provide a norm on Mn(V/W ). Letting π : V → V/W be the canonical quotient map,
we have that for each n ∈ N , πn : Mn(V ) → Mn(V/W ) is a quotient map, and for
v ∈ Mn(V/W ),
‖v‖ = inf‖v‖ : v ∈ Mn(V ), πn(v) = v.
Proposition ([47]) 1.2.3. If W is a closed subspace of an operator space V , then V/W is
an operator space and the canonical quotient map π : V → V/W is a complete contraction.
For operator spaces V and W , we may identify an n×n matrix ϕ = [ϕij] of completely
bounded maps ϕij : V → W with a completely bounded map ϕ : V → Mn(W ) by
letting ϕ(v) = [ϕi,j(v)]. Conversely, any completely bounded mapping ϕ : V → Mn(W )
is determined in this manner by the matrix of completely bounded maps ϕ = [ϕij ], where
ϕij = EiϕE∗j , Ei = [0 . . . 1i . . . 0] ∈ M1,n. We use the resulting linear identification
(1.2.1) Mn(CB(V,W )) ∼= CB(V, Mn(W ))
and the completely bounded norm on the second space to define a norm on Mn(CB(V, W )).
Thus by definition we have that (1.2.1) is a natural isometry.
6
Theorem ([23]) 1.2.4. Given operator spaces V and W , the space of all completely
bounded maps CB(V,W ) is an operator space. Furthermore, if W is complete, then
CB(V, W ) is a complete operator space.
From (1.2.1) and the isometry CB(V, C ) = B(V, C ) = V ′, the Banach dual space V ′ of
V is an operator space via the identification
Mn(V ′) = CB(V,Mn).
We will use the symbol V † to indicate the dual operator space of V ([6, 25]). It is important
to remember that V † = V ′ as Banach spaces, but Mn(V †) is not defined as the Banach
dual of Mn(V ), i.e., we must distinguish Mn(V †) from Mn(V ′) = Mn(V )′. Nevertheless,
the matrix norm on Mn(V ) does determine that on Mn(V †), since given f ∈ Mn(V †), we
have that
‖f‖ = sup‖fn(v)‖ : v ∈ Mn(V ), ‖v‖ ≤ 1,
and the norm on Mn(V †) determines that on Mn(V ), since if v ∈ Mn(V ),
‖v‖ = sup‖fn(v)‖ : f ∈ CB(V, Mn), ‖f‖cb ≤ 1.
We let θ : V → V †† be the canonical inclusion defined by
〈θ(v), f 〉 = 〈f, v〉,
and for v ∈ Mn(V ), we let v = θn(v).
Theorem ([4, 25]) 1.2.5. Given an operator space V , the canonical inclusion θ : V →
V †† is completely isometric.
For the Banach dual of Mn(V ), we use the notation Mn(V ′) and identify Mn(V ′) with
Mn(V )′, the prime representing classical duality given by the duality pairing
〈[fij ] , [xij ]〉 =∑
i,j
〈fij , xij〉.
We have
Theorem ([4]) 1.2.6. If V is an operator space, then the second dual operator space
matrix norm structure coincides with the classical second dual structure. That is, V †† ∼= V ′′
completely isometrically.
7
1.3. Operator Space Tensor Products
We will need the use of three operator space tensor products as aids in achieving our
results on matrix ordered operator spaces. They are the projective, injective and Haagerup
tensor products. We now look at their definitions and some of their key properties.
Definition 1.3.1. Given operator spaces V and W , we define the (operator space)
projective norm ‖·‖∧ for u ∈ M n(V ⊗W ) by
‖u‖∧ = inf‖α‖ ‖v‖ ‖w‖ ‖β‖ : u = α(v ⊗ w)β
where α ∈ Mn,pq, v ∈ Mp(V ), w ∈ Mq(W ), β ∈ Mpq,n, p, q ∈ N .
Proposition ([6, 25]) 1.3.2. For each n ∈ N , ‖·‖∧ is a norm on M n(V ⊗W ), and with
these matrix norms V ⊗W is an operator space.
We let V ⊗∧W denote the operator space V ⊗W with the projective tensor norm ‖·‖∧,
and let V∧⊗ W denote its completion. We call these spaces the algebraic and complete
projective tensor products of V and W , respectively. For elements in the complete projective
tensor product, we have the following.
Proposition ([24]) 1.3.3. An element u in Mn(V∧⊗W ) satisfies ‖u‖∧ < 1 if and only
if it has the form
u = α(v ⊗ w)β
where α ∈ Mn,∞2 , β ∈ M∞2,n, v ∈ M∞(V ), w ∈ M∞(W ), and ‖α‖ ‖v‖ ‖w‖ ‖β‖ < 1. One
may further assume that v ∈ K∞(V ) and w ∈ K∞(W ).
Included in the following is the fact that the projective tensor product is both commu-
tative and associative.
Proposition ([6, 25]) 1.3.4. Given operator spaces V , W and X, we have the completely
isometric isomorphisms:
V∧⊗W ∼= W
∧⊗ V,(1.3.1)
(V∧⊗W )
∧⊗X ∼= V
∧⊗ (W
∧⊗X),(1.3.2)
CB(V∧⊗W,X) ∼= CB(V, CB(W, X)), and(1.3.3)
(V∧⊗W )† ∼= CB(V,W †).(1.3.4)
8
Theorem ([29]) 1.3.5. For any operator space V , the pairing
M n(V ′)×M n(V ) → C : (f, v) 7→ 〈f, v〉 =∑
fi,j(vi,j)
determines isometric isomorphisms
Tn(V )′ ∼= Mn(V †) = CB(V,Mn)
and
Mn(V )′ ∼= Tn(V †).
Theorem ([29]) 1.3.6. For any operator space V and n ∈ N , we have a natural isometric
isomorphism
Tn(V ) ∼= Tn
∧⊗ V.
Definition 1.3.7. Given operator spaces V and W , we define the (operator space)
injective norm ‖·‖∨ for u ∈ M n(V ⊗W ) by
(1.3.5) ‖u‖∨ = sup‖(f ⊗ g)n(u)‖ : f ∈ Mp(V†), g ∈ Mq(W
†), ‖f‖ , ‖g‖ ≤ 1.
Proposition ([6]) 1.3.8. Suppose V and W are operator spaces. Then the injective
norms (1.3.5) provide an operator space structure on V ⊗W and are determined by the
natural embedding
θ : V ⊗W → CB(V †, W ).
We let V ⊗∨ W denote the operator space V ⊗W with the injective tensor norm ‖·‖∨,
and let V∨⊗W denote its completion. These are called the algebraic and complete injective
(or spatial) tensor products of V and W , respectively. The idea of “spatial” comes from
the following proposition:
Proposition ([42]) 1.3.9. Given completely isometric inclusions V → B(H) and W →
B(K), the corresponding map
V∨⊗W → B(H ⊗K)
is completely isometric.
The injective tensor product is also both commutative and associative.
9
Proposition ([29]) 1.3.10. Given operator spaces V , W and X , we have the completely
isometric isomorphisms
V∨⊗W ∼= W
∨⊗ V(1.3.6)
and
(V∨⊗W )
∨⊗X ∼= V
∨⊗ (W
∨⊗X).(1.3.7)
Proposition (cf. [27]) 1.3.11. Given an operator space V , there is a natural complete
isometry
Mn(V ) ∼= Mn
∨⊗ V.
Proposition ([24]) 1.3.12. Let V be any operator space. For any index set I, we have
the completely isometric isomorphism
KI(V ) ∼= KI
∨⊗ V.
The third tensor norm, and most important for our work, is the Haagerup tensor norm.
Definition 1.3.13. Given operator spaces V and W , for v ∈ Mn,r(V ) and w ∈ Mr,n(W ),
we define
v ¯ w =
[∑
k
vik ⊗ wkj
]∈ M n(V ⊗W ).
This is a definition that is designed to be analogous to matrix multiplication. We note
that if α is a matrix of scalars, we get v ¯ (αw) = (vα)¯ w, and that if u ∈ M n(V ⊗W ),
there exist v ∈ Mn,r(V ) and w ∈ Mr,n(W ) for some r ∈ N such that u = v ¯ w.
Definition 1.3.14. For u ∈ M n(V ⊗W ), we define the Haagerup tensor norm ‖·‖h by
(1.3.8) ‖u‖h = inf‖v‖ ‖w‖ : u = v ¯ w where v ∈ Mn,r(V ), w ∈ Mr,n(W ), r ∈ N .
Theorem ([43, 47]) 1.3.15. For each n ∈ N , ‖·‖h is a norm on M n(V ⊗W ), and with
these matrix norms V ⊗W is an operator space.
We let V ⊗h W denote the operator space V ⊗W with the Haagerup tensor norm ‖·‖h,
and let V ⊗h W denote its completion. We call these spaces the algebraic and complete
Haagerup tensor products of V and W , respectively.
10
Proposition ([6]) 1.3.16. Let V and W be operator spaces. Then the map
ϕ : Mm,1(V ) ⊗h M1,n(W )→ Mm,n(V ⊗h W )
given by ϕ(v ⊗ w) = v ¯w is a completely isometric isomorphism.
To see what is happening here, suppose u = v ¯ w ∈ Mm,n(V ⊗h W ) with v = [vij] ∈
Mm,r(V ) and w = [wjk] ∈ Mr,n(W ). Since
Mm,r(V ) ∼= M1,r(Mm,1(V )) and Mr,n(W ) ∼= Mr,1(M1,n(W )),
we have
u = [ v1 . . . vr ]¯
w1...
wr
=
∑
j
vj ⊗ wj
with vj =
v1j
...vmj
∈ Mm,1(V ) and wj = [ wj1 . . . wjn ] ∈ M1,n(W ). It is then easy to
see that the norm of u in Mm,n(V ⊗h W ) is the same as that in Mm,1(V )⊗h M1,n(W ).
An important result for our work is that the infimum in (1.3.8) is actually attained for
elements in the algebraic Haagerup tensor product.
Theorem ([27]) 1.3.17. Given operator spaces V and W and u ∈ Mm,n(V ⊗h W ) an
algebraic element of the tensor product, there exist an integer p and elements v ∈ Mm,p(V )
and w ∈ Mp,n(W ) for which u = v ¯ w and ‖u‖h = ‖v‖ ‖w‖.
Note. The result in [27] is for m=n, but the basic part of the proof is for the case u ∈
V ⊗h W , with the matrix levels following from Proposition 1.3.16.
The Haagerup norm is associative, but not commutative.
Proposition ([43]) 1.3.18. Given operator spaces V , W and Z, we have the complete
isometry
(V ⊗h W )⊗h Z ∼= V ⊗h (W ⊗h Z).
Using this associative property, Theorem 1.3.17 extends easily to tensor products of 3
or more spaces X1, . . . , Xt. With u = x1 ¯ · · · ¯ xt, we may take x2, . . . , xt−1 as square
matrices of the same size by adding appropriate rows or columns of 0’s, i.e., xj ∈ Ms(Xj)
for some s and j = 2, . . . , tt−1.
The next theorem shows that the Haagerup tensor product satisfies the injective prop-
erty.
11
Theorem ([43]) 1.3.19. Let V and W be operator spaces with V1 ⊆ V and W1 ⊆ W
subspaces. Then the inclusion of V1 ⊗h W1 into V ⊗h W is a complete isometry.
We now look at the norm representation for elements in the complete Haagerup tensor
product, and note that Kn,∞(V ), K∞,n(V ) and K∞(V ) are complete when V is.
Theorem 1.3.20. Given operator spaces X1, . . . , Xt (t ≥ 2), u ∈ Mn(X1 ⊗h · · · ⊗h Xt)
if and only if
u = x1 ¯ · · · ¯ xt
where x1 ∈ Kn,∞(X1), xt ∈ K∞,n(Xt), and xi ∈ K∞(Xi), 2 ≤ i ≤ t − 1. We call such a
representation a standard representation. Further,
‖u‖h = inf∥∥x1
∥∥ · · ·∥∥xt
∥∥ : u = x1 ¯ · · · ¯ xt, x1 ∈ Kn,∞(X1), xt ∈ K∞,n(Xt),
xi ∈ K∞(Xi)(2 ≤ i ≤ t− 1).
Proof. [=⇒] If u ∈ Mn(X1⊗h · · ·⊗hXt), then u = limj→∞
sj where sj ∈ Mn(X1⊗h · · ·⊗hXt).
Given ε > 0, and by choosing a subsequence if necessary, we may assume that
‖u− s1‖h <ε2
4and ‖sj+1 − sj‖h <
ε2
2tj+2.
Then ‖s1‖h < ‖u‖h + ε2
4and u = s1 +
∞∑j=1
(sj+1 − sj). From Theorem 1.3.17, we may
assume that
s1 = x11 ¯ · · · ¯ xt
1 and sj+1 − sj = x1j+1 ¯ · · · ¯ xt
j+1
where x1j ∈ Mn,nj (X1), xt
j ∈ Mnj ,n(Xt), xij ∈ Mnj (Xi) (2 ≤ i ≤ t − 1),
∥∥x11
∥∥ = ‖xt1‖ <
(‖u‖h + ε2
4
)1/2
,∥∥x2
1
∥∥ = · · · =∥∥xt−1
1
∥∥ = 1,∥∥x1
j+1
∥∥ =∥∥xt
j+1
∥∥ < ε2j+1 , and
∥∥x2j+1
∥∥ = · · · =∥∥xt−1
j+1
∥∥ = 12j .
Let x1 = [x11 x1
2 . . . ], xt =
xt1
xt2...
, and, for 2 ≤ i ≤ t− 1, xi =
xi1
xi2
. . .
.
Using the fact that
(1.3.9)(‖u‖h +
ε2
4
)1/2
≤(‖u‖h + ε ‖u‖1/2
h +ε2
4
)1/2
=
[(‖u‖1/2
h +ε
2
)2]1/2
= ‖u‖1/2h +
ε
2,
12
we get
∥∥x1∥∥ ≤
∞∑
j=1
∥∥x1j
∥∥ < ‖u‖1/2h +
ε
2+
∞∑
j=2
ε
2j= ‖u‖1/2
h + ε and
∥∥xt∥∥ ≤
∞∑
j=1
∥∥xtj
∥∥ < ‖u‖1/2h +
ε
2+
∞∑
j=2
ε
2j= ‖u‖1/2
h + ε.
Then, since∥∥xi
∥∥ = 1 for 2 ≤ i ≤ t− 1, we have that
u = x1 ¯ · · · ¯ xt =
∞∑
j=1
x1j ¯ · · · ¯ xt
j
with∥∥x1
∥∥ · · · ‖xt‖ <(‖u‖1/2
h + ε)2
= ‖u‖h + 2ε ‖u‖1/2h + ε2. Letting ε → 0, we have
‖u‖h ≥ inf∥∥x1
∥∥ · · ·∥∥xt
∥∥ : u = x1 ¯ · · · ¯ xt, x1 ∈ Kn,∞(X1), xt ∈ K∞,n(Xt),
xi ∈ K∞(Xi)(2 ≤ i ≤ t− 1).
Since the reverse inequality is clear by Definition 1.3.14, we get that
‖u‖h = inf∥∥x1
∥∥ · · ·∥∥xt
∥∥ : u = x1 ¯ · · · ¯ xt, x1 ∈ Kn,∞(X1), xt ∈ K∞,n(Xt),
xi ∈ K∞(Xi)(2 ≤ i ≤ t− 1).
[⇐=] For all p ∈ N and i = 1, . . . , t, let xip = Epx
iEp and up = x1p ¯ · · · ¯ xt
p. Since
xi ∈ K∞(Xi) by hypothesis for i = 1, . . . , t,∥∥xi − xi
p
∥∥ → 0 for i = 1, . . . , t. We then get
that
u− up = (x1 − x1p)¯ x2 ¯ · · · ¯ xt
+ x1p ¯ (x2 − x2
p) ¯ x3 ¯ · · · ¯ xt
+ · · ·+ x1p ¯ · · · ¯ xt−1
p ¯ (xt − xtp)
and
‖u− up‖h ≤∥∥(x1 − x1
p)¯ x2 ¯ · · · ¯ xt∥∥
h
+∥∥x1
p ¯ (x2 − x2p)¯ x3 ¯ · · · ¯ xt
∥∥h
+ · · ·+∥∥x1
p ¯ · · · ¯ xt−1p ¯ (xt − xt
p)∥∥
h
≤∥∥x1 − x1
p
∥∥ ∥∥x2∥∥ · · ·
∥∥xt∥∥
+∥∥x1
p
∥∥ ∥∥x2 − x2p
∥∥ ∥∥x3∥∥ · · ·
∥∥xt∥∥
+ · · ·+∥∥x1
p
∥∥ · · ·∥∥xt−1
p
∥∥∥∥xt − xtp
∥∥ → 0 as p →∞.
13
Definition 1.3.21. Given operator spaces V and W , a matix norm ‖·‖µ on V ⊗W is
an operator space cross norm if
‖v ⊗ w‖µ = ‖v‖ ‖w‖
for every v ∈ Mm(V ) and w ∈ Mn(W )(m,n ∈ N ).
Given two operator space cross norms ‖·‖µ and ‖·‖ν on V ⊗W , we write ‖·‖µ ≤ ‖·‖ν if
‖u‖µ ≤ ‖u‖ν for all u ∈ M n(V ⊗W )(n ∈ N ).
Proposition ([6]) 1.3.22. ‖·‖∨, ‖·‖∧ and ‖·‖h are operator space cross norms such that
‖·‖∨ ≤ ‖·‖h ≤ ‖·‖∧ .
Given an operator space cross norm ‖·‖µ on V ⊗W , there is a natural operator space
structure on V † ⊗ W † obtained by identifying V † ⊗ W † with an operator subspace of
(V ⊗µ W )†. This induced matrix norm ‖·‖µ′ on V † ⊗W † is given by
‖f‖µ′ = sup‖ [fkl(uij)] ‖ : u ∈ Mn(V ⊗µ W ), ‖u‖µ ≤ 1, n ∈ N
for f ∈ Mm(V † ⊗W †). We call ‖·‖µ′ the dual matrix norm of ‖·‖µ.
Proposition ([6]) 1.3.23. The dual matrix norm of ‖·‖∧ is ‖·‖∨, i.e., ‖·‖∧′ = ‖·‖∨.
The theorem which follows shows that the Haagerup tensor product is self-dual, i.e.,
that ‖·‖h′ = ‖·‖h.
Theorem ([27]) 1.3.24. Given operator spaces V and W , the natural embedding
V † ⊗h W † → (V ⊗h W )†
is completely isometric.
In the final theorem of this section on tensor products, we list several special properties
that are particular to the case where at least one of the operator spaces is a column or row
Hilbert space. We will draw heavily on these in the sequel.
14
Theorem ([5, 27]) 1.3.25. Given an operator space V and Hilbert spaces H and K,
we have the following complete isometries:
B(H, K) ∼= CB(Hc,Kc)(1.3.10)
B(K, H) ∼= CB(Hr, Kr)(1.3.11)
(Hc)† ∼= Hr(1.3.12)
(Hr)† ∼= Hc(1.3.13)
V ⊗h Hc∼= V
∧⊗Hc(1.3.14)
Hr ⊗h V ∼= Hr
∧⊗ V(1.3.15)
Hc ⊗h V ∼= Hc
∨⊗ V(1.3.16)
V ⊗h Hr∼= V
∨⊗ Hr(1.3.17)
Hr
∨⊗Kc
∼= Kc ⊗h Hr∼= K(H,K)(1.3.18)
Hr
∧⊗Kc
∼= Hr ⊗h Kc∼= B(K, H)†(1.3.19)
Hr
∧⊗Hc
∼= Hr ⊗h Hc∼= B(H,H)† ∼= B(H)† ∼= T (H)(1.3.20)
(Kc)† ⊗h V ⊗h Hc
∼= V∧⊗B(H, K)†(1.3.21)
((Kc)† ⊗h V ⊗h Hc)
† ∼= CB(V,B(H, K))(1.3.22)
15
2. Matrix Ordered Operator Spaces
In this chapter we develop the concept of the matrix ordered operator space. Previously,
we saw that every operator space is completely isometric to a subspace of B(H) for some
Hilbert space H . From the theory of Banach spaces, every Banach space E is isometric to
a subspace of C(X), where X = (E′)1, the closed unit ball of the dual of E. We have that
C(X) is a commutative C∗-algebra and a concrete operator space on l2(X). For all n ∈ N ,
the operator matrix norm on Mn(C(X)) is the unique C∗-algebra norm on the same space
given by
‖ [fij] ‖n = sup‖ [fij(t)] ‖n : t ∈ X
for all [fij ] ∈ Mn(C(X)).
These operator matrix norms on C(X) then provide an operator space structure on E.
We refer to E with this operator space structure as MIN(E) and denote the norms by
‖·‖MIN. Suppose ||| · ||| is any other matrix norm that provides E with an operator space
structure. Since any bounded linear mapping ϕ from an operator space into a commutative
C∗-algebra is completely bounded with ‖ϕ‖cb = ‖ϕ‖, we get that the identity mapping
id : (E, ||| · |||) → (E, ‖·‖MIN), which is an isometry, is a complete contraction. Thus
||| · ||| ≥ ‖·‖MIN, and so ‖·‖MIN is the smallest operator space norm on E, explaining the
choice of the name. Then, given any two Banach spaces E and F and bounded linear map
ϕ : E → F , this map is completely bounded with ‖ϕ‖cb = ‖ϕ‖ when considered as a map
ϕ : MIN(E) → MIN(F ).
We remark at this point that whereas the bounded linear maps are the morphisms in
the category of Banach spaces, the completely bounded linear maps are the morphisms in
the category of operator spaces, and these latter maps take into account the matrix norm
structure of operator spaces. Thus, in conjuction with the relationships noted above, it
would seem natural for matrix order structures on operator spaces to be related to the
order structures of Banach spaces, and this is indeed the case.
2.1. Ordered Real Banach Spaces
We begin by reviewing some necessary concepts from the theory of ordered real Banach
spaces. Throughout this section, (E, P, ‖·‖) will indicate an ordered real Banach space E
16
with (positive) cone P (not necessarily proper) and norm ‖·‖. When we have this situation,
we can define a dual cone P ′ on the dual space E′ by
P ′ = f ∈ E′ : f (p) ≥ 0 for all p ∈ P.
We then have that (E′, P ′, ‖·‖) is an ordered real Banach space where ‖·‖ here indicates
the dual norm. We call P ′ the dual cone to P .
Proposition 2.1.1. Given an ordered real Banach space (E,P, ‖·‖), the dual cone P ′ is
‖·‖-closed.
Proposition 2.1.2. Let (E,P, ‖·‖) be an ordered real Banach space. Then P is a ‖·‖-
closed set if and only if the following condition is satisfied: If x0 ∈ E and f (x0) ≥ 0 for
all f ∈ P ′, then x0 ∈ P .
Corollary 2.1.3. Let (E,P, ‖·‖) be an ordered real Banach space. Then P = P ′′ ∩E.
Definition 2.1.4. Let (E, P, ‖·‖) be an ordered real Banach space. We say that P is
generating if E = P − P
We note that if a cone P is generating for an ordered real Banach space (E,P, ‖·‖), then
P is “large” enough that each element of E can be written as a linear combination of two
of its elements. We next define the real Banach space equivalent of the key concept in this
thesis.
Definition 2.1.5. The norm ‖·‖ on an ordered real Banach space (E,P, ‖·‖) is called a
regular (or Riesz) norm if it satisfies the following two conditions:
(1) ‖·‖ is absolute-monotone: −u ≤ x ≤ u in E implies ‖x‖ ≤ ‖u‖.
(2) For each x ∈ E with ‖x‖ < 1 there exists u ∈ E with ‖u‖ < 1 such that −u ≤ x ≤ u.
Proposition 2.1.6. If the norm ‖·‖ on an ordered real Banach space (E, P, ‖·‖) is
regular, then the cone P is both generating and proper.
Proof. Given x ∈ E, x 6= 0, and ε > 0,∥∥∥ x‖x‖+ε
∥∥∥ < 1, so there exists y ∈ E such that
−y ≤ x‖x‖+ε
≤ y. But then −y(‖x‖+ ε) ≤ x ≤ y(‖x‖+ ε), so
x =1
2(y(‖x‖+ ε) + x)−
1
2(y(‖x‖+ ε)− x)
17
is a linear combination of two positive elements. Thus P is generating.
If x,−x ∈ P , then −0 ≤ x ≤ 0, so ‖x‖ ≤ 0, which implies x = 0. Thus P is proper.
Definition 2.1.7. Let (E,P, ‖·‖) be an ordered real Banach space.
(1) A ⊆ E is absolutely order convex if a ∈ A and −a ≤ x ≤ a implies that x ∈ A.
(2) A ⊆ E is absolutely dominated if for every a ∈ A there exists x ∈ A such that
−x ≤ a ≤ x.
(3) A ⊆ E is solid if it is both absolutely order convex and absolutely dominated.
Proposition 2.1.8. Let U be the open unit ball of (E,P, ‖·‖). Then ‖·‖ is a regular
norm if and only if U is solid.
We can now state the main theorem, due to E.B. Davies, that we are interested in from
the theory of real Banach spaces. We shall refer to it often. A development leading up
the the proof of this theorem, which includes the proofs of the other propositions in this
section, is given as an appendix.
Theorem (Davies) 2.1.9. Let (E,P, ‖·‖) be an ordered real Banach space, and let U
and Σ denote its open and closed unit balls. Let (E′, P ′, ‖·‖) be the real Banach dual space
with dual cone P ′, and let U ′ and Σ′ denote its open and closed unit balls. Consider:
(1) ‖·‖ is a regular norm in (E, P, ‖·‖);
(2) U is solid in (E, P, ‖·‖);
(3) ‖·‖ is a regular norm in (E′, P ′, ‖·‖);
(4) U ′ is solid in (E′, P ′, ‖·‖);
(5) Σ′ is solid in (E′, P ′, ‖·‖).
Then (1) ⇐⇒ (2) =⇒ (3) ⇐⇒ (4) ⇐⇒ (5).
Further, if P is closed, then (3) =⇒ (1), so (1)—(5) are mutually equivalent.
It is important to note here that whereas regularity of norm is always equivalent to the
open unit ball being solid, in the case of dual spaces with dual cones, regularity is also
equivalent to the closed unit ball being solid. For example, let X be a compact Hausdorff
space, CR (X) the real valued continuous functions on X, and MR (X) the signed (finite)
Radon measures on X . Then CR (X)′ = MR (X). The positive cone on CR (X) consists of
18
the non-negative continuous functions and the dual cone in MR (X) consists of the (finite)
positive measures, corresponding to the (bounded) positive linear functionals on CR (X)
by the Riesz representation theorem.
If −g ≤ f ≤ g in CR (X), then for all x ∈ X, −g(x) ≤ f(x) ≤ g(x), so |f(x)| ≤ |g(x)|.
Thus ‖f‖ ≤ ‖g‖. Also, if given f ∈ CR (X), − |f | ≤ f ≤ |f | and ‖ |f | ‖ = ‖f‖. Thus the
usual supremum norm is regular.
For −µ ≤ ν ≤ µ in MR (X), we have that
ν =1
2(µ + ν)− 1
2(µ− ν).
From measure theory, 12(µ + ν) ≥ ν+ and 1
2(µ − ν) ≥ ν−. Thus
‖ν‖ = |ν| (X) = ν+(X) + ν−(X) ≤ 1
2(µ(X) + ν(X)) +
1
2(µ(X)− ν(X))
= µ(X) = |µ| (X) = ‖µ‖ .
Since we are in a dual space, let us now suppose that ν ∈ MR (X) with ‖ν‖ ≤ 1. By Jordan
decomposition, ν = ν+ − ν− where |ν| = ν+ + ν−. We then have that − |ν| ≤ ν ≤ |ν| and
‖ |ν| ‖ = ‖ν‖ ≤ 1. Thus MR (X) has a regular dual norm.
2.2. Ordered Complex Banach Spaces
We next extend the notion of order from real to complex Banach spaces. Suppose VR
is a real vector space. Then we can define a complex vector space
V = VR + i VR
by using the natural addition and scalar multiplication. Then we define an involution ∗
on V by
(a + ib)∗ = a− ib.
Recall that an involution is a conjugate linear map v 7→ v∗ such that v∗∗ = v. We refer to
a complex vector space with involution as a ∗-vector space.
Coming from the other direction, suppose V is a ∗-vector space. The self-adjoint ele-
ments of V , the elements v ∈ V such that v = v∗, form a real subspace of V which we
denote as Vsa. For v ∈ V , we define
Re v =1
2(v + v∗) and Im v =
1
2i(v − v∗).
19
These are self-adjoint elements of V such that v = Re v + i Im v, yielding
V = Vsa + i Vsa, Vsa ∩ i Vsa = 0.
This establishes that there is a one-to-one correspondence between real vector spaces and
∗-vector spaces.
We say that a complex vector space V is partially ordered if it is a ∗-vector space
together with a (not necessarily proper) cone V + ⊆ Vsa, and we write v ≥ w (or w ≤ v) if
v − w ∈ V +. Since this cone is entirely within the self-adjoint elements of V , it defines a
cone on the corresponding real vector space, and vice-versa. Thus there is also a one-to-one
correspondence between partially ordered real vector spaces and partially ordered complex
vector spaces.
If VR is a normed space, we wish to define a norm on V = VR + i VR in such a way that
‖v‖ = ‖v∗‖ for all v ∈ V . There are many equivalent ways of doing this. For instance, one
could define ‖a + ib‖ = max‖a‖ , ‖b‖ or ‖a + ib‖ = (‖a‖p+ ‖b‖p
)1/p for 1 ≤ p < ∞.
On the other hand, if V is a normed ∗-vector space, the norm on V determines a norm
on Vsa. But each such norm is equivalent to some norm ||| · ||| for which |||v||| = |||v∗|||.
For example, one could define |||v||| = max‖v‖ , ‖v∗‖.
Thus, without loss of generality, for an involutive Banach space we may assume that the
involution is an isometry. This also means that the self-adjoint elements form a norm-closed
set.
Assumption. Throughout the remainder of this thesis we shall assume, without further
statement, that for all involutive Banach spaces, the involution is an isometry.
If V and W are any two ∗-vector spaces and ϕ : V → W is linear, we define ϕ∗ : V → W
by ϕ∗(v) = ϕ(v∗)∗. This provides an involution on L(V,W ), the vector space of linear maps
from V to W . We say ϕ ∈ L(V,W ) is self-adjoint if ϕ∗ = ϕ, and this is the case if and
only if ϕ : Vsa → Wsa. Restricting ourselves to B(E,F ), the bounded linear operators
from an involutive Banach space E to an involutive Banach space F , we get that B(E,F )
is an involutive Banach space.
If V and W are partially ordered complex vector spaces, we say a linear map ϕ ∈
L(V,W ) is positive if ϕ ∈ L(V,W )sa and ϕ(V +) ⊆ W+. If ϕ is a linear isomorphism from
20
V onto W , we say that ϕ is an order isomorphism if ϕ and ϕ−1 are positive. The cone of
positive maps provides a partial ordering for L(V,W ). Letting C + = [0,∞), we get that
V d = L(V, C ) is a ∗-vector space that is partially ordered by (V d)+. For the case where E
and F are partially ordered (and thus also involutive) Banach spaces, this partial ordering
restricts natuarally to B(E,F ). We get that the dual cone to E+ is
E′+
= E ′ ∩ (Ed)+ = f ∈ E′sa : f (x) ≥ 0 for all x ∈ E+.
We then get the following immediate analogs to statements in Section 2.1.
Proposition 2.2.1. Given an ordered complex Banach space (E,E+, ‖·‖), the dual cone
E′+
is ‖·‖-closed.
Proposition 2.2.2. Let (E,E+, ‖·‖) be an ordered complex Banach space. Then E+ is
a ‖·‖-closed set if and only if the following condition is satisfied: If x0 ∈ E and f(x0) ≥ 0
for all f ∈ E′+, then x0 ∈ E+.
Corollary 2.2.3. Let (E,E+, ‖·‖) be an ordered complex Banach space. Then E+ =
E′′+ ∩E.
Definition 2.2.4. Let (E,E+, ‖·‖) be an ordered complex Banach space. We say that
E+ is generating if Esa = E+ − E+
Thus, if a cone E+ is generating for an ordered complex Banach space (E,E+, ‖·‖),
each element of E can be written as a linear combination of four elements of E+.
Definition 2.2.5. The norm ‖·‖ on an ordered complex Banach space (E, E+, ‖·‖) is
called a regular (or Riesz) norm if its restriction to the real Banach space Esa is regular.
Proposition 2.2.6. If the norm ‖·‖ on an ordered complex Banach space (E,E+, ‖·‖)
is regular, then the cone E+ is both generating and proper.
Definition 2.2.7. Let (E,E+, ‖·‖) be an ordered complex Banach space.
(1) A ⊆ Esa is absolutely order convex if a ∈ A and −a ≤ x ≤ a implies that x ∈ A.
(2) A ⊆ Esa is absolutely dominated if for every a ∈ A there exists x ∈ A such that
−x ≤ a ≤ x.
(3) A ⊆ Esa is solid if it is both absolutely order convex and absolutely dominated.
21
Proposition 2.2.8. Let U be the open unit ball of the ordered complex Banach space
(E,E+, ‖·‖). Then ‖·‖ is a regular norm if and only if U ∩Esa is solid.
Theorem 2.2.9. Let (E,E+, ‖·‖) be an ordered complex Banach space, and let U and Σ
denote its open and closed unit balls. Let (E′, E′+
, ‖·‖) be the complex Banach dual space
with dual cone E′+, and let U ′ and Σ′ denote its open and closed unit balls. Consider:
(1) ‖·‖ is a regular norm in (E, E+, ‖·‖);
(2) U ∩Esa is solid in (E,E+, ‖·‖);
(3) ‖·‖ is a regular norm in (E′, E′+
, ‖·‖);
(4) U ′ ∩ E′sa is solid in (E′, E′+, ‖·‖);
(5) Σ′ ∩E′sa is solid in (E′, E′+
, ‖·‖).
Then (1) ⇐⇒ (2) =⇒ (3) ⇐⇒ (4) ⇐⇒ (5).
Further, if E ′+ is closed, then (3) =⇒ (1), so (1)—(5) are mutually equivalent.
Returning to the previous example, we now let C(X) denote the complex-valued, contin-
uous functions on a compact Hausdorff space X and M (X) the complex Radon measures
on the same set. Then M (X) = C(X)′, the norm on C(X) is regular, and the norm on
M (X) is a regular dual norm.
For another example, consider the spaces Lp(µ) where 1 ≤ p ≤ ∞. Here the self-
adjoint and positive elements are those functions that are real-valued and positive a.e.(µ),
respectively. If f and g are self-adjoint elements of Lp(µ) with −g ≤ f ≤ g, then −g(x) ≤
f (x) ≤ g(x) a.e.(µ), so |f(x)| ≤ |g(x)| a.e.(µ). It is then clear that ‖f‖p ≤ ‖g‖p. Also,
if f is self-adjoint in Lp(µ) with ‖f‖ < 1, it is clear that we have∥∥ |f |
∥∥ = ‖f‖ < 1 and
− |f | ≤ f ≤ |f |. This shows that the spaces Lp(µ) are regular for 1 ≤ p ≤ ∞. We will see
later that the Schatten class spaces Sp, 1 ≤ p ≤ ∞, are also regular.
2.3. Matrix Ordered Operator Spaces
We now generalize the above to the operator space case. If an operator space V has an
involution, then, for all n ∈ N , there is a natural involution on Mn(V ) given by
[vij ]∗
=[v∗ji
].
This leads to a definition.
22
Definition 2.3.1. An involutive operator space is an operator space with an involution
such that, for each n ∈ N , Mn(V ) is an involutive Banach space with the natural involution,
i.e., the involution on Mn(V ) is an isometry.
If V and W are involutive operator spaces and ϕ : V → W is self-adjoint, then it follows
that ϕn : M n(V ) → M n(W ) is also self-adjoint for all n ∈ N .
Proposition 2.3.2. If V and W are involutive operator spaces, then CB(V, W ) is an
involutive operator space.
Proof. Since CB(V,W ) is an operator space and clearly a ∗-vector space, we need only
prove that the involution on each Mn(CB(V,W )) = CB(V,Mn(W )) is an isometry. Let
[fij] ∈ Mn(CB(V,W )) = CB(V,Mn(W )). Since V and W are involutive operator spaces,
∥∥ [fij ]∗ ∥∥
cb=
∥∥ [f∗ji
] ∥∥cb
= sup∥∥(
[f∗ji
])m([vkl])
∥∥ : [vkl] ∈ Mm(V ), ‖ [vkl] ‖ ≤ 1,m ∈ N
= sup∥∥∥
[f∗ji(vkl)
](k,j),(l,i)
∥∥∥ : [vkl] ∈ Mm(V ), ‖ [vkl] ‖ ≤ 1,m ∈ N
= sup∥∥∥
[fji(v
∗kl)∗]
(k,j),(l,i)
∥∥∥ : [vkl] ∈ Mm(V ), ‖ [vkl] ‖ ≤ 1,m ∈ N
= sup∥∥∥
[fji(v
∗kl)∗]∗
(k,j),(l,i)
∥∥∥ : [vkl] ∈ Mm(V ), ‖ [vkl] ‖ ≤ 1,m ∈ N
= sup∥∥∥ [fij(v
∗lk)](k,j),(l,i)
∥∥∥ : [vkl] ∈ Mm(V ), ‖ [vkl] ‖ ≤ 1,m ∈ N
= sup∥∥([fij ])m([vkl]
∗)∥∥ : [vkl]
∗ ∈ Mm(V ),∥∥ [vkl]
∗∥∥ = ‖ [vkl] ‖ ≤ 1, m ∈ N
= ‖ [fij ] ‖ .
Then Mn(CB(V,W )) = CB(V, Mn(W )) is an involutive Banach space for each n ∈ N and
thus CB(V,W ) is an involutive operator space.
Proposition 2.3.3. If V is an involutive operator space, then the induced involutions on
both Mn(V †) and Mn(V ′) are isometries for all n ∈ N . Thus V † is an involutive operator
space.
Proof. Since V † = CB(V, C ), V † is an involutive operator space by the previous proposi-
tion.
For Mn(V ′), since V is involutive as an operator space, Mn(V ) is involutive as a Banach
space. It then follows directly that the Banach dual Mn(V ′) is also involutive as a Banach
space.
23
Definition ([9]) 2.3.4. A complex vector space V is matrix ordered if
(1) V is a ∗-vector space (hence so is M n(V ) for all n ≥ 1),
(2) each M n(V ), n ≥ 1, is partially ordered by a cone M n(V )+ ⊆ M n(V )sa, and
(3) if γ = [γij ] ∈ M m,n, then γ∗M m(V )+γ ⊆ M n(V )+.
Definition 2.3.5. An involutive operator space V is called a matrix ordered operator
space if it is a matrix ordered (vector) space and, in addition, the cones Mn(V )+ are closed
in the norm topology for all n ∈ N .
Given a matrix ordered operator space V and a closed subspace W of V that is self-
adjoint (i.e., w ∈ W implies w∗ ∈ W ), we get a corresponding quotient operator space
V/W with canonical quotient map π : V → V/W . We define an involution on V/W (and
thus on each Mn(V/W )) by π(v)∗ = π(v∗). We also define positive cones on Mn(V/W )
for each n ∈ N by
Mn(V/W )+ = (πn(Mn(V )+))−‖·‖.
Proposition 2.3.6. If V is a matrix ordered operator space and W is a closed, self-
adjoint subspace of V , then V/W is a matrix ordered operator space.
Proof. Since Mn(V ) is an involutive Banach space for all n ∈ N , the same is true for each
Mn(V/W ). All that remains to be shown is property (3) of Definition 2.3.4.
Suppose u ∈ Mm(V/W )+ and γ ∈ Mm,n. Then u = lims
us where us ∈ πm(Mm(V )+),
so us = vs + Mm(W ) where vs ∈ Mm(V )+. We then have that
γ∗usγ = γ∗vsγ + γ∗Mm(W )γ = γ∗vsγ + Mn(W ) = πn(γ∗vsγ) ∈ πn(Mn(V )+)
and
‖γ∗uγ − γ∗usγ‖ ≤ ‖γ∗‖ ‖u− us‖ ‖γ‖ → 0.
Thus V/W is a matrix ordered operator space.
Definition 2.3.7. Let V and W be matrix ordered vector spaces and let ϕ : V → W
be a linear map. We call ϕ completely positive if ϕn : M n(V ) → M n(W ) is positive for
each n. We denote the set of completely positive maps from V to W by CP (V, W ). A
24
linear isomorphism ϕ of V onto W is a complete order isomorphism if both ϕ and ϕ−1 are
completely positive.
Several theorems over the next few pages, as indicated, come from an unpublished
manuscript of Takashi Itoh ([32]). However, he takes the notion of matrix ordered operator
space in a less restrictive sense than is done here. He does not require that Mn(V ) be an
involutive Banach space and that Mn(V )+ be closed for each n ∈ N .
For X a matrix ordered vector space, we consider the two completely positive maps Φ
and Ψ defined by
Φ : M n(X) → X, [xij ] 7→n∑
i,j=1
xij
Ψ : X 7→ M n(X), x 7→
x . . . x. . . . . . . . .x . . . x
.
Lemma ([32]) 2.3.8. Suppose that X is a matrix ordered space. Given [fij] ∈ M n(X ′),
the following are equivalent:
(1) X → M n(C ) : x 7→ [fij(x)] is completely positive.
(2) M n(X)→ M n(C ) : [xij ] 7→ [fij(xij)] is positive.
(3) M n(X)→ M n(C ) : [xij ] 7→ [fij(xij)] is completely positive.
(4) M n(X)→ C : [xij ] 7→∑
fij(xij) is positive.
(5) M n(X)→ C : [xij ] 7→∑
fij(xij) is completely positive.
Proof. [(1) =⇒ (2)] Given [xkl] ∈ M n(X)+, we have that[[fij(xkl)]i,j
]k,l≥ 0 as an n2×n2
matrix. Letting γ = [ e11 e22 . . . enn ] ∈ M n,n2 where eij (i, j = 1, . . . , n) is a matrix
unit, we have
[fij(xij)] = γ [fij(xkl)] γ∗ ≥ 0.
[(2) =⇒ (3)] Identify M m(M n(X)) with M nm(X). If[[
x(i,k),(j,l)
]i,j
]k,l∈ M m(M n(X))+,
then for any λik ∈ C (i = 1, . . . , n; k = 1, . . . , m) with
λ =
λ11 λ1m
. . . . . .. . .
λn1 λnm
∈ M n,mn(C )
25
we have that
∑
k,l
[λikx(i,k),(j,l)λjl
]i,j
= λ[[
x(i,k),(j,l)
]i,j
]k,l
λ∗ ∈ M n(X)+.
This implies that
λ[[
fij(x(i,k),(j,l))]i,j
]k,l
λ∗ ∈ M n(C )+
by the assumption. Thus we have
∑
i,j,k,l
fij(x(i,k),(j,l))λikλjl = Φ
(λ
[[fij(x(i,k),(j,l))
]i,j
]k,l
λ∗)≥ 0.
Hence we obtain that[[
fij(x(i,k),(j,l))]i,j
]k,l∈ M m(M n(C ))+.
[(3) =⇒ (1)] It is clear that the given map is the composition of Ψ and the completely
positive map in (3), i.e.,
x −→
x . . . x. . . . . . . . .x . . . x
−→ [fij(x)] .
[(3) =⇒ (5)] It is immediate that the given map is the composition of the completely
positive map in (3) and Φ, i.e.,
[xij ] −→ [fij(xij)] −→∑
fij(xij).
[(5) =⇒ (4)] This is trivial.
[(4) =⇒ (2)] Given [xij ] ∈ M n(X)+, we have for any λi (i = 1, . . . , n) that
[λixij λj
]=
λ1
. . .
λn
[xij ]
λ1
. . .
λn
.
It follows that∑
fij(xij)λiλj ≥ 0. Hence we obtain that [fij(xij)] ∈ M n(C )+.
Based on the equivalence of (1) and (4) in Lemma 2.3.8, for a matrix ordered operator
space V and for each n ∈ N , we define cones on Mn(V †) by
Mn(V †)+ = CB(V, Mn) ∩ CP (V,Mn) = CB(V, Mn)+.
Further, for matrix ordered operator spaces V and W and for all n ∈ N , we define cones
on Mn(CB(V,W )) by
Mn(CB(V,W ))+ = CB(V,Mn(W ))+ = CB(V, Mn(W )) ∩ CP (V,Mn(W )).
26
Proposition 2.3.9. Let V and W be matrix ordered operator spaces. Then CB(V, W )
is a matrix ordered operator space.
Note. Itoh ([32]) states a similar proposition for his less restrictive sense.
Proof. First, CB(V,W ) is an involutive operator space by Proposition 3.1.2.
We next need to show that, given γ ∈ Mm,n and ϕ = [ϕij] ∈ Mm(CB(V,W ))+, γ∗ϕγ ∈
Mn(CB(V, W ))+. Directly from the definition of operator space,
‖γ∗ϕγ‖cb ≤ ‖γ∗‖ ‖ϕ‖cb ‖γ‖ < ∞,
so γ∗ϕγ ∈ Mn(CB(V,W )). Then, for [vij ] ∈ Mp(V )+,
(γ∗ϕγ)p([vij ]) = [γ∗ϕ(vij)γ] = (γ∗ ⊗ Ip) [ϕ(vij)] (γ ⊗ Ip) ≥ 0
since ϕ is completely positive. Thus γ∗ϕγ ∈ Mn(CB(V,W ))+.
It now only remains to show that the cones CB(V,W )+ are closed for all n ∈ N .
Let ϕ ∈ (CB(V,Mn(W ))+)−‖·‖cb . Then there exists (ϕα) in CB(V,Mn(W ))+ such that
ϕα → ϕ in cb-norm. Suppose m ∈ N and v ∈ Mm(V )+. Then
‖(ϕα)m(v)− ϕm(v)‖ = ‖ [(ϕα)m − ϕm] (v)‖ ≤ ‖(ϕα)m − ϕm‖‖v‖ ≤ ‖ϕα − ϕ‖cb ‖v‖ → 0.
Since each (ϕα)m(v) ∈ Mm(Mn(W ))+ and Mm(Mn(W ))+ is closed, we have ϕm(v) ∈
Mm(Mn(W ))+. Thus ϕ ∈ CB(V,Mn(W ))+, so CB(V, Mn(W ))+ is closed.
Then Mn(CB(V,W )) is a matrix ordered operator space.
We define positive cones for Mn(V∧⊗W ) by
Mn(V∧⊗W )+ = α(v⊗w)α∗ ∈ Mn(V
∧⊗W ) : v ∈ Mp(V )+, w ∈ Mq(W )+, α ∈ Mn,pq−‖·‖∧ .
Proposition ([32]) 2.3.10. Suppose V and W are matrix ordered operator spaces. Then
V∧⊗W is a matrix ordered operator space.
Note. Despite the fact that Itoh is working in a less restrictive sense, in this instance each
cone, as defined above, is closed, and, based on the definition below, the involution is an
isometry at each matrix level.
27
Proof. There is a natural involution on Mn(V∧⊗W ) that is defined by u∗ = β∗(v∗⊗w∗)α∗
for any u = α(v ⊗ w)β ∈ Mn(V∧⊗W ). Since the involutions on Mn(V ) and Mn(W ) are
isometries for all n ∈ N , the same is true for Mn(V∧⊗W ).
Given algebraic elements ui = αi(vi ⊗ wi)α∗i ∈ Mn(V
∧⊗ W )+(i = 1, 2) such that
vi ∈ Mpi(V )+, wi ∈ Mqi(V )+, then v1 ⊕ v2 ∈ Mp1+p2(V )+ and w1 ⊕ w2 ∈ Mq1+q2(W )+.
Thus we have
u1 + u2 = [α1 0 0 α2 ]
([v1 00 v2
]⊗
[w1 00 w2
])[α1 0 0 α2 ]
∗ ∈ Mn(V∧⊗W )+.
It is easy to see that tMn(V∧⊗W )+ ⊆ Mn(V
∧⊗W )+ for any t > 0. Therefore Mn(V
∧⊗W )
is partially ordered.
To see condition (3) of Definition 2.3.4, given u = α(v ⊗ w)α∗ ∈ Mm(V∧⊗ W )+ and
γ ∈ Mn,m, then we have γuγ∗ = (γα)(v ⊗ w)(γα)∗ ∈ Mn(V∧⊗W )+.
Proposition ([32]) 2.3.11. Suppose V and W are matrix ordered operator spaces. Then
θ : V∧⊗W → W
∧⊗ V
is a ∗-preserving, completely isometric, complete order isomorphism.
Proof. In view of Proposition 1.3.4, we need only yet prove that θ is ∗-preserving and a
complete order isomorphism.
Given α(v ⊗w)β ∈ Mn(V∧⊗W ), we denote [αn,ki] by αs for α = [αn,ik ] ∈ Mn,∞2 . It is
easy to see that θ(α(v ⊗ w)β) = αs(w ⊗ v)βs. Then we have
θ((α(v ⊗ w)β)∗) = θ((β∗(v∗ ⊗ w∗)α∗)
= βs∗(w∗ ⊗ v∗)αs∗
= (αs(w ⊗ v)βs)∗
= θ(α(v ⊗ w))β)∗.
Thus, given an algebraic element α(v ⊗ w)α∗ ∈ Mn(V∧⊗W )+, we obtain that
θ(α(v ⊗w)α∗) = αs(w ⊗ v)αs∗ ∈ Mn(W∧⊗ V )+.
28
Theorem ([32]) 2.3.12. Suppose V , W and X are matrix ordered operator spaces.
Then the completely isometric isomorphism CB(V∧⊗W,X) ∼= CB(V,CB(W,X)) is a ∗-
preserving complete order isomorphism.
Proof. Let ϕ ∈ CB(V∧⊗W,X) correspond to ϕ ∈ CB(V, CB(W, X)) such that ϕ(v)(w) =
ϕ(v ⊗ w) for v ∈ V and w ∈ W .
Given [ϕij ] ∈ Mn(CB(V∧⊗W,X)), [vkl] ∈ Mm(V ) and [wst] ∈ Mp(W ), we have
(˜ [ϕij ]
∗[vkl]
)[wst] =
[ϕ∗ji(vkl ⊗ wst)
]
= [ϕij(v∗lk ⊗ w∗ts)]
∗
=(
˜ [ϕij]∗[vkl]
)[wst] .
Next we show the order isomorphism. Let [ϕij ] ∈ CB(V∧⊗W,X)+, [vij ] ∈ Mn(V )+ and
[wkl] ∈ Mm(W )+. Then we have
[vij ⊗ wkl] = Inm([vij ]⊗ [wkl])Inm ∈ Mnm(V∧⊗W )+.
Hence we obtain [ϕ(vij)(wkl)] = [ϕ(vij ⊗ wkl)] ∈ Mnm(X)+.
Conversely, let ϕ ∈ CB(V,CB(W, X))+ and α(v⊗w)α∗ ∈ Mn(V∧⊗W )+. Then we have
ϕ(α(v ⊗ w)α∗) = α [ϕ(vij)(wkl)] α∗ ∈ Mn(X)+.
Denoting this order isomorphism by ', we get
Mn(CB(V∧⊗W,X))+ ' CB(V
∧⊗W,Mn(X))+ ' CB(V, CB(W, Mn(X)))+
' CB(V, Mn(CB(W, X)))+ ' Mn(CB(V,CB(W,X)))+.
This implies that ' is a complete order isomorphism.
Corollary ([32]) 2.3.13. Suppose that V and W are matrix ordered operator spaces.
Then the matrix ordered operator spaces CB(V,W †), CB(W, V †), (V∧⊗W )† and (W
∧⊗V )†
are ∗-preserving, completely isometric, and completely order isomorphic.
Proof. This follows from Proposition 2.3.11 and Theorem 2.3.12.
29
Theorem 2.3.14. If V is a matrix ordered operator space, then V † is. For all n ∈ N ,
the cones in Mn(V †) and Mn(V ′) consist of the same elements (even though the norms
are in most cases different), as do the cones in Mn(V ††) and Mn(V ′′). If V † is a matrix
ordered operator space and Mn(V )+ = Mn(V ) ∩ Mn(V ††)+ (i.e., Mn(V )+ is closed and
has Mn(V †)+ as its dual cone), then V is a matrix ordered operator space. The complete
isometry V → V †† is a ∗-preserving complete order isomorphism onto its range.
Proof. Let v ∈ V †† be the image of v ∈ V . For f ∈ V †, v∗(f) = v(f∗) = f∗(v) =
f (v∗). Then v is self-adjoint ⇐⇒ v∗ = v ⇐⇒ v∗(f ) = v(f ) for all f ∈ V † ⇐⇒ f(v∗) =
f (v) for all f ∈ V † ⇐⇒ f (v∗ − v) = 0 for all f ∈ V † ⇐⇒ v∗ = v ⇐⇒ v is self-adjoint.
Thus Vsa = V ∩ (V ††)sa.
Assuming V to be a matrix ordered operator space, it only remains to show condition (3)
of Definition 2.3.4 to prove that V † is a matrix ordered operator space. Let γ ∈ Mm,n and
[ϕij] ∈ Mn(V †)+. For [vkl] ∈ Mr(V )+,
(γ∗ [ϕij ] γ)r ([vkl]) =[(γ∗ [ϕij ] γ) (vkl)
]k,l
=
∑
i,j
γipγjqϕij
p,q
(vkl)
k,l
=
∑
i,j
γipγjqϕij(vkl)
p,q
k,l
=[(γ∗ [ϕij(vkl)] γ)
]k,l
=
(γ∗ ⊗ Ir) [ϕij(vkl)](k,i),(l,j) (γ ⊗ Ir) ∈ M+mr
since [ϕij] ∈ CB(V,Mn)+. Thus V † is matrix ordered.
If V † is a matrix ordered operator space, then so is V †† by the above. It follows
that V is then a matrix ordered operator space by restriction, where, for all n ∈ N ,
Mn(V )+ = V ∩ (V ††)+.
The equivalence of the cones follows directly from the equivalence of (1) and (4) in
Lemma 2.3.8. The last statement is a direct result of Theorem 1.2.5, Lemma 2.3.9, and
the above.
Definition 2.3.15. If V is a matrix ordered operator space and I is an infinite index
set, define v∗ ∈ MI(V ) as[v∗ji
]where v = [vij ] ∈ MI(V ),
MI(V )+ = v ∈ MI(V )sa : for all ∆ ∈ I,∆ finite, E∆vE∆ = v∆ ∈ M∆(V )+,
30
and
M∞(V )+ = v ∈ M∞(V )sa : EnvEn ∈ Mn(V )+ for all n ∈ N .
We define KI(V )+ and K∞(V )+ similarly.
Proposition 2.3.16. For an operator space V , the involution on M∞(V ) (resp., K∞(V ))
is an isometry if and only if the involution on Mn(V ) is an isometry for all n ∈ N .
Proof. [=⇒] This is clear by restriction.
[⇐=] For [vij ] ∈ M∞(V ),
∥∥ [v∗ji
] ∥∥ = limn
∥∥En
[v∗ji
]En
∥∥ = limn‖En [vij ]En‖ = ‖ [vij ] ‖ ,
so the involution on M∞(V ) is an isometry. For K∞(V ), we still only need that the
involution is a closed operation. But we have this since the involution on Mn(V ) is an
isometry and
∥∥ [v∗ji
]− En
[v∗ji
]En
∥∥ = ‖ [vij ]− En [vij ]En ‖ → 0.
Proposition 2.3.17. Given an operator space V that is matrix ordered as a vector space,
K∞(V )+ (resp., M∞(V )+) is closed if and only if Mn(V )+ is closed for all n ∈ N .
Proof. [⇐=] Let v ∈ [K∞(V )+]−
(resp. [M∞(V )+]−
). Then v = limα
vα where vα ∈
K∞(V )+ (resp. M∞(V )+). For all n ∈ N and for all α, EnvαEn ∈ Mn(V )+, so EnvEn ∈
Mn(V )+ also since ‖EnvEn − EnvαEn‖ = ‖En(v − vα)En‖ ≤ ‖v − vα‖ → 0 and Mn(V )+
is closed. Thus v ∈ K∞(V )+ (resp. M∞(V )+), so K∞(V )+ (resp. M∞(V )+) is closed.
[=⇒] Suppose v ∈ (Mn(V )+)−. Then v ∈ (K∞(V )+)− ⊆ (M∞(V )+)−. Then v ∈
K∞(V )+ ⊆ M∞(V )+, so v = EnvEn ∈ Mn(V )+. Thus Mn(V )+ is closed.
Definition 2.3.18. Given an operator space X, we define the conjugate space X∗ such
that
(1) X∗ = x∗ : x ∈ X,
(2) (x + y)∗ = x∗ + y∗ and αx∗ = (αx)∗ for all α ∈ C , and
(3) for x = [xij] ∈ Mm,n(X),[x∗ji
]∈ Mn,m(X∗) and ‖x∗‖ =
∥∥ [x∗ji
] ∥∥ = ‖x‖.
31
Corollary 2.3.19. Given an operator space X, X ∼= X∗∗ is a complete isometry.
Proof. Define ϕ : X → X∗∗ by ϕ(x) = x∗∗. This is easily seen to be a bijective linear map.
For each n ∈ N and [xij] ∈ Mn(X),
‖ϕn([xij])‖n = ‖ [ϕ(xij)] ‖n =∥∥ [
x∗∗ij] ∥∥
n=
∥∥ [x∗ji
] ∥∥n
= ‖ [xij] ‖n .
Thus ϕ is a complete isometry.
Corollary 2.3.20. Given an operator space X, X∗ is an operator space.
Proof. For v∗ ∈ M n(X∗) and w∗ ∈ M m(X∗),
‖v∗ ⊕w∗‖ = ‖(v ⊕w)∗‖ = ‖v ⊕w‖ = max‖v‖ , ‖w‖ = max‖v∗‖ , ‖w∗‖.
For v∗ ∈ M n(X∗) and α, β ∈ Mn,
‖αv∗β‖ = ‖(β∗vα∗)∗‖ = ‖β∗vα∗‖ ≤ ‖β∗‖ ‖v‖ ‖α∗‖ = ‖α‖ ‖v∗‖ ‖β‖ .
Thus X∗ meets the requirements for the definition of an operator space.
Proposition 2.3.21. Suppose X is an operator space and V is an involutive operator
space. Then X∗ ⊗h V ⊗h X is an involutive operator space where the involution is given
by (ξ∗ ⊗ v ⊗ x)∗ = x∗ ⊗ v∗ ⊗ ξ, with extension to the completion Mn(X∗ ⊗h V ⊗h X) by
continuity for each n ∈ N .
Proof. Clearly X∗⊗h V ⊗h X is an operator space. If u = ξ∗¯v¯η ∈ Mn(X∗⊗h V ⊗h X),
then u∗ = η∗¯ v∗ ¯ ξ. But ‖ξ∗‖‖v‖‖η‖ = ‖η∗‖‖v∗‖‖ξ‖. It follows that ‖u‖ = ‖u∗‖, so the
involution is an isometry on Mn(X∗⊗h V ⊗h X), the isometry extending to the completion
by continuity.
We get special representations for self-adjoint elements in X∗⊗h V ⊗h X . We look first
at the algebraic tensor product.
Lemma ([33]) 2.3.22. Suppose X is an operator space and V is an involutive operator
space. If u ∈ Mn(X∗ ⊗h V ⊗h X)sa, then u has a representation
u = x∗ ¯ V x
32
where x ∈ Mp,n(X) and V ∈ Mp(V )sa. Furthermore,
‖u‖h = inf‖x‖2 ‖V ‖ : u = x∗ ¯ V x with x ∈ Mp,n(X), V ∈ Mp(V )sa, p ∈ N .
Proof. Suppose u ∈ Mn(X∗ ⊗h V ⊗h X)sa. Given ε > 0, by Theorem 1.3.20 there exist
c, d ∈ Mq,n(X) and U ∈ Mq(V ) such that
u = d∗ ¯ U c and ‖u‖h ≤ ‖d∗‖ ‖U ‖ ‖c‖ < ‖u‖h + ε.
Then, for all λ > 0,
u =1
2(u + u∗)
=1
2(λd∗ ¯ U λ−1c + λ−1c∗ ¯ U ∗ ¯ λd)
=1√2
[λd∗ λ−1c∗ ]¯[
0 U U ∗ 0
]¯ 1√
2
[λd
λ−1c
].
We get the desired representation by letting x = 1√2
[λd
λ−1c
]and V =
[0 U
U ∗ 0
].
For the norm of u, since
∥∥∥∥1√2
[λd
λ−1c
] ∥∥∥∥2
≤ 1
2(λ2 ‖d∗‖2 + λ−2 ‖c‖2) and
minλ>0
1
2(λ2 ‖d∗‖2 + λ−2 ‖c‖2) = ‖d∗‖ ‖c‖ ,
there exists λ0 > 0 such that
‖u‖h ≤∥∥∥∥
1√2
[λd
λ−1c
] ∥∥∥∥2 ∥∥∥∥
[0 U
U ∗ 0
] ∥∥∥∥ ≤ ‖d∗‖ ‖U ‖ ‖c‖ < ‖u‖h + ε.
This then establishes the desired norm condition.
Proposition 2.3.23. Suppose X is an operator space and V is an involutive operator
space. If u ∈ Mn(X∗ ⊗h V ⊗h X)sa, then u has a representation
u = x∗ ¯ V x
where x ∈ K∞,n(X) and V ∈ K∞(V )sa. Furthermore,
‖u‖h = inf‖x‖2 ‖V ‖ : u = x∗ ¯ V x with x ∈ K∞,n(X),V ∈ M∞(V )sa.
33
Proof. If u ∈ Mn(X∗ ⊗h V ⊗h X)sa, then, since the set of self-adjoint elements is closed,
u = limj→∞
sj where sj ∈ Mn(X∗⊗h V ⊗h X)sa. Given ε > 0, and by choosing a subsequence
if necessary, we may assume that
‖u− s1‖h <ε2
4and ‖sj+1 − sj‖h <
ε2
23j+2.
Then ‖s1‖h < ‖u‖h + ε2
4and u = s1 +
∞∑j=1
(sj+1 − sj). Since s1 and sj+1 − sj (j ∈ N ) are
self-adjoint, from Lemma 2.3.22, we may assume that
s1 = x∗1 ¯ v1 ¯ x1 and sj+1 − sj = x∗j+1 ¯ vj+1 ¯ xj+1
where xj ∈ Mnj ,n(X), vj ∈ Mnj (V )sa, ‖x1‖ <(‖u‖h + ε2
4
)1/2
, ‖v1‖ = 1, ‖xj+1‖ < ε2j+1 ,
and ‖vj+1‖ = 12j .
Let x =
x1
x2...
and V =
v1
v2
. . .
. Recalling from (1.3.9) that
(‖u‖h +
ε2
4
)1/2
≤ ‖u‖1/2h +
ε
2,
we get
‖x‖ ≤∞∑
j=1
‖xj‖ < ‖u‖1/2h +
ε
2+
∞∑
j=2
ε
2j= ‖u‖1/2
h + ε.
Then
u = x∗ ¯ V x =
∞∑
j=1
x∗j ¯ vj ¯ xj = [x∗1 x∗2 . . . ]¯
v1
v2
. . .
¯
x1
x2...
with ‖x∗‖ ‖V ‖ ‖x‖ <(‖u‖1/2
h + ε)2
= ‖u‖h + 2ε ‖u‖1/2h + ε2 and V ∈ K∞(V )sa. Letting
ε → 0, we have
‖u‖h ≥ inf‖x∗‖ ‖V ‖ ‖x‖ : u = x∗ ¯ V x, x ∈ K∞,n(X),V ∈ K∞(V )sa.
Since the reverse inequality is clear by Definition 1.3.14, we get that
‖u‖h = inf‖x∗‖ ‖V ‖ ‖x‖ : u = x∗ ¯ V x, x ∈ K∞,n(X),V ∈ K∞(V )sa.
Next we define positive cones for X∗ ⊗h V ⊗h X.
34
Definition 2.3.24. Suppose X is an operator space and V is a matrix ordered operator
space. We define
(1) Pn(X∗ ⊗h V ⊗h X) =
x∗ ¯ V x ∈ Mn(X∗ ⊗h V ⊗h X) : x ∈ Mp,n(X), V ∈ Mp(V )+, p ∈ N , and
(2) Mn(X∗ ⊗h V ⊗h X)+ = Pn(X∗ ⊗h V ⊗h X)−‖·‖h .
Proposition 2.3.25. Suppose X is an operator space and V is a matrix ordered operator
space. Then Mn(X∗ ⊗h V ⊗h X)+ is a cone in Mn(X∗ ⊗h V ⊗h X) for all n ∈ N .
Proof. If u = x∗ ¯ U x, v = y∗ ¯ V y ∈ Pn(X∗ ⊗h V ⊗h X), then
u + v = [x∗ y∗ ]¯[
U 00 V
]¯
[xy
]∈ Pn(X∗ ⊗h V ⊗h X).
Since the positive scalar property is clear, Pn(X∗ ⊗h V ⊗h X) is a cone. Since the closure
of a cone is a cone, we have that Mn(X∗ ⊗h V ⊗h X)+ is a cone.
Proposition 2.3.26. Suppose V is a matrix ordered operator space and X is an operator
space. Then X∗ ⊗h V ⊗h X is a matrix ordered operator space.
Proof. The only item still needing proof is the scalar multiplication property. Suppose
u ∈ Mm(X∗ ⊗h V ⊗h X)+ and γ ∈ Mm,n. Then u = lims
us = lims
x∗s ¯ V s ¯ xs where
us = x∗s ¯ V s ¯ xs ∈ Pm(X∗ ⊗h V ⊗h X) for all s ∈ N . It follows that
γ∗usγ = γ(x∗s ¯ V s ¯ xs)γ = (xsγ)∗ ¯ V s ¯ (xsγ) ∈ Pn(X∗ ⊗h V ⊗h X)
and
‖γ∗uγ − γ∗usγ‖h = ‖γ∗‖ ‖u− us‖h ‖γ‖ → 0,
so γ∗uγ ∈ Mn(X∗ ⊗h V ⊗h X)+.
Definition 2.3.27. Suppose X is an operator space and V is a matrix ordered operator
space. Then we define
P∞n (X∗ ⊗h V ⊗h X) = x∗ ¯ V x ∈ Mn(X∗ ⊗h V ⊗h X) : x ∈ K∞,n(X),V ∈ K∞(V )+.
35
Proposition 2.3.28. Suppose X is an operator space and V is a matrix ordered operator
space. Then
Mn(X∗ ⊗h V ⊗h X)+ = P∞n (X∗ ⊗h V ⊗h X)−‖·‖h .
Proof. If u ∈ Mn(X∗ ⊗h V ⊗h X)+, then
u ∈ Pn(X∗ ⊗h V ⊗h X)−‖·‖h ⊆ P∞n (X∗ ⊗h V ⊗h X)−‖·‖h .
Now suppose u ∈ P∞n (X∗ ⊗h V ⊗h X). Then
u = x∗ ¯ V x
with x = [xij] ∈ K∞,n(X) and V = [vij] ∈ K∞(V )+, so
u = limp→∞
[ x∗1 . . . x∗p ]¯
v11 . . . v1p
.... . .
...vp1 . . . vpp
¯
x1...
xp
∈ Pn(X∗ ⊗h V ⊗h X)−‖·‖h
where xi = [ xi1 . . . xin ]. Thus
P∞n (X∗ ⊗h V ⊗h X) ⊆ Pn(X∗ ⊗h V ⊗h X)−‖·‖h ,
which implies
P∞n (X∗ ⊗h V ⊗h X)−‖·‖h ⊆ Pn(X∗ ⊗h V ⊗h X)−‖·‖h = Mn(X∗ ⊗h V ⊗h X)+.
Although in general we do not know that Mn(X∗ ⊗h V ⊗h X)+ = P∞n (X∗ ⊗h V ⊗h X),
we will soon be able to show this in the case where X = Hc. As we shall see in the next
chapter, we also get some other nice properties for this specific choice for X.
36
3. Matrix Regularity and Duality
Our main objective in this chapter is to define the concept of a matrix regular (or
matricial Riesz) operator space and then show that a matrix ordered operator space is
matrix regular if and only if its operator dual is. In order to do this, we first need to focus
on operator spaces of the form X∗ ⊗h V ⊗h X where X = Hc. We do this in the first
section of this chapter where we also define matrix regularity, and then look at the duality
of matrix regularity in the second section.
3.1. Matrix Regularity
We first determine the nature of X∗ for X = Hc.
Proposition 3.1.1. Given a Hilbert space H, (Hc)∗ = Hr where h∗ = h = 〈· | h〉.
Proof. Since properties (1) and (2) are clear, we need only check property (3) of Defini-
tion 2.3.18. Suppose [hij ] ∈ Mm,n(Hc) = B(C n,Hm). Then [hij ]∗
=[h∗ji
]=
[hji
]∈
Mn,m(Hr) = B(Hm, C n). Let h =
h1...
hn
∈ Hm and α =
α1...
αn
∈ C n with ‖h‖ ≤ 1 and
‖α‖ ≤ 1. Then also α =
α1...
αn
∈ C n with ‖α‖ ≤ 1. Since
|〈h | [hij ] α〉| =∣∣∣∑
ij
αj〈hi | hij〉∣∣∣ =
∣∣⟨α |[hji
]h⟩∣∣ ,
we have ‖ [hij ] ‖ =∥∥ [hij ]
∗∥∥.
Corollary 3.1.2. Given a Hilbert space H, (Hc)† ∼= (Hc)
∗ is a complete isometry.
Proof. This follows directly from Proposition 3.1.1 and Theorem 1.3.25 (1.3.12).
The following lemma relating to Hilbert-Schmidt operators is a useful tool in the proof
of our next theorem.
Lemma ([55, Proposition III.G.12]) 3.1.3. Let H and K be Hilbert spaces and let
T : H → K be a linear operator. Then the following are equivalent:
(1) T is a Hilbert-Schmidt operator.
37
(2) For some orthonormal basis hjj∈J in the space H we have that∑j∈J
‖T (hj)‖2 <
∞.
(3) For every orthonormal basis hjj∈J in the space H we have that∑j∈J
‖T (hj)‖2 <
∞.
Theorem 3.1.4. Suppose V is an operator space and H a Hilbert space. For u ∈
Hr ⊗h V ⊗h Hc, u has a representation (called a standard Hilbert representation)
u =
∞∑
i,j=1
ξ∗i ⊗ vij ⊗ ηj = [ ξ∗1 ξ∗2 . . . ]¯ V
η1
η2...
= ξ∗ ¯ V η
where ξ∗i and ηj are nonzero orthogonal sets in H,∑i
‖ξ∗i ‖2
< ∞,∑j
‖ηj‖2 < ∞, and
V ∈ K∞(V ). Further,
‖u‖h = inf(∑
j
∥∥ξ∗j∥∥2
)1/2
‖V ‖(∑
i
‖ηi‖2)1/2
where the inf is taken over all standard Hilbert representations.
If u is self-adjoint, then it has a standard Hilbert representation with ξ = η and V ∈
K∞(V )sa (called a standard Hilbert self-adjoint representation). In this case,
‖u‖h = inf(∑
i
‖ηi‖2)‖V ‖
with the inf taken over all standard self-adjoint Hilbert representations.
Note. The theorem applies irrespective of dim H. If only finitely many ξi and ηj are
nonzero, then, because of the injectivity of the Haagerup tensor product (Theorem 1.3.19),
finite sums may be used with V finite dimensional.
Proof. Given u ∈ Hr ⊗h V ⊗h Hc, by Theorem 1.3.20 we can write
u = β∗ ¯ U α = [β∗1 β∗2 . . . ]¯ U
α1
α2...
=
∑
i,j
β∗i ⊗ uij ⊗ αj
with β∗ ∈ K1,∞(Hr), U ∈ K∞(V ), and α ∈ K∞,1(Hc). Then∑i
‖αi‖2 < ∞ and
∑i
‖β∗i ‖2
< ∞. If u is self-adjoint, Proposition 2.3.23 allows us to take β = α and
U ∈ K∞(V )sa.
38
It is also clear from the rightmost representation that if any β∗i = 0, then none of the
elements of row i of U actually appear in the sum in a significant way. If any αj = 0, then
none of the elements of column j of U appear significantly in the sum. Thus, by removing
these zero elements from the sets β∗i and αj along with the corresponding rows and
columns of U , we may assume that all β∗i and all αj are nonzero.
The set span(αi ∪ βi) is a closed subspace of H . Since the αi and βi are both
countable sets, we may, without loss of generality, assume that H = l2 with standard
orthonormal basis ei. Then
αi =∑
j
aijej =∑
j
〈αi | ej〉 ej ,
and∑i,j
|aij |2 =∑i,j
|〈αi | ej〉|2 =∑i
‖αi‖2 < ∞, so A = [aij ] ∈ B(l2) is Hilbert-Schmidt.
Then A factors as
A = S1T1 =
a11
‖Ae1‖a12
‖Ae2‖ . . .a21
‖Ae1‖a22
‖Ae2‖ . . .
......
. . .
‖Ae1‖
‖Ae2‖. . .
.
To see this, we have first from Lemma 3.1.3 that∑j
‖Aej‖2 < ∞. Then T1 ∈ B(l2) and is
Hilbert-Schmidt by the same Lemma. For γ =
γ1
γ2...
∈ l2,
‖S1γ‖2 =
∞∑
i=1
∣∣∣∞∑
j=1
aij
‖Aej‖γj
∣∣∣2
≤∞∑
i=1
∞∑
j=1
∣∣∣ γj
‖Aej‖aij
∣∣∣2
=
∞∑
j=1
|γj |2
‖Aej‖2( ∞∑
i=1
|aij|2)
=
∞∑
j=1
|γj |2 = ‖γ‖2 .
Thus ‖S1‖ ≤ 1, so S1 ∈ B(l2) also.
Let
η1
η2...
= T1
e1
e2...
=
‖Ae1‖ e1
‖Ae2‖ e2
...
. Then the ηi are orthogonal and nonzero,
(3.1.1) ‖η‖2 =
∞∑
i=1
‖ηi‖2 =
∞∑
i=1
‖Aei‖2 = ‖A‖2 =∑
i,j
|aij |2 =∑
i
‖αi‖2 = ‖α‖2 < ∞,
39
and
α1
α2...
= S1
η1
η2...
.
Similarly, with
βi =∑
j
bijej =∑
j
〈βi | ej〉 ej and
β∗i =∑
j
bije∗j =
∑
j
〈ej | βi〉 e∗j ,
B = [βij] = S2T2 =
b11
‖Be1‖b12
‖Be2‖ . . .b21
‖Be1‖b22
‖Be2‖ . . .
......
. . .
‖Be1‖
‖Be2‖. . .
where ‖S2‖ ≤ 1,
ξ1
ξ2...
= T2
e1
e2...
=
‖Be1‖ e1
‖Be2‖ e2
...
, the ξi and thus ξ∗i are orthogonal
and nonzero,
(3.1.2) ‖ξ∗‖2 =
∞∑
i=1
‖ξ∗i ‖2
=
∞∑
i=1
‖Bei‖2 = ‖B‖2 =∑
i,j
|bij|2 =∑
i
‖βi‖2 = ‖β‖2 < ∞,
and
[β∗1 β∗2 . . . ] = [ ξ∗1 ξ∗2 . . . ]S∗2 .
Let ξ∗ = [ ξ∗1 ξ∗2 . . . ] and η = [ η1 η2 . . . ]tr
. Then
U = [β∗1 β∗2 . . . ]¯ U
α1
α2...
= ξ∗S∗2 ¯ U S1η = ξ∗ ¯ S∗2U S1 ¯ η.
Let V = S∗2U S1.
Now U ∈ K∞(V ) = V∨⊗K∞, so ‖U − U s‖ → 0 where U s ∈ V ⊗∨ K∞. Then S∗2U sS1 ∈
V ⊗∨ K∞ and
‖S∗2U S1 − S∗2U sS1‖ = ‖S∗2(U −U s)S1‖ ≤ ‖S∗2‖ ‖U −U s‖ ‖S1‖ → 0,
so V ∈ K∞(V ) since K∞(V ) is closed. Further, if u is self-adjoint, we have U ∈ K∞(V )sa
and S2 = S1, so V = S∗1U S1 ∈ K∞(V )sa.
40
Thus
‖V ‖ = ‖S∗2U S1‖ ≤ ‖U ‖ ,
and using also (3.1.1) and (3.1.2),
‖ξ∗‖ ‖V ‖ ‖η‖ ≤ ‖β∗‖ ‖U ‖‖α‖ ,
so the norm condition holds in view of Theorem 1.3.20, or Proposition 2.3.23 in the self-
adjoint case.
Lemma 3.1.5. Suppose V is a matrix ordered operator space, H is a Hilbert space, ξ∗i
is an orthogonal set in Hr with supi‖ξ∗i ‖ < ∞, and ηi is an orthogonal set in Hc with
supi‖ηi‖ < ∞. Define mappings ϕξ∗,η
ij : Hr ⊗h V ⊗h Hc → V by
ϕξ∗,ηij (l∗ ⊗ v ⊗ k) = 〈ξi | l〉 〈k | ηj〉 v.
These are bounded linear maps, and also positive if ηi = ξi for all i ∈ N . Then the mapping[ϕξ∗,η
ij
]: Hr ⊗h V ⊗h Hc → M∞(V ) defined by
[ϕξ∗,η
ij
](l∗ ⊗ v ⊗ k) =
[ϕξ∗,η
ij (l∗ ⊗ v ⊗ k)]
is a bounded linear map, and also positive if ηi = ξi for all i ∈ N .
Proof. The linearity for all of these mappings is clear since V is a linear space and inner
products are linear in the first variable and conjugate linear in the second. For boundedness
and positivity, we just provide a proof for[ϕξ∗,η
ij
]since the proofs for the individual maps
are similar.
Suppose
u = l∗ ¯ V k = [ l∗1 l∗2 . . . ]¯ [vij ]¯
k1
k2...
with l∗ ∈ K1,∞(Hr), k ∈ K∞,1(Hc), and V ∈ K∞(V ).
∥∥∥[ϕξ∗,η
ij
](u)
∥∥∥ =
∥∥∥∥∥
[ϕξ∗,η
ij
(∑
s,t
l∗s ⊗ vst ⊗ ky
)] ∥∥∥∥∥
=
∥∥∥∥∥
[∑
s,t
〈ξi | ls〉 〈kt | ηj〉 vst
] ∥∥∥∥∥
41
=
∥∥∥∥∥∥∥
〈ξ1 | l1〉 〈ξ1 | l2〉 . . .〈ξ2 | l1〉 〈ξ2 | l2〉 . . .
......
. . .
¯ [vst]¯
〈k1 | η1〉 〈k1 | η2〉 . . .〈k2 | η1〉 〈k2 | η2〉 . . .
......
. . .
∥∥∥∥∥∥∥= ‖L¯ V K‖ .
Then, since the ηj are orthogonal,
‖K‖2 = ‖ [〈kt | ηj〉] ‖2 = sup∥∥∥ [〈kt | ηj〉]
γ1
γ2...
∥∥∥2
:∑
j
‖γj‖2 ≤ 1
= sup∑
t
∣∣∣∑
j
〈kt | γjηj〉∣∣∣2
:∑
j
‖γj‖2 ≤ 1
= sup∑
t
∣∣⟨kt
∣∣ ∑
j
γjηj
⟩∣∣2 :∑
j
‖γj‖2 ≤ 1
≤ sup∑
t
‖kt‖2∥∥∑
j
γjηj
∥∥2:∑
j
‖γj‖2 ≤ 1
= sup∥∥∑
j
γjηj
∥∥2(∑
t
‖kt‖2) :∑
j
‖γj‖2 ≤ 1
= ‖k‖2 sup⟨∑
j
γjηj
∣∣ ∑
i
γiηi
⟩:∑
j
‖γj‖2 ≤ 1
= ‖k‖2 sup∑
j
|γj |2 ‖ηj‖2 :∑
j
‖γj‖2 ≤ 1
≤ ‖k‖2 supj‖ηj‖2 < ∞,
so ‖K‖ ≤ ‖k‖ supj‖ηj‖ < ∞. Similarly, ‖L‖ ≤ ‖l∗‖ sup
i‖ξ∗i ‖ < ∞. Then
∥∥∥[ϕξ∗,η
ij
](u)
∥∥∥ = ‖L ¯ V K‖ ≤ ‖L‖ ‖V ‖ ‖K‖ ≤ ‖l∗‖ ‖k‖ ‖V ‖ supj‖ηj‖ sup
i‖ξi‖ < ∞,
so[ϕξ∗,η
ij
](Hr ⊗h V ⊗h Hc) ∈ M∞(V ) and
[ϕξ∗,η
ij
]is bounded.
Now suppose ηi = ξi for all i ∈ N and u ∈ P1(Hr ⊗h V ⊗h Hc). Then
u = k∗ ¯ V k = [ k∗1 . . . k∗p ]¯
v11 . . . v1p
.... . .
...vp1 . . . vpp
¯
k1...
kp
,
so
[ϕη∗,η
ij
](u) =
〈η1 | k1〉 . . . 〈η1 | kp〉〈η2 | k1〉 . . . 〈η2 | kp〉
......
. . .
¯ V
〈k1 | η1〉 〈k1 | η2〉 . . .
......
. . .
〈kp | η1〉 〈kp | η2〉 . . .
.
42
Since 〈ηj | kt〉 = 〈kt | ηj〉 for all j ∈ N and 1 ≤ t ≤ p and V ∈ Mp(V )+,[ϕη∗,η
ij
](u) ∈
M∞(V )+. Since[ϕη∗,η
ij
]is continuous and M∞(V )+ is closed, we have
[ϕη∗,η
ij
](u) ∈
M∞(V )+ for u ∈ (Hr ⊗h V ⊗h Hc)+. We conclude that
[ϕη∗,η
ij
]is positive.
Theorem 3.1.6. Suppose V is an operator space and H a Hilbert space. If u ∈
Hr ⊗h V ⊗h Hc has a standard Hilbert representation
u =
∞∑
i,j=1
ξ∗i ⊗ vij ⊗ ηj = [ ξ∗1 ξ∗2 . . . ]¯ V
η1
η2...
= ξ∗ ¯ V η,
then V is uniquely determined in the sense that if u = ξ∗ ¯ V η = ξ∗ ¯W η are two
standard Hilbert representations, it follows that V = W .
Proof. Suppose
u = ξ∗ ¯ V η = ξ∗ ¯W η
are two standard Hilbert representations, or, equivalently,
u =∑
s,t
ξ∗s ⊗ vst ⊗ ηt =∑
s,t
ξ∗s ⊗ wst ⊗ ηt.
Then, for i, j ∈ N ,
ϕξ∗,ηij
(∑
s,t
ξ∗s ⊗ vst ⊗ ηt
)=
∑
s,t
ϕξ∗,ηij (ξ∗s ⊗ vst ⊗ ηt)
=∑
s,t
〈ξi | ξs〉 〈ηt | ηj〉 vkl
= ‖ξi‖2 ‖ηj‖2 vij .
Similarly,
ϕξ∗,ηij
(∑
s,t
ξ∗s ⊗wst ⊗ ηt
)= ‖ξi‖2 ‖ηj‖2 wij .
Since all the ξi and ηi are nonzero, vij = wij . Thus V = W .
Theorem 3.1.7. Let V be a matrix ordered operator space and let H be a Hilbert space.
Then P∞1 (Hr ⊗h V ⊗h Hc) is closed, i.e.,
(Hr ⊗h V ⊗h Hc)+ = P∞1 (Hr ⊗h V ⊗h Hc).
43
Proof. If u ∈ (Hr ⊗h V ⊗h Hc)+, then u has a standard self-adjoint Hilbert representation
u = x∗ ¯ V x where V ∈ K∞(V )sa. Thus, referring to the previous proof,
[ϕx∗,x
ij
](u) =
‖x1‖2
‖x2‖2. . .
[vij ]
‖x1‖2
‖x2‖2. . .
∈ M∞(V )+
since[ϕx∗,x
ij
]is positive. For all n ∈ N ,
En
[ϕx∗,x
ij
](u)En =
‖x1‖2
. . .
‖xn‖2
En [vij ]En
‖x1‖2
. . .
‖xn‖2
∈ Mn(V )+.
Thus
En [vij ]En =
1‖x1‖2
. . .1
‖xn‖2
(En
[ϕx∗,x
ij
](u)En)
1‖x1‖2
. . .1
‖xn‖2
∈ Mn(V )+,
so V = [vij ] ∈ M∞(V )+. But since V ∈ K∞(V )sa, we must have V ∈ K∞(V )+. Then
x∗ ¯ V x ∈ P∞1 (Hr ⊗h V ⊗h Hc).
Consider u ∈ Mn(Hr⊗hV ⊗hHc). From Theorem 1.3.20, we know u has a representation
u = ξ∗ ¯ V ¯ η where ξ∗ =[ξ∗ji
]∈ Kn,∞(Hr), V = [vij] ∈ K∞(V ), and η = [ηij] ∈
K∞,n(Hc). But this is the same representation (ignoring norms) as that for
u′ = [ ξ∗1 ξ∗2 . . . ]¯ V
η1
η2...
∈ (Hn)r ⊗h V ⊗h (Hn)c
where ξ∗i =
ξ∗i1...
ξ∗in
and ηi = [ ηi1 . . . ηin ]. We can work just as easily in the opposite
direction. We also have
M1,n(Hc) = M1,n
∨⊗Hc = (C n)r
∨⊗Hc and
(Hn)c = B(C ,Hn) = Mn,1(Hc) = Mn,1
∨⊗Hc = (C n)c
∨⊗Hc.
Consider the map ϕ : (C n)r → (C n)c which maps a row vector to its corresponding
column vector. Clearly ϕ is invertible. From a lemma of Effros and Haagerup ([17];
44
the lemma is stated for operator systems, but the proof works equally well for operator
spaces), both ϕ and ϕ−1 are completely bounded with ‖ϕ‖cb ,∥∥ϕ−1
∥∥cb≤ n. Thus there
exists a completely bounded isomorphism between M1,n(Hc) = (C n)r
∨⊗Hc and (Hn)c =
(C n)c
∨⊗ Hc. Thus there is a natural one-to-one correspondence between the elements of
M∞,n(Hc) and M∞,1((Hn)c). With a corresponding result for Hr, we have a natural
correspondence between the elements of Mn(Hr ⊗h V ⊗h Hc) and (Hn)r ⊗h V ⊗h (Hn)c.
In this correspondence, positive elements also correspond. Noting the “equivalence” of the
norms, we get the following corollary:
Corollary 3.1.8. Let V be a matrix ordered operator space and let H be a Hilbert space.
Then P∞n (Hr ⊗h V ⊗h Hc) is closed, i.e., Mn(Hr ⊗h V ⊗h Hc)+ = P∞n (Hr ⊗h V ⊗h Hc).
We now proceed to define the key new concept in this thesis, the matrix regular operator
space, which expands Definitions 2.1.5 and 2.2.5 to the case of matrix ordered operator
spaces.
Definition 3.1.9. Suppose V is a matrix ordered operator space. We say V is a matrix
regular (or matricial Riesz) operator space if for each n ∈ N , the norm ‖·‖n on the ordered
complex Banach space Mn(V ) is a regular norm, i.e., if for each n ∈ N and for all v ∈
Mn(V )sa,
(1) u ∈ Mn(V )sa and −u ≤ v ≤ u imply that ‖v‖n ≤ ‖u‖n, and
(2) ‖v‖n < 1 implies that there exists u ∈ Mn(V )sa such that ‖u‖n < 1
and −u ≤ v ≤ u.
Note. In both (1) and (2) above, u is actually positive.
First examples of matrix regular operator spaces include all C∗-algebras, and in fact
all operator systems (linear self-adjoint subspaces of B(H) for some Hilbert space H that
contain the unit of B(H) ). To see this, we first recall that if A is a C∗-algebra, then so
is Mn(A), and it is a well-known C∗-algebra fact that if a and b are self-adjoint elements
with −b ≤ a ≤ b, then ‖a‖ ≤ ‖b‖. It is also well known that for each self-adjoint element
a of a C∗-algebra A with identity that ‖a‖ id±a ∈ A+. Since∥∥ ‖a‖ id
∥∥ = ‖a‖, we have
then established the matrix regularity of operator systems and C∗-algebras with identity.
For the case where a C∗-algebra has no identity, if a is self-adjoint, then a2 ∈ A+, as is
45
|a| = (a2)1/2, a+ = 12 (|a|+ a) and a− = 1
2 (|a| − a). Since a = a+ − a− and |a| = a+ + a−,
we get that − |a| ≤ a ≤ |a|. Since∥∥ |a|
∥∥2=
∥∥ |a|2∥∥ =
∥∥a2∥∥ = ‖a‖2, if ‖a‖ < 1, then
‖ |a| ‖ < 1 and we conclude that we have matrix regularity in this case also.
The following is an equivalent formulation of the definition:
Theorem 3.1.10. Suppose V is a matrix ordered operator space. Then V is matrix
regular if and only if the following condition holds:
For all u ∈ Mn(V ), ‖u‖ < 1 if and only if there exist u1, u2 ∈ Mn(V )+, ‖u1‖ < 1 and
‖u2‖ < 1, such that
[u1 uu∗ u2
]∈ M2n(V )+.
Proof. Several times in this proof we shall use the following fact:
(3.1.3) If
[a bc d
]∈ M2n(V )+, then so is
[a −b−c d
]∈ M2n(V )+.
This is because [a −b−c d
]=
[In 00 −In
] [a bc d
] [In 00 −In
].
[=⇒] We assume that V is matrix regular.
(⇒) Suppose u ∈ Mn(V ) and ‖u‖ < 1. Then
[0 uu∗ 0
]∈ M2n(V )sa and
∥∥∥∥[
0 uu∗ 0
]∥∥∥∥ < 1.
Since V is matrix regular, there exists
[u1 u3
u∗3 u2
]∈ M2n(V )sa with
∥∥∥∥[
u1 u3
u∗3 u2
]∥∥∥∥ < 1 and
[u1 u3 ± u
u∗3 ± u∗ u2
]∈ M2n(V )+.
Clearly, u1, u2 ∈ Mn(V )+, ‖u1‖ < 1, and ‖u2‖ < 1. Now
[u1 −u3 ∓ u
−u∗3 ∓ u∗ u2
]∈ M2n(V )+ by (3.1.3), so
[u1 uu∗ u2
]=
1
2
([u1 −u3 + u
−u∗3 + u∗ u2
]+
[u1 u3 + u
u∗3 + u∗ u2
])∈ M2n(V )+.
(⇐) Suppose there exist u1, u2 ∈ Mn(V )+ such that ‖u1‖ < 1, ‖u2‖ < 1, and
[u1 uu∗ u2
]∈ M2n(V )+.
Since [u1 −u−u∗ u2
]∈ M2n(V )+ by (3.1.3),
46
we get [u1 00 u2
]±
[0 uu∗ 0
]∈ M2(V )+.
Since V is matrix regular,
∥∥∥∥[
0 uu∗ 0
]∥∥∥∥ ≤∥∥∥∥[
u1 00 u2
]∥∥∥∥ < 1,
so ‖u‖ < 1.
[⇐=]
(1) Suppose u, w ∈ Mn(V )sa with −u ≤ w ≤ u. Then u±w ≥ 0, so
[u −iwiw u
]=
1
2
([1i
](u + w) [ 1 −i ] +
[1−i
](u−w) [ 1 i ]
)∈ M2n(V )+.
Suppose ‖w‖ > ‖u‖. Then, ‖w‖ = ‖u‖+ ε for some ε > 0, so
[u/‖w‖ −iw/‖w‖iw/‖w‖ u/‖w‖
]∈ M2n(V ) + .
Since ‖u/‖w‖ ‖ = ‖u‖/‖w‖ < 1, we have by hypothesis that ‖−iw/‖w‖ ‖ < 1.
But ‖−iw/‖w‖ ‖ = 1, a contradiction. Thus ‖w‖ ≤ ‖u‖.
(2) Suppose w ∈ Mn(V )sa with ‖w‖ < 1. Then there exist w1, w2 ∈ Mn(V )+ with
‖w1‖ < 1, ‖w2‖ < 1, and
[w1 ww w2
]∈ M2n(V )+. Then also
[w1 −w−w w2
]∈
M2n(V )+ by (3.1.3), so
w1 + w2
2± w = [ (1/
√2 )In 1/(
√2 )In ]
[w1 ±w±w w2
] [(1/√
2 )In
(1/√
2 )In
]∈ Mn(V )+.
Let u =w1 + w2
2. Then u ± w ∈ Mn(V )+ and ‖u‖ < 1. Thus V is matrix
regular.
The following theorem relates the matrix order structure of a matrix ordered operator
space V to that of K∞(V ).
Theorem 3.1.11. Suppose V is a matrix ordered operator space. Then V is matrix
regular if and only if K∞(V ) is regular.
Proof. [=⇒] Suppose w ∈ K∞(V )sa, ‖w‖ < 1. Then ‖w‖ < 1 − ε for some ε > 0. By
definition, there exists a sequence (∆n) of finite subsets of N such that ∆1 ⊆ ∆2 ⊆ . . .
47
and wn = E∆nwE∆n → w in norm. Hence there exists a subsequence (wnk ) with wnk ∈
M∆nk(V )sa such that
‖wn1‖ < 1− ε, ‖wn2 − wn1‖ < ε/2, . . . ,∥∥wnk+1 − wnk
∥∥ < ε/2k, . . . .
Then, for all k ∈ N there exists uk ∈ M∆nk(V )+ such that
u1 ± wn1 ≥ 0, uk ± (wnk − wnk−1) ≥ 0, ‖u1‖ < 1− ε, and ‖uk‖ < ε/2k−1.
Hence u =∞∑
k=1
uk ∈ K∞(V )+,
‖u‖ ≤∞∑
k=1
‖uk‖ < 1− ε + (ε/2 + ε/4 + ε/8 + · · · ) = 1, and
u±w = (u1 ± wn1 ) +
∞∑
k=2
[uk ± (wnk − wnk−1)
]≥ 0.
Now suppose u, w ∈ K∞(V )sa, u± w ≥ 0. Then, for all ∆ ⊆ N , ∆ finite,
E∆(u± w)E∆ = E∆uE∆ ± E∆wE∆ ≥ 0,
so ‖E∆wE∆‖ ≤ ‖E∆uE∆‖ for all ∆ ⊆ N . We conclude that ‖w‖ ≤ ‖u‖.
[⇐=] Let w ∈ Mn(V )sa, ‖w‖ < 1. Then w ∈ K∞(V )sa also. Since K∞(V ) is regular,
there exists u ∈ K∞(V )+ such that u ± w ≥ 0 and ‖u‖ < 1. But then En(u ± w)En ≥ 0,
so EnuEn ± w ≥ 0. Hence EnuEn ∈ Mn(V )+ and ‖EnuEn‖ ≤ ‖u‖ < 1.
Secondly, if −u ≤ w ≤ u in Mn(V ), then −u ≤ w ≤ u in K∞(V ), so ‖w‖ ≤ ‖u‖.
Thus V is matrix regular.
3.2. Matrix Regularity and Duality
Having defined matrix regularity, we proceed in this section to establish its duality
properties. The following theorem is the main tool that allows us to establish this duality.
Theorem 3.2.1. If V is a matrix regular operator space, then the Haagerup norm on
Hr ⊗h V ⊗h Hc is a regular norm.
48
Proof. Suppose u, w ∈ (Hr⊗h V ⊗h Hc)sa with −u ≤ w ≤ u. Then u and w have standard
self-adjoint Hilbert representations
u = a∗ ¯ U a =∑
i,j∈α
a∗i ⊗ uij ⊗ aj and
w = b∗ ¯W b =∑
i,j∈β
b∗i ⊗ wij ⊗ bj
where∑j∈α
‖aj‖2 = 1,∑j∈β
‖bj‖2 = 1, U ∈ K∞(V )+ (since u must be positive), W ∈
K∞(V )sa, and ‖u‖h ≤ ‖U ‖ < ‖u‖h + ε for given ε > 0.
The set
ai
‖ai‖
is an orthonormal set in H . Rename this orthonormal set as cii∈α
and extend to an orthonormal basis cii∈I for H , I being the index set for H . Since
we may assume that H is separable here because of the injectivity of the Haagerup norm
(Theorem 1.3.19), we have that I = N . For i ∈ N \α or j ∈ N \α, define uij = 0.
Then
u =∑
i∈α
∑
j∈α
a∗i ⊗ uij ⊗ aj =∑
i∈α
∑
j∈α
a∗i‖ai‖
⊗ ‖ai‖ ‖aj‖uij ⊗aj
‖aj‖
=∑
i∈α
∑
j∈α
c∗i ⊗ uij ⊗ cj =∑
i∈N
∑
j∈N c∗i ⊗ uij ⊗ cj
where uij =
‖ai‖ ‖aj‖uij , for i, j ∈ α
0, for i ∈ N \α or j ∈ N \α.
Written formally in terms of the same orthonormal basis,
w =∑
i∈N
∑
j∈N
c∗i ⊗wij ⊗ cj .
That this expression for w is well-defined will become clear as the proof proceeds.
We recall from Lemma 3.1.5 the existence of the positive bounded linear map
[ϕc∗,c
ij
]: Hr ⊗h V ⊗h Hc → M∞(V )
defined by [ϕc∗,c
ij
](l∗ ⊗ v ⊗ k) =
[ϕc∗,c
ij (l∗ ⊗ v ⊗ k)]
where
ϕc∗,cij (l∗ ⊗ v ⊗ k) = 〈ci | l〉 〈k | cj〉 v.
49
Since
−u ≤ w ≤ u,
we get that
−[ϕc∗,c
ij (u)]≤
[ϕc∗,c
ij (w)]≤
[ϕc∗,c
ij (u)]
or
(3.2.1) − [uij ] ≤ [wij ] ≤ [uij ] .
Let f ∈ (V †)+. Since positive linear functionals are completely positive, f is completely
positive, so f∞ is positive. Applying f∞ to (3.2.1), we have
− [f (uij)] ≤ [f (wij)] ≤ [f(uij)] .
Let Eiji,j∈N be a matrix unit of M∞. For i′ ∈ N \α,
Ei′i′ [f(uij)± f (wij)] Ei′i′ ∈ M∞(V )+,
so f(ui′i′)± f(wi′i′) ≥ 0. Since f (ui′i′) = 0, f(wi′i′) = 0. For j ′ ∈ N \α,
(Ei′i′ + Ej′j′) [f(uij)± f (wij)] (Ei′i′ + Ej′j′) ∈ M∞(V )+,
so [f(ui′i′)± f(wi′i′) ±f (wi′j′)
±f(wi′j′) 0
]∈ M+
2 .
We get a similar result for i′ ∈ N \α. In each case, using determinants, we get f (wi′j′) = 0.
Since V has a regular norm, so does V ′ by Theorem 2.2.9. Then (V ′)+ is generating, so
each g ∈ V ′ is a linear combination of at most four positive linear functionals. Since f ∈ V ′
is arbitrary, wij = 0 for i ∈ N \α or j ∈ N \α. Thus we may assume that α = N . We then
can rewrite w as
w =∑
i,j∈N a∗i ⊗ ‖a∗i ‖ ‖aj‖wij ⊗ aj =
∑
i,j∈N a∗i ⊗ w′ij ⊗ aj = a∗ ¯W ′ ¯ a.
Now applying[ϕa∗,a
ij
]to −u ≤ w ≤ u, we get that
− [uij] ≤[w′ij
]≤ [uij ] or − U ≤ W ′ ≤ U
50
in M∞(V ). But this space is regular since V is matrix regular, so ‖W ′‖ ≤ ‖U ‖, this
inequality quaranteeing that all previous representations of w were well-defined. Then,
since ‖a‖2 =∞∑
j=1
‖aj‖2 = 1, we have
‖w‖h ≤ ‖a‖2 ‖W ′‖ ≤ ‖U ‖ < ‖u‖h + ε.
Since ε is arbitrary, we get that ‖w‖h ≤ ‖u‖h. Thus we have the first of the two necessary
conditions for regularity.
For the second, suppose w ∈ (Hr⊗h V ⊗h Hc)sa with ‖w‖h < 1. Then there exists ε > 0
such that ‖w‖h < 1− ε. There exists an orthogonal set xi in Hc such that
‖x‖ =
∥∥∥∥∥∥
x1
x2...
∥∥∥∥∥∥=
( ∞∑
i=1
‖xi‖2)1/2
= 1
and W ∈ K∞(V )sa such that
w = x∗ ¯W x and ‖w‖h ≤ ‖W ‖ < ‖w‖h + ε < 1.
Since V is matrix regular, K∞(V ) is regular by Theorem 3.1.11. We have W self-
adjoint and ‖W ‖ < 1, so there exists U ⊂ K∞(V )+ such that ‖U ‖ < 1 and U ±W ≥ 0.
Let u = x∗ ¯ U x. Then
‖u‖h ≤ ‖U ‖ < 1 and u± w = x∗ ¯ (U ±W )¯ x ≥ 0.
Thus the second condition is also met, so Hr ⊗h V ⊗h Hc has a regular norm.
For the case where V is a matrix regular operator space, we have now established
that Hr ⊗h V ⊗h Hc is regular. But to establish its matrix regularity, we need to take
a back door approach. This is the first instance of where the promised heavy use of
Theorem 2.2.9 enters the picture. We use Theorem 2.2.9 to establish first the regularity of
(Hr⊗h V ⊗hHc)†. We then work to establish the matrix regularity of this latter space, and
then finally apply Theorem 2.2.9 in the opposite direction at each matrix level to ascertain
the matrix regularity of Hr ⊗h V ⊗h Hc.
Corollary 3.2.2. If V is a matrix regular operator space, then (Hr ⊗h V ⊗h Hc)′ =
(Hr ⊗h V ⊗h Hc)† has a regular norm.
Proof. This follows directly from Theorems 3.2.1 and 2.2.9.
51
Theorem 3.2.3. If V is a matrix ordered operator space, then
CB(V,B(H)) ∼= (Hr ⊗h V ⊗h Hc)†
is a ∗-preserving, completely isometric, complete order isomorphism.
Proof. That CB(V, B(H)) ∼= (Hr ⊗h V ⊗h Hc)† is a complete isometry follows from The-
orem 1.3.25 ((1.3.12) and (1.3.22)). We have that ϕ ∈ CB(V,B(H)) corresponds to
ϕ ∈ (Hr ⊗h V ⊗h Hc)† via
〈ϕ(v)η | ξ〉 = ϕ(ξ∗ ⊗ v ⊗ η).
Since
ϕ∗(ξ∗ ⊗ v ⊗ η) = 〈ϕ∗(v)η | ξ〉 = 〈ϕ(v∗)∗η | ξ〉
= 〈η | ϕ(v∗)ξ〉 = 〈ϕ(v∗)ξ | η〉 = ϕ(η∗ ⊗ v∗ ⊗ ξ)
= ϕ((ξ∗ ⊗ v ⊗ η)∗) = ϕ∗(ξ∗ ⊗ v ⊗ η),
we conclude that the complete isometry is ∗-preserving.
Next, suppose that ϕ = [ϕst] ∈ Mn(CB(V,B(H)))+ = CB(V,Mn(B(H)))+. In order
to show that ϕ = [ϕst] ∈ Mn((Hr ⊗h V ⊗h Hc)†)+ = CB(Hr ⊗h V ⊗h Hc,Mn)+, we need
to show that ϕm(u) ∈ M+mn for every u ∈ Mm(Hr⊗h V ⊗h Hc)
+. But since M+mn is closed,
we need only show that ϕm(u) ∈ M+mn for every element u ∈ Pm(Hr ⊗h V ⊗h Hc). For
such u,
u = c∗ ¯ V c
=
c∗11 . . . c∗p1
.... . .
...c∗1m . . . c∗pm
¯
v11 . . . v1p
.... . .
...vp1 . . . vpp
¯
c11 . . . c1m
.... . .
...cp1 . . . Cpm
=
p∑
i,j=1
cik ⊗ vij ⊗ cjl
k,l
with V ∈ Mp(V )+.
For α1, . . . , αm ∈ C n, αi =
αi1...
αin
,
⟨ϕm
p∑
i,j=1
cik ⊗ vij ⊗ cjl
k,l
α1...
αm
|
α1...
αm
⟩=
52
⟨
p∑
i,j=1
ϕ(cik ⊗ vij ⊗ cjl)
k,l
α1...
αm
|
α1...
αm
⟩=
∑
i,j,k,l
〈ϕ(cik ⊗ vij ⊗ cjl)αl | αk〉 =
∑
i,j,k,l
⟨
ϕ11(cik ⊗ vij ⊗ cjl) . . . ϕ1n(cik ⊗ vij ⊗ cjl)...
. . ....
ϕn1(cik ⊗ vij ⊗ cjl) . . . ϕnn(cik ⊗ vij ⊗ cjl)
αl1...
αln
|
αk1...
αkn
⟩=
∑
i,j,k,l,s,t
ϕst(c∗ik ⊗ vij ⊗ cjl)αltαks =
∑
i,j,k,l,s,t
〈ϕst(vij)αltcjl | αkscik〉 =
∑
i,j,k,l
⟨
ϕ11(vij) . . . ϕ1n(vij)...
. . ....
ϕn1(vij) . . . ϕnn(vij)
αl1cjl
...αlncjl
|
αk1cik...
αkncik
⟩
=
where αlcjl = [αl1cjl . . . αlncjl ]tr
∑
k,l
⟨
[ϕst(v11)] . . . [ϕst(v1p)]...
. . ....
[ϕst(vp1)] . . . [ϕst(vpp)]
αlc1l...
αlcpl
|
αkc1k...
αkcpk
⟩
=
∑
k,l
⟨
ϕ(v11) . . . ϕ(v1p)...
. . ....
ϕ(vp1) . . . ϕ(vpp)
αlc1l...
αlcpl
|
αkc1k...
αkcpk
⟩=
∑
k,l
⟨ϕp(V )
αlc1l...
αlcpl
|
αkc1k...
αkcpk
⟩
=
where bl = [αlc1l . . . αlcpl ]tr
⟨
ϕp(V ) . . . ϕp(V )...
. . ....
ϕp(V ) . . . ϕp(V )
b1...
bm
|
b1...
bm
⟩≥ 0
since ϕp(V ) and thus
ϕp(V ) . . . ϕp(V )...
. . ....
ϕp(V ) . . . ϕp(V )
are positive. This establishes that ϕ ∈
CB(Hr ⊗h V ⊗h Hc,Mn)+.
Finally, suppose ϕ = [ϕij ] ∈ Mn(CB(Hr⊗hV ⊗hHc, C ))+ = CB(Hr⊗hV ⊗hHc,Mn)+.
To show ϕ = [ϕij] ∈ Mn(CB(V, B(H)))+ = CB(V, Mn(B(H)))+, suppose V = [vkl] ∈
Mm(V )+, h1, . . . , hm ∈ Mn,1(H), hi =
hi1...
hin
, that ei are the standard basis vectors of
53
C n, written as columns, and e =
e1...
en
.
Then
⟨ϕm(V )
h1...
hm
|
h1...
hm
⟩=
∑
k,l
〈ϕ(vkl)hl | hk〉 =
∑
k,l
⟨
ϕ11(vkl) . . . ϕ1n(vkl)...
. . ....
ϕn1(vkl) . . . ϕnn(vkl)
hl1...
hln
|
hk1...
hkn
⟩=
∑
i,j,k,l
〈ϕij(vkl)hlj | hki〉 =∑
i,j,k,l
ϕij(h∗ki ⊗ vkl ⊗ hlj) =
∑
i,j,k,l
⟨
ϕ11(h∗ki ⊗ vkl ⊗ hlj) . . . ϕ1n(h∗ki ⊗ vkl ⊗ hlj)
.... . .
...ϕn1(h
∗ki ⊗ vkl ⊗ hlj) . . . ϕnn(h∗ki ⊗ vkl ⊗ hlj)
ej | ei
⟩=
∑
i,j,k,l
〈ϕ(h∗ki ⊗ vkl ⊗ hlj)ej | ei〉 =
∑
k,l
⟨
ϕ(h∗k1 ⊗ vkl ⊗ hl1) . . . ϕ(h∗k1 ⊗ vkl ⊗ hln)...
. . ....
ϕ(h∗kn ⊗ vkl ⊗ hl1) . . . ϕ(h∗kn ⊗ vkl ⊗ hln)
e1...
en
|
e1...
en
⟩=
∑
k,l
⟨ϕn
h∗k1...
h∗kn
¯ vkl ¯ [hl1 . . . hln ]
e1...
en
|
e1...
en
⟩=
∑
k,l
⟨ϕn
((h∗k)tr ¯ vkl ¯ (hl)
tr)e | e
⟩=
⟨ϕmn
(h∗1)tr ¯ v11 ¯ (h1)
tr . . . (h∗1)tr ¯ v1m ¯ (hm)tr
.... . .
...(h∗m)tr ¯ vm1 ¯ (h1)
tr . . . (h∗m)tr ¯ vmm ¯ (hm)tr
e...e
|
e...
em
⟩=
⟨ϕmn
(h∗1)tr
. . .
(h∗m)tr
¯ V
(hl)tr
. . .
(hm)tr
e...e
|
e...e
⟩≥ 0
since ϕ is completely positive. Thus ϕ ∈ CB(V, Mn(B(H)))+, and the complete order
isomorphism is established.
Theorem 3.2.4. If V is a matrix regular operator space and H is a Hilbert space, then
CB(V, B(H)) is a matrix regular operator space.
54
Proof. From Proposition 2.3.9, for all n ∈ N , CB(V,B(Hn)) is a matrix ordered operator
space. Also for all n ∈ N , ((Hn)r ⊗h V ⊗h (Hn)c)† is a regularly normed operator space
by Corollary 3.2.2. But then by Theorem 3.2.3, Mn(CB(V,B(H))) ∼= CB(V,B(Hn)) has
a regular norm. Thus CB(V, B(H)) is a matrix regular operator space.
Corollary 3.2.5. If V is a matrix regular operator space, then so is V †.
Proof. From Theorem 2.3.14, we know that V † is a matrix ordered operator space. Since
V † = CB(V, C ), the result follows from the above theorem.
The above corollary yields half of the stated goal of this chapter. We now proceed to
develop the converse.
Theorem 3.2.6. Suppose V is a matrix regular operator space and H is a Hilbert space.
Then Hr ⊗h V ⊗h Hc is a matrix regular operator space.
Proof. Since V is matrix regular, we have from Theorem 3.2.4 that CB(V,B(H)) is matrix
regular. It follows from Theorem 3.2.3 that (Hr ⊗h V ⊗h Hc)† is matrix regular. By
Corollary 3.2.5, (Hr ⊗h V ⊗h Hc)†† is matrix regular, so (Hr ⊗h V ⊗h Hc)
′′ is matrix
regular by Theorems 1.2.6 and 2.3.14. Then, for each n ∈ N , we use Theorem 2.2.9 twice
to show that since Mn((Hr ⊗h V ⊗h Hc)′′) = Mn((Hr ⊗h V ⊗h Hc)
′)′ is regular, first
Mn((Hr ⊗h V ⊗h Hc)′) = Mn(Hr ⊗h V ⊗h Hc)
′ is regular, and then Mn(Hr ⊗h V ⊗h Hc)
is regular. Thus Hr ⊗h V ⊗h Hc is matrix regular.
Theorem 3.2.7. Suppose V is a matrix ordered operator space and V † is matrix regular.
Then V is a matrix regular operator space.
Proof. Since V † is matrix regular, V †† is matrix regular by Corollary 3.2.5 and then V ′′
is matrix regular by Theorems 1.2.6 and 2.3.14. Then, as in the above theorem, for each
n ∈ N , we use Theorem 2.2.9 twice to show that since Mn(V ′′) = Mn(V ′)′ is regular, first
Mn(V ′) = Mn(V )′ is regular, and then Mn(V ) is regular. Thus V is matrix regular.
Theorem 3.2.8. Suppose V is a matrix ordered operator space and H is a Hilbert space.
If CB(V,B(H)) is matrix regular, then so is Hr ⊗h V ⊗h Hc.
Proof. From Theorem 3.2.3, since CB(V,B(H)) is matrix regular, (Hr ⊗h V ⊗h Hc)† is
matrix regular. The remainder of the proof is exactly the same as in Theorem 3.2.6.
55
Theorem 3.2.9. Suppose V is a matrix ordered operator space and H is a Hilbert space.
If CB(V,B(H)) is matrix regular, then so is V .
Proof. We prove that V † is matrix regular. The result then follows from Theorem 3.2.7.
Suppose f ∈ Mn(V †)sa = CB(V,Mn)sa such that ‖f‖cb < 1. Noting that CB(V,Mn)
matrix regular implies that CB(V,Mnm) is matrix regular for all m ∈ N , without loss of
generality we may assume that dim H ≥ n and C n → H . Then f ∈ CB(V,B(H)). Since
CB(V, B(H)) is regular, there exists g ∈ CB(V,B(H))+ with ‖g‖ < 1 and −g ≤ f ≤ g.
Let p be the projection of B(H) onto Mn. It is clear that p ∈ CB(B(H),Mn)+. Let
h = p g. Then ‖h‖cb ≤ ‖p‖cb ‖g‖cb < 1 and −p g ≤ p f ≤ p g, so −h ≤ f ≤ h.
Now suppose f, g ∈ Mn(V †)sa = CB(V,Mn)sa with −g ≤ f ≤ g. But f, g ∈
CB(V, B(H)) also, and with the same norms. Then, since CB(V, B(H)) is regular,
‖f‖cb ≤ ‖g‖cb. Thus V † is matrix regular.
56
4. Extensions
In the previous chapter, we defined the concept of matrix regularity and established the
equivalence of matrix regularity in an operator space and its operator dual. Along the way,
we were able to show that if an operator V is matrix regular, then so are Hr ⊗h V ⊗h Hc
and CB(V,B(H)). In this chapter we primarily expand on these two results. In the first
section of this chapter, we will see that we can replace Hc in the first result by any operator
space. In the second section, we will find that in the second result B(H) can be replaced
by any other injective von Neumann algebra. In that section we shall also look at two
quotient tensor products involving von Neumann algebras.
4.1. Expanding the Scope
In order to achieve the generalization that allows us to replace Hc by any operator space
X in Theorem 3.2.6, we begin by reversing the positions of the column and row Hilbert
spaces.
Lemma 4.1.1. Suppose V is a matrix regular operator space and H is a Hilbert space.
Then Hc ⊗h V ⊗h Hr is a matrix regular operator space.
Proof. We first note that since
Mn,∞(Hc) = B(C ∞,Hn) = M1,∞((Hn)c) and
M∞,n(Hr) = B(Hn, C ∞) = M∞,1((Hn)r),
we have that
Mn(Hc ⊗h V ⊗h Hr) = (Hn)c ⊗h V ⊗h (Hn)r.
Thus, to prove the matrix regularity of Hc ⊗h V ⊗h Hr , it suffices to prove regularity in
the general case since Hn is just another Hilbert space.
For this, without loss of generality, we assume that H = l2. By Propositions 1.3.10 and
1.3.12 and Theorem 1.3.25, we have the following chain of complete isometries:
Hc ⊗h V ⊗h Hr∼= Hc
∨⊗ V
∨⊗ Hr
∼= V∨⊗Hc
∨⊗ Hr
∼= V∨⊗K(H) ∼= V
∨⊗K∞ ∼= K∞(V ).
57
The basic correspondence between Hc ⊗h V ⊗h Hr and K∞(V ) is as follows:
u = [ c1 c2 . . . ]¯ [uij ]¯
d∗1d∗2...
∈ Hc ⊗h V ⊗h Hr
with ci =
ci1
ci2...
∈ M∞,1, ‖ [ c1 c2 . . . ] ‖ < ∞, [uij ] ∈ K∞(V ), d∗j = [ dj1 dj2 . . . ] ∈
M1,∞ and
∥∥∥∥∥∥∥
d∗1d∗2...
∥∥∥∥∥∥∥< ∞ corresponds to
∞∑
i,j=1
cisdjtuij
s,t
= [vst] ∈ K∞(V ).
If u ∈ Hc ⊗h V ⊗h Hr, then u, which can be written as
u = en ¯ V n ¯ en∗ = [ e1 . . . en ]¯
v11 . . . v1n...
. . ....
vn1 . . . vnn
¯
e∗1...
e∗n
where the ei are the usual basis vectors of C n, corresponds to
u =
v11 . . . v1n...
. . ....
vn1 . . . vnn
∈ Mn(V ).
This algebraic correspondence is clearly a ∗-preserving order isometry.
Since (Hc⊗h V ⊗h Hr)+ = P1(Hc⊗h V ⊗h Hr)
−‖·‖h and u ∈ K∞(V )+ if and only if u =
lims
us where each us ∈ Mns(V )+ for some ns ∈ N , this ∗-preserving order isometry extends
to the completions. Since V is matrix regular, K∞(V ) is regular by Theorem 3.1.11. Thus
we conclude that Hc ⊗h V ⊗h Hr is regular.
Lemma 4.1.2. Suppose V is a matrix regular operator space and H is a Hilbert space.
Then T (H) ⊗h V ⊗h T (H) is a matrix regular operator space.
Proof. By Theorem 1.3.25 (1.3.20), T (H) ∼= Hr ⊗h Hc is a complete isometry. Thus
T (H)⊗h V ⊗h T (H) ∼= (Hr ⊗h Hc)⊗h V ⊗h (Hr ⊗h Hc) ∼= Hr ⊗h (Hc ⊗h V ⊗h Hr)⊗h Hc
are complete order isometries. By applying Lemma 4.1.1 and Theorem 3.2.6 in succession,
we have that Hr ⊗h (Hc⊗h V ⊗h Hr)⊗h Hc is matrix regular. Then T (H)⊗h V ⊗h T (H)
is also matrix regular.
58
Lemma 4.1.3. Suppose V is a matrix regular operator space and H is a Hilbert space.
Then B(H)⊗h V ⊗h B(H) is a matrix regular operator space.
Proof. First, suppose w ∈ Mn(B(H) ⊗h V ⊗h B(H))sa with ‖w‖h < 1. Then w can be
represented as
w = c∗ ¯W c
where c ∈ M∞,n(B(H)), ‖c‖ = 1, and W ∈ K∞(V )sa, ‖W ‖ < 1. By Theorem 3.1.11,
K∞(V ) is regular, so there exists U ∈ K∞(V ) such that ‖U ‖ < 1 and −U ≤ W ≤ U . Let
u = c∗ ¯ U c. Then ‖u‖h < 1 and
u± w = c∗ ¯U c± c∗ ¯W c = c∗ ¯ (U ±W )¯ c ∈ Mn(B(H)⊗h V ⊗h B(H))+.
Now suppose −u ≤ w ≤ u in Mn(B(H)⊗h V ⊗h B(H))sa. To complete the proof that
B(H) ⊗h V ⊗h B(H) is matrix regular, we must show that ‖w‖h ≤ ‖u‖h. To do this, we
recall that T (H) = B(H)† ∼= Hr ⊗h Hc and, using Theorems 1.3.19 and 1.3.24, note that
we have complete isometries
B(H)⊗h V ⊗h B(H) → B(H)⊗h V †† ⊗h B(H)
= T (H)† ⊗h V †† ⊗h T (H)† → (T (H) ⊗h V † ⊗h T (H))†
where the inclusion
B(H)⊗h V ⊗h B(H) → (T (H) ⊗h V † ⊗h T (H))† = CB(T (H) ⊗h V † ⊗h T (H), C )
is realized by
(d ⊗ v ⊗ c)(a⊗ f ⊗ b) = d(a)v(f )c(b)
on elementary tensors where v is the image of v ∈ V in V ††. But (T (H)⊗h V † ⊗h T (H))†
is matrix regular by Lemma 4.1.2 and Corollary 3.2.5. Suppose that the inclusion is
completely order preserving. Then −u ≤ w ≤ u in (T (H) ⊗h V † ⊗h T (H))†. Since
this space is regular, we get that ‖w‖ ≤ ‖u‖ in (T (H) ⊗h V † ⊗h T (H))†, implying that
‖w‖h ≤ ‖u‖h and establishing matrix regularity. Thus it remains to show that the inclusion
B(H)⊗h V ⊗h B(H) → (T (H) ⊗h V † ⊗h T (H))† = CB(T (H) ⊗h V † ⊗h T (H), C )
59
is completely order preserving.
We begin with algebraic elements. For
u = c∗ ¯ v ¯ c =
c∗11 . . . c∗p1
.... . .
...c∗1n . . . c∗pn
¯
v11 . . . v1p
.... . .
...vp1 . . . vpp
¯
c11 . . . c1n
.... . .
...cp1 . . . cpn
=
p∑
i,j=1
c∗iγ ⊗ vij ⊗ cjδ
γ,δ
= [Cγδ] ∈ Pn(B(H)⊗h V ⊗h B(H)),
we have that u ∈ CB(T (H)⊗h V † ⊗h T (H),Mn). For
X = a∗ ¯ f ¯ a =
a∗11 . . . a∗q1
.... . .
...a∗1m . . . a∗qm
¯
f11 . . . f1q
.... . .
...fq1 . . . fqq
¯
a11 . . . a1m
.... . .
...aq1 . . . aqm
=
[q∑
s,t=1
a∗sα ⊗ fst ⊗ atβ
]
α,β
= [Aαβ ] ∈ Pm(T (H)⊗h V † ⊗h T (H)),
we get
um(X) =
u(A11) . . . u(A1m)...
. . ....
u(Am1) . . . u(Amm)
α,β
=
[Cγδ(A11)] . . . [Cγδ(A1m)]...
. . ....
[Cγδ(Am1)] . . . [Cγδ(Amm)]
α,β
=
(∑
ij
c∗iγ ⊗ vij ⊗ cjδ
)(∑
s,t
a∗sα ⊗ fst ⊗ atβ
)
(α,γ),(β,δ)
=
∑
i,j,s,t
c∗iγ(a∗sα)vij(fst)cjδ(atβ)
(α,γ),(β,δ)
=
[[∑
s,t
[ c∗iγ(a∗sα) ]γ,i [ vij(fst) ]i,j [ cjδ(atβ) ]j,δ
]]
γ,δ
=
c∗(a∗11) . . . c∗(a∗q1)...
. . ....
c∗(a∗1m) . . . c∗(a∗qm)
v(f11) . . . v(f1q)...
. . ....
v(fq1) . . . v(fqq)
c(a11) . . . c(a1m)...
. . ....
c(aq1) . . . c(aqm)
∈ M+
mn
since c∗(a∗sα) = c(asα)∗ and
v(f11) . . . v(f1q)...
. . ....
v(fq1) . . . v(fqq)
=
f11(v) . . . f1q(v)...
. . ....
fq1(v) . . . fqq(v)
= fq(v) ∈ M+
pq
60
by virtue of the positivity of f and v.
Thus u ∈ CB(T (H) ⊗h V ⊗h T (H),Mn)+ since u is completely bounded and M+mn is
closed for all m ∈ N .
For u ∈ Mn(B(H)⊗hV ⊗hB(H))+, u = lims
us where each us ∈ Pn(B(H)⊗hV ⊗hB(H)),
so, for X ∈ Mm(T (H)⊗h V † ⊗h T (H))+,
u(X) = lims
us(X) ∈ M+mn
for X ∈ (T (H) ⊗h V † ⊗h T (H))+, again since M+mn is closed.
Thus Mn(B(H)⊗h V ⊗h B(H))+ → CB(T (H)⊗h V †⊗h T (H),Mn)+ , so the inclusion
B(H)⊗h V ⊗h B(H) → CB(T (H) ⊗h V † ⊗h T (H), C ) is completely order preserving.
Thus B(H) ⊗h V ⊗h B(H) is matrix regular.
Lemma 4.1.4. Suppose V is a matrix regular operator space and A is a C∗-algebra.
Then A⊗h V ⊗h A is a matrix regular operator space.
Proof. Let π be a faithful ∗-representation of A on B(H) for some Hilbert space H . Then
A⊗h V ⊗h A → B(H)⊗h V ⊗h B(H)
is a complete order isometry.
The result then follows exactly as in the proof of Lemma 4.1.3.
Theorem 4.1.5. Suppose V is a matrix regular operator space and X is an operator
space. Then X∗ ⊗h V ⊗h X is a matrix regular operator space.
Proof. Since X is an operator space, it is completely isometrically isomorphic to a norm
closed subspace of B(H) for some Hilbert space H . Then
X∗ ⊗h V ⊗h X → B(H)⊗h V ⊗h B(H)
is a complete order isometry.
The result then follows exactly as in the proof of Lemma 4.1.3.
61
4.2. Von Neumann Algebras
We now turn our focus to von Neumann algebras as we continue to expand our class of
matrix regular operator spaces. We will show that we may replace B(H) in several of our
previous results by a von Neumann algebra R provided that, in most cases, R is injective.
We will also look at the matrix order properties of two quotient tensor products, which
we shall refer to as R′-module tensor products. We begin by recalling some well-known
properties of von Neumann algebras.
Definition 4.2.1. Let A be a C∗-algebra. We say A is injective if whenever given C∗-
algebras B and C such that B ⊆ C and a completely positive mapping ϕ : B → A, then
there exists a completely positive mapping ψ : C → A such that ψ |B = ϕ. A von Neumann
algebra is injective if it is injective as a C∗-algebra.
Proposition 4.2.2. For every Hilbert space H, B(H) is injective.
Theorem 4.2.3. Let R ⊆ B(H) be a von Neumann algebra. Then R is injective if and
only if there exists a projection E : B(H)→ R such that ‖E‖ = 1.
Proposition 4.2.4. Every finite dimensional von Neumann algebra is injective.
Theorem 4.2.5. Let R ⊆ B(H) be a von Neumann algebra. Then R is injective if and
only if its commutant R′ is injective.
Definition 4.2.6. Let R ⊆ B(H) and S ⊆ B(K) be von Neumann algebras. Then
R ⊗ S is a ∗-algebra on H ⊗ K. The von Neumann algebra tensor product R−⊗ S is
the von Neumann algebra generated by R ⊗ S in B(H ⊗ K), i.e., R−⊗ S = (R ⊗ S)′′ =
(R⊗ S)−WOT with the closure in the weak operator topology.
Theorem 4.2.7. Let R ⊆ B(H) and S ⊆ B(K) be von Neumann algebras. Then R−⊗ S
is injective if and only if R and S are injective.
Theorem 4.2.8. If V is a matrix regular operator space and R ⊆ B(H) is an injective
von Neumann algebra, the CB(V,R) is a matrix regular operator space.
Proof. Since R, as a von Neumann algebra, is a C∗-algebra, it is a matrix ordered operator
space. Then, by Theorem 2.3.9, CB(V,R) is a matrix ordered operator space.
62
Suppose ψ ∈ CB(V,R)sa with ‖ ψ ‖cb < 1. Let idR denote the identity map on R, ι the
inclusion of R into B(H), γ = ι ψ , and E : B(H) → R with ‖E‖ = 1 the projection
guaranteed by Theorem 4.2.3. Then E ∈ CB(B(H), R)+.
B(H)A A
A A C
E
V
h
h
h
h j γ
w
ψR w
idR
u
y ι
R
But γ ∈ CB(V,B(H))sa with ‖γ‖cb < 1, so, by Theorem 3.2.4, we have that there exists
α ∈ CB(V, B(H))+ with ‖α‖cb < 1 and −α ≤ γ ≤ α. Then, since E is positive, we have
−Eα ≤ Eγ ≤ Eα. Now Eγ = Eιψ = ψ and ϕ = Eα ∈ CB(V,R) with −ϕ ≤ ψ ≤ ϕ. Also,
‖ϕ‖cb = ‖Eα‖cb ≤ ‖E‖cb ‖α‖cb < 1.
Now suppose ϕ, ψ ∈ CB(V,R)sa with −ϕ ≤ ψ ≤ ϕ. Then ϕ, ψ ∈ CB(V, B(H))sa
with −ϕ ≤ ψ ≤ ϕ, ‖ϕ‖CB(V,R) = ‖ϕ‖CB(V,B(H)) and ‖ ψ ‖CB(V,R) = ‖ ψ ‖CB(V,B(H)). By
Theorem 3.2.4, ‖ ψ ‖CB(V,B(H)) ≤ ‖ϕ‖CB(V,B(H)), so ‖ ψ ‖CB(V,R) ≤ ‖ϕ‖CB(V,R). Thus ‖·‖cb
is a regular norm on CB(V,R).
Now Mn(R) = R ⊗Mn = (R ⊗Mn)′′ = R−⊗Mn is an injective von Neumann algebra
by Proposition 4.2.2 and Theorem 4.2.7 for all n ∈ N . Thus the completely bounded norm
on Mn(CB(V,R)) = CB(V,Mn(R)) is a regular norm. Thus CB(V, R) is a matrix regular
operator space.
In order to prove the converse of the above theorem, we need to begin with the following
proposition so as to be able to define two quotients.
Proposition ([16]) 4.2.9. Suppose R ⊆ B(H) is a von Neumann algebra. Then H is a
left R-module, and we may regard H as a right R-module by letting 〈ξ∗r, η〉 = 〈ξ∗, rη〉, or
equivalently, ξ∗r = (r∗ξ)∗ for r ∈ R, ξ∗ ∈ H and η ∈ H.
Notation. Suppose R ⊆ B(H) is a von Neumann algebra and V is an operator space.
Let
R(Hr ⊗h Hc) = spanξ∗r ⊗ η − ξ∗ ⊗ rη : ξ∗ ∈ Hr, η ∈ Hc, r ∈ R.
We have that R(Hr ⊗h Hc) ⊆ Hr ⊗h Hc and that R(Hr ⊗h Hc) is self-adjoint. Let
R(Hr ⊗h Hc) = R(Hr ⊗h Hc)−‖·‖h .
63
Then R(Hr ⊗h Hc) is a closed self-adjoint subspace of Hr ⊗h Hc.
Similarly, let
R(Hr ⊗h V ⊗h Hc) = spanξ∗r ⊗ v ⊗ η − ξ∗ ⊗ v ⊗ rη : ξ∗ ∈ Hr, η ∈ Hc, v ∈ V, r ∈ R,
and since R(Hr ⊗h V ⊗h Hc) ⊆ Hr ⊗h V ⊗h Hc, let
R(Hr ⊗h V ⊗h Hc) = R(Hr ⊗h V ⊗h Hc)−‖·‖h .
Then R(Hr ⊗h V ⊗h Hc) is a closed self-adjoint subspace of Hr ⊗h V ⊗h Hc.
Based on Theorem 1.3.25, we can replace any occurence of “h” in the above with “∧”
without changing the composition of the sets, and in conjunction with the commutative
property of the operator projective tensor product (Proposition 1.3.4), we also get that
R(Hr ⊗h V ⊗h Hc) ∼= R(V∧⊗ Hr
∧⊗Hc)
is a complete isometry where
R(V∧⊗ Hr
∧⊗Hc) = R(V ⊗∧ Hr ⊗∧ Hc)
−‖·‖∧
= spanv ⊗ ξ∗r ⊗ η − v ⊗ ξ∗ ⊗ rη : ξ∗ ∈ Hr, η ∈ Hc, v ∈ V, r ∈ R−‖·‖∧ .
Theorem 4.2.10. Suppose R ⊆ B(H) is a von Neumann algebra and V is an operator
space. Then
V∧⊗ (Hr
∧⊗Hc/R′(Hr
∧⊗Hc)) ∼= V
∧⊗ Hr
∧⊗Hc/R′(V
∧⊗ Hr
∧⊗Hc)
∼= Hr
∧⊗ V
∧⊗Hc/R′(Hr
∧⊗ V
∧⊗Hc)
∼= Hr ⊗h V ⊗h Hc/R′(Hr ⊗h V ⊗h Hc)
are complete isometries.
Notation. We denote Hr
∧⊗Hc/R′(Hr
∧⊗Hc) by Hr
∧⊗R′Hc and Hr
∧⊗V
∧⊗Hc/R′(Hr
∧⊗V
∧⊗Hc)
by Hr
∧⊗R′ V
∧⊗R′ Hc and refer to them as R′-module projective tensor products. We denote
the corresponding R′-module Haagerup tensor products similarly.
Proof. The second complete isometry is just an application of the commutative property of
the operator projective tensor product, and the third results directly from Theorem 1.3.25,
64
so we concentrate on the first. For notational simplicity we let J = R′(Hr
∧⊗ Hc) and
K = R′(V∧⊗ Hr
∧⊗Hc).
Define ϕ : V∧⊗ (Hr
∧⊗Hc/J) → V
∧⊗ Hr
∧⊗Hc/K by
ϕ(v ⊗ (ξ∗ ⊗ η + J)) = v ⊗ ξ∗ ⊗ η + K.
Then
(4.2.1) ϕn(α(v ⊗ (γ(ξ∗ ⊗ η)δ + Mq(J)))β) = α(v ⊗ γ(ξ∗ ⊗ η)δ)β + Mn(K)
with α ∈ Mn,pq, v ∈ Mp(V ), γ ∈ Mq,rs, ξ∗ ∈ Mr(Hr), η ∈ Ms(Hc), δ ∈ Mrs,q , and
β ∈ Mpq,n, where n, p, q, r, s may take on the value ∞.
Suppose for u ∈ V∧⊗ (Hr
∧⊗Hc/J) that
u = α1(v1 ⊗ (γ1(ξ∗1 ⊗ η1)δ1 + Mq1
(J)))β1 = α2(v2 ⊗ (γ2(ξ∗2 ⊗ η2)δ2 + Mq2
(J)))β2.
Let q = q1 + q2. Then
α1
00−α2
tr
([v1 00 v2
]⊗
[γ1(ξ
∗1 ⊗ η1)δ1 + Mq1
(J) Mq1,q2(J)
Mq2,q1(J) γ2(ξ∗2 ⊗ η2)δ2 + Mq2
(J)
])
β1
00β2
= 0
=⇒
α1
00−α2
tr
([v1 00 v2
]⊗
[γ1(ξ
∗1 ⊗ η1)δ1 0
0 γ2(ξ∗2 ⊗ η2)δ20
]+ Mq(J)
)
β1
00β2
= 0
=⇒ (by application of ϕn)
[α1 0 0 −α2 ]
([v1 00 v2
]⊗
[γ1(ξ
∗1 ⊗ η1)δ1 0
0 γ2(ξ∗2 ⊗ η2)δ20
])
β1
00β2
∈ Mn(K)
=⇒
α1(v1 ⊗ (γ1(ξ∗1 ⊗ η1)δ1))β1 + Mn(K) = α2(v2 ⊗ (γ2(ξ
∗2 ⊗ η2)δ2))β2 + Mn(K) = u′ = ϕ(u).
65
Thus ϕ is well-defined.
Next, we define ψ : V∧⊗ Hr
∧⊗Hc/K → V
∧⊗ (Hr
∧⊗Hc/J) by
ψ (v ⊗ ξ∗ ⊗ η + K) = v ⊗ (ξ∗ ⊗ η + J).
Then
(4.2.2) ψ n(α(v ⊗ γ(ξ∗ ⊗ η)δ)β + Mn(K)) = α(v ⊗ (γ(ξ∗ ⊗ η)δ + Mq(J)))β
with, again, α ∈ Mn,pq, v ∈ Mp(V ), γ ∈ Mq,rs, ξ∗ ∈ Mr(Hr), η ∈ Ms(Hc), δ ∈ Mrs,q , and
β ∈ Mpq,n, where n, p, q, r, s may take on the value ∞.
Suppose u′ ∈ K. Then u′ = limi
u′i where u′i ∈ R′(V ⊗∧ Hr ⊗∧ Hc), i.e.,
u′i =
ni∑
j=1
(vj ⊗ ξ∗j r′j ⊗ ηj − vj ⊗ ξ∗j ⊗ r′jηj).
Given ε > 0, by choosing a subsequence if necessary, we may assume that ‖u′ − u′1‖∧ < ε2
and∥∥u′k+1 − u′k
∥∥∧ < ε
2k+1 . Then ‖u′1‖∧ < ‖u′‖∧ + ε2 and
u′ = u′1 +
∞∑
k=1
(u′k+1 − u′k).
Since ‖u′1‖∧ +∞∑
k=1
∥∥u′k+1 − u′k∥∥∧ < ‖u′‖∧ + ε
2 +∞∑
k=1
ε2k+1 < ‖u′‖∧ + ε, the infinite sum
representation for u′ is well-defined. Then u′ can be represented as
u′ =
∞∑
λ=1
(vλ ⊗ ξ∗λr′λ ⊗ ηλ − vλ ⊗ ξ∗λ ⊗ r′ληλ + K),
and we get that
ψ (u′) =
∞∑
λ=1
(vλ ⊗ (ξ∗λr′λ ⊗ ηλ + J)− vλ ⊗ (ξ∗λ ⊗ r′ληλ + J)
)
=
∞∑
λ=1
(vλ ⊗ (ξ∗λr′λ ⊗ ηλ − ξ∗λ ⊗ r′ληλ + J)
)
=
∞∑
λ=1
vλ ⊗ J = 0.
Thus we have that ψ is also well-defined.
66
From Equations (4.2.1) and (4.2.2), it is clear that ϕ ψ = idV⊗∧Hr⊗∧Hc/K and ψ ϕ =
idV⊗∧(Hr⊗∧Hc/J). Thus ϕ and ψ are bijections.
Next we consider norms. Let u = α(v⊗(γ(ξ∗⊗η)δ+Mq(J)))β ∈ Mn(V∧⊗(Hr
∧⊗Hc/J)).
Let ε1, ε2, ε3 > 0 be given. Let h = γ(ξ∗ ⊗ η)δ and A = α(v ⊗ h)β. We have
u′ = ϕn(u) = A + Mn(K) and ‖u′‖ = ‖ϕn(u)‖ = ‖A + Mn(K)‖ .
There exists k ∈ Mn(K) such that
u′ = A + k + Mn(K) and ‖u′‖ ≤ ‖A + k‖ < ‖u′‖+ ε1.
But A+k ∈ V∧⊗ (Hr
∧⊗Hc), so there are α1 ∈ Mn,p1q1 , v1 ∈ Mp1(V ), h1 ∈ Mq1(Hr⊗∧Hc),
and β1 ∈ Mp1q1,n such that
A + k = α1(v1 ⊗ h1)β1,
u′ = α1(v1 ⊗ h1)β1 + Mn(K), and
‖A + k‖ ≤ ‖α1‖ ‖v1‖ ‖h1‖ ‖β1‖ < ‖A + k‖+ ε2.
There also exist γ1 ∈ Mq1,r1s1 , ξ∗1 ∈ Mr1(Hr), η1 ∈ Ms1 (Hc), and δ1 ∈ Mr1s1,q1 such
that
h1 = γ1(ξ∗1 ⊗ η1)δ1,
u′ = α1(v1 ⊗ γ1(ξ∗1 ⊗ η1)δ1)β1 + Mn(K), and
‖h1‖ ≤ ‖γ1‖ ‖ξ∗1‖‖η1‖‖δ1‖ < ‖h1‖+ ε3.
Thus u = ϕ−1n (u′) = α1(v1 ⊗ (γ1(ξ
∗1 ⊗ η1)δ1+ < Mq1(J)))β1 and
‖u‖ ≤ ‖α1‖ ‖v1‖‖γ1‖‖ξ∗1‖ ‖η1‖ ‖δ1‖ ‖β1‖ < ‖α1‖ ‖v1‖ ‖β1‖ (‖h1‖+ ε3)
= ‖α1‖ ‖v1‖‖β1‖ ‖h1‖+ ‖α1‖ ‖v1‖ ‖β1‖ ε3 < ‖A + k‖+ ε2 + ‖α1‖ ‖v1‖ ‖β1‖ ε3
< ‖u′‖+ ε1 + ε2 + ‖α1‖ ‖v1‖ ‖β1‖ ε3.
Thus ‖u‖ ≤ ‖u′‖ since ‖α1‖ , ‖v1‖ , ‖β1‖ is bounded and independent of ε3 and the εi
are arbitrary.
67
Again, let ε1, ε2, ε3 > 0 be given. Without loss of generality, we may assume
‖u‖ ≤ ‖α‖ ‖v‖ ‖h + Mq(J)‖‖β‖ < ‖u‖+ ε1.
There exists j ∈ Mq(J) such that
‖h + Mq(J)‖ ≤ ‖h + j‖ < ‖h + Mq(J)‖+ ε2 and u = α(v ⊗ (h + j + Mq(J)))β,
and there exist γ1 ∈ Mq,r1s1 , ξ∗1 ∈ Mr1(Hr), η1 ∈ Ms1(Hc), and δ1 ∈ Mr1s1,q such that
h + j = γ1(ξ∗1 ⊗ η1)δ1,
u = α(v ⊗ (γ1(ξ∗1 ⊗ η1)δ1 + Mq(J)))β, and
‖h + j‖ ≤ ‖γ1‖ ‖ξ∗1‖‖η1‖‖δ1‖ < ‖h + j‖+ ε3.
Then u′ = α(v ⊗ γ1(ξ∗1 ⊗ η1)δ1)β + Mn(K) and
‖u′‖ ≤ ‖α‖ ‖v‖‖γ1‖‖ξ∗1‖ ‖η1‖ ‖δ1‖ ‖β‖
< ‖α‖ ‖v‖‖β‖ (‖h + j‖+ ε3)
< ‖α‖ ‖v‖‖β‖ (‖h + Mq(J)‖+ ε2 + ε3)
< ‖u‖+ ε1 + ‖α‖ ‖v‖ ‖β‖ (ε2 + ε3).
Since ‖α‖ , ‖v‖ , ‖β‖ is bounded and independent of ε2 and ε3 and the εi are arbitrary,
we get ‖u′‖ ≤ ‖u‖.
Thus we conclude that ϕ is a complete isometry, which completes the proof of the
theorem.
In order to study further properties of these R′-module tensor products, we need to know
that the cone on a von Neumann algebra R is actually a dual cone. Given a von Neumann
algebra R, we define a cone on R†, the predual of R, by
R+† = R†+ ∩ R†.
The following Lemma is used to show that this is the proper definition for a cone on R†.
Lemma ([48], Lemma 1.7.2) 4.2.11. Let R be a von Neumann algebra. For every
r ∈ Rsa\R+, there exists f ∈ R+† such that f (r) < 0.
68
Proposition 4.2.12. Let R be a von Neumann algebra. Then R+ is the dual cone of
R+† .
Proof. Let P be the dual cone of R+† , i.e.,
P = r ∈ Rsa : r(f ) ≥ 0 for all f ∈ R+† .
For r ∈ R+, r(f ) = f(r) ≥ 0 for all f ∈ R+† . Thus R+ ⊆ P . By Lemma 4.2.11,
R+ = P .
Corollary 4.2.13. Let V be a matrix ordered operator space and R a von Neumann
algebra. Then
CB(V, R) ∼= (V∧⊗R†)
†
is a ∗-preserving, completely isometric, complete order isomorphism.
Proof. This follows directly from Corollary 2.3.13 and Proposition 4.2.12.
Proposition ([16]) 4.2.14. Suppose R ⊆ B(H) is a von Neumann algebra. Then
R† ∼= Hr
∧⊗R′ Hc is a complete isometry.
Theorem 4.2.15. Let V be a matrix ordered operator space and R ⊆ B(H) a von Neu-
mann algebra. Then
CB(V,R) ∼= (Hr ⊗hR′ V ⊗h
R′ Hc)†
is a ∗-preserving, completely isometric, complete order isomorphism.
Proof. Again we let J = R′(Hr ⊗h Hc) and K = R′(Hr ⊗h V ⊗h Hc). First, we have that
CB(V, R) ∼= (V∧⊗R†)
† ∼= (V∧⊗(Hr
∧⊗Hc/J))† ∼= (Hr⊗hV ⊗hHc/K)† ∼= (Hr⊗h
R′ V ⊗hR′Hc)
†
are complete isometries by Corollary 4.2.13, Proposition 4.2.14 and Theorem 4.2.10.
We have that ϕ ∈ CB(V,R) corresponds to ϕ ∈ (Hr ⊗h V ⊗h Hc/K)† via
〈ϕ(v)η | ξ〉 = ϕ(ξ∗ ⊗ v ⊗ η + K) = ϕ(ξ∗ ⊗ v ⊗ η),
69
this following from the facts that CB(V,R) ⊆ CB(V, B(H)) and
ϕ(ξ∗r′ ⊗ v ⊗ η − ξ∗ ⊗ v ⊗ r′η) = 〈ϕ(v)η | (ξ∗r′)∗〉 − 〈ϕ(v)r′η | ξ〉
=⟨ϕ(v)η | r′∗ξ
⟩− 〈ϕ(v)r′η | ξ〉
= 〈r′ϕ(v)η | ξ〉 − 〈ϕ(v)r′η | ξ〉 = 0
for r′ ∈ R′, ξ∗ ∈ Hr, η ∈ Hc and ϕ ∈ CB(V,R).
Then
ϕ∗(ξ∗ ⊗ v ⊗ η + K) = 〈ϕ∗(v)η | ξ〉 = 〈ϕ(v∗)∗η | ξ〉
= 〈η | ϕ(v∗)ξ〉 = 〈ϕ(v∗)ξ | η〉 = ϕ(η∗ ⊗ v∗ ⊗ ξ + K)
= ϕ((ξ∗ ⊗ v ⊗ η + K)∗) = ϕ∗(ξ∗ ⊗ v ⊗ η + K),
so we conclude that the complete isometry is ∗-preserving.
Next, suppose that ϕ = [ϕst] ∈ Mn(CB(V,R))+ = CB(V,Mn(R))+. In order to show
that ϕ = [ϕst] ∈ Mn(CB(Hr ⊗h V ⊗h Hc/K, C ))+ = CB(Hr ⊗h V ⊗h Hc/K,Mn)+, we
need to show that ϕm(u) ∈ M+mn for every u ∈ Mm(Hr ⊗h V ⊗h Hc/K)+.
However, we have that u ∈ Mm(Hr⊗hV ⊗hHc/K)+ if and only if u = lims
us where us ∈
π(Mm(Hr ⊗h V ⊗h Hc)
+). For each s ∈ N , us = πm(vs), vs ∈ Mm(Hr⊗h V ⊗h Hc). Then
there exists ks ∈ Mm(K) such that vs +ks ∈ Mm(Hr⊗h V ⊗h Hc)+ and πm(vs +ks) = us.
Let xs = vs + ks. Then xs = limt
xst where xst ∈ Pm(Hr ⊗h V ⊗h Hc). Then we have that
‖u− πm(xst)‖ = ‖u− us + us − πm(xst)‖
≤ ‖u− us‖+ ‖us − πm(xst)‖
= ‖u− us‖+ ‖πm(xs)− πm(xst)‖
≤ ‖u− us‖+ ‖xs − xst‖ → 0
as s, t →∞, since π is completely contractive.
Thus, since M+mn is closed, we need only show that ϕm(u) ∈ M+
mn for every element
u ∈ πm(Pm(Hr ⊗h V ⊗h Hc)). For such u,
u = c∗ ¯ V c + Mm(K)
70
=
c∗11 . . . c∗p1
.... . .
...c∗1m . . . c∗pm
¯
v11 . . . v1p
.... . .
...vp1 . . . vpp
¯
c11 . . . c1m
.... . .
...cp1 . . . Cpm
+ Mm(K)
=
p∑
i,j=1
cik ⊗ vij ⊗ cjl + K
k,l
.
For α1, . . . , αm ∈ C n, αi =
αi1...
αin
,
⟨ϕm
p∑
i,j=1
cik ⊗ vij ⊗ cjl + K
k,l
α1...
αm
|
α1...
αm
⟩=
⟨
p∑
i,j=1
ϕ(cik ⊗ vij ⊗ cjl + K)
k,l
α1...
αm
|
α1...
αm
⟩=
∑
i,j,k,l
〈ϕ(cik ⊗ vij ⊗ cjl + K)αl | αk〉 =
∑
i,j,k,l
⟨
ϕ11(cik ⊗ vij ⊗ cjl + K) . . . ϕ1n(cik ⊗ vij ⊗ cjl + K)...
. . ....
ϕn1(cik ⊗ vij ⊗ cjl + K) . . . ϕnn(cik ⊗ vij ⊗ cjl + K)
αl1...
αln
|
αk1...
αkn
⟩=
∑
i,j,k,l,s,t
ϕst(c∗ik ⊗ vij ⊗ cjl + K)αltαks =
∑
i,j,k,l,s,t
〈ϕst(vij)αltcjl | αkscik〉 =
∑
i,j,k,l
⟨
ϕ11(vij) . . . ϕ1n(vij)...
. . ....
ϕn1(vij) . . . ϕnn(vij)
αl1cjl
...αlncjl
|
αk1cik...
αkncik
⟩=
where αlcjl = [ αl1cjl . . . αlncjl ]tr
∑
k,l
⟨
[ϕst(v11)] . . . [ϕst(v1p)]...
. . ....
[ϕst(vp1)] . . . [ϕst(vpp)]
αlc1l...
αlcpl
|
αkc1k...
αkcpk
⟩=
∑
k,l
⟨
ϕ(v11) . . . ϕ(v1p)...
. . ....
ϕ(vp1) . . . ϕ(vpp)
αlc1l...
αlcpl
|
αkc1k...
αkcpk
⟩=
∑
k,l
⟨ϕp(V )
αlc1l...
αlcpl
|
αkc1k...
αkcpk
⟩=
where bl = [αlc1l . . . αlcpl ]tr
71
⟨
ϕp(V ) . . . ϕp(V )...
. . ....
ϕp(V ) . . . ϕp(V )
b1...
bm
|
b1...
bm
⟩≥ 0
since ϕp(V ) and thus
ϕp(V ) . . . ϕp(V )...
. . ....
ϕp(V ) . . . ϕp(V )
are positive. This establishes that ϕ ∈
CB(Hr ⊗h V ⊗h Hc/K, Mn)+.
Finally, let ϕ = [ϕij ] ∈ Mn(CB(Hr⊗hV ⊗hHc/K, C ))+ = CB(Hr⊗hV ⊗hHc/K,Mn)+.
To show ϕ = [ϕij] ∈ Mn(CB(V,R))+ = CB(V,Mn(R))+, suppose V = [vkl] ∈ Mm(V )+,
h1, . . . , hm ∈ Mn,1(H), hi =
hi1...
hin
, that ei are the standard basis vectors of C n,
written as columns, and e =
e1...
en
.
Then
⟨ϕm(V )
h1...
hm
|
h1...
hm
⟩=
∑
k,l
〈ϕ(vkl)hl | hk〉 =
∑
k,l
⟨
ϕ11(vkl) . . . ϕ1n(vkl)...
. . ....
ϕn1(vkl) . . . ϕnn(vkl)
hl1...
hln
|
hk1...
hkn
⟩=
∑
i,j,k,l
〈ϕij(vkl)hlj | hki〉 =∑
i,j,k,l
ϕij(h∗ki ⊗ vkl ⊗ hlj) =
∑
i,j,k,l
⟨
ϕ11(h∗ki ⊗ vkl ⊗ hlj) . . . ϕ1n(h∗ki ⊗ vkl ⊗ hlj)
.... . .
...ϕn1(h
∗ki ⊗ vkl ⊗ hlj) . . . ϕnn(h∗ki ⊗ vkl ⊗ hlj)
ej | ei
⟩=
∑
i,j,k,l
〈ϕ(h∗ki ⊗ vkl ⊗ hlj)ej | ei〉 =
∑
k,l
⟨
ϕ(h∗k1 ⊗ vkl ⊗ hl1) . . . ϕ(h∗k1 ⊗ vkl ⊗ hln)...
. . ....
ϕ(h∗kn ⊗ vkl ⊗ hl1) . . . ϕ(h∗kn ⊗ vkl ⊗ hln)
e1...
en
|
e1...
en
⟩=
∑
k,l
⟨ϕn
h∗k1...
h∗kn
¯ vkl ¯ [hl1 . . . hln ]
e1...
en
|
e1...
en
⟩=
∑
k,l
⟨ϕn
((h∗k)tr ¯ vkl ¯ (hl)
tr)e | e
⟩=
72
⟨ϕmn
(h∗1)tr ¯ v11 ¯ (h1)
tr . . . (h∗1)tr ¯ v1m ¯ (hm)tr
.... . .
...(h∗m)tr ¯ vm1 ¯ (h1)
tr . . . (h∗m)tr ¯ vmm ¯ (hm)tr
e...e
|
e...
em
⟩=
⟨ϕmn
(h∗1)tr
. . .
(h∗m)tr
¯ V
(hl)tr
. . .
(hm)tr
e...e
|
e...e
⟩≥ 0
since ϕ is completely positive. Thus ϕ ∈ CB(V,Mn(R))+, and the complete order isomor-
phism is established.
Corollary 4.2.16. Suppose V is a matrix regular operator space and R ⊆ B(H) is an
injective von Neumann algebra. Then both Hr ⊗hR′ V ⊗h
R′ Hc and (Hr ⊗hR′ V ⊗h
R′ Hc)† are
matrix regular operator spaces.
Proof. This follows directly from Theorems 3.2.7, 4.2.8, and 4.2.15.
Theorem 4.2.17. Suppose V is a matrix ordered operator space and R ⊆ B(H) is an
injective von Neumann algebra. If CB(V,R) is matrix regular, then so is V .
Proof. We prove V † is matrix regular, the result then following from Theorem 3.2.7.
Suppose f ∈ Mn(V †)sa = CB(V,Mn)sa such that ‖f‖cb < 1. Now Mn ⊆ Mn(R), so
f ∈ CB(V,Mn(R))sa = MnCB(V,R)sa. Since CB(V,R) is matrix regular, there exists
g ∈ MnCB(V,R)+ = CB(V,Mn(R))+ such that ‖g‖cb < 1 and −g ≤ f ≤ g. But
g ∈ CB(V,Mn(R))+ ⊆ CB(V,B(Hn))+. Since B(Hn) is injective and Mn ⊆ B(Hn),
there exists a projection p : B(Hn) → Mn such that ‖p‖ = 1. Let h = p g. Then
‖h‖cb < 1 and −p g ≤ p f ≤ p g, so −h ≤ f ≤ h.
Now suppose f, g ∈ Mn(V †)sa = CB(V,Mn)sa with −g ≤ f ≤ g. But f, g ∈
CB(V, Mn(R))sa with the same completely bounded norms. Since CB(V,Mn(R)) is regu-
lar, ‖f‖cb ≤ ‖g‖cb. Thus V † is matrix regular, giving us that V is also matrix regular.
73
5. Matrix Regularity of Schatten Class and Lp-spaces
In this chapter, we will use the method of complex interpolation to define “natural”
operator space structures on the Schatten class spaces Sp and the commutative Lp-spaces.
Using these operator space structures, we will show that these spaces are matrix regular.
We will also use similar techniques to show that some generalized Schatten class spaces
are matrix regular.
5.1. Generalized Schatten Class Spaces
In this section we shall study spaces Sp[E] and Sp[K;E] where E is an operator space
and K is a Hilbert space. These are related to the Schatten class spaces Sp and Sp(K),
and are spaces that have all of the natural properties of E-valued lp spaces. Our goal
is to establish that such spaces are matrix regular with naturally defined cones whenever
the space E is matrix regular. The necessary background work on the spaces Sp[E] and
Sp[K;E] follows the work of Gilles Pisier ([44, 45]).
We begin by recalling the Schatten classes Sp. These spaces can be considered as non-
commutative analogs of lp. First, S∞ = K∞, the space of compact operators on l2 with
the operator norm. Then, for 1 ≤ p < ∞,
Sp = T ∈ K∞ : tr |T |p < ∞
where |T | = (T ∗T )1/2. Each of these spaces is a Banach space with norm given by ‖T‖p =
(tr |T |p)1/p.
Similarly, if K is any Hilbert space, S∞(K) is the space of all compact operators on K
with the operator norm, and, for 1 ≤ p < ∞,
Sp(K) = T : K → K : tr |T |p < ∞.
Again, each of these spaces is a Banach space with norm given by ‖T‖p = (tr |T |p)1/p.
In order to define the spaces Sp[E] and Sp[L;E], we will need to introduce complex
interpolation for operator spaces. We begin first, however, with complex interpolation for
Banach spaces ([3, 8]).
Given complex Banach spaces E0 and E1, the couple (E0, E1) is called compatible if
there exists a Hausdorff complex topological vector space X such that E0 and E1 are
74
subspaces of X . Then we can form the spaces E0 ∩ E1 and E0 + E1. These are Banach
spaces with norms given by
‖x‖E0∩E1= max‖x‖E0
, ‖x‖E1 and ‖x‖E0+E1
= infx=x0+x1
‖x0‖E0+ ‖x1‖E1
.
To define an interpolation space (E0, E1)θ for 0 ≤ θ ≤ 1, we consider the space F of all
functions
f : 0 ≤ Re z ≤ 1 → E0 + E1
which are bounded and continuous on 0 ≤ Re z ≤ 1, analytic on 0 < Re z < 1, and,
moreover, have the property that the functions t 7→ f(j + it) (j = 0, 1) from R to Ej are
continuous and tend to 0 as |t| → ∞. F is a Banach space with norm
‖f‖F = maxsup ‖f(it)‖E0, sup ‖f (1 + it)‖E1
.
The interpolation space Eθ = (E0, E1)θ, 0 ≤ θ ≤ 1, consists of all x ∈ E0 + E1 such
that x = f(θ) for some f ∈ F . With norm given by
‖x‖θ = inf‖f‖F : f (θ) = x, f ∈ F ,
Eθ is a Banach space.
In stating that Eθ is an interpolation space, we mean that Eθ is an intermediate space
between E0 and E1, i.e., E0 ∩ E1 ⊆ Eθ ⊆ E0 + E1, and that for any bounded linear map
T : E0 + E1 → E0 + E1 with T (E0) ⊆ E0 and T (E1) ⊆ E1, we have that T (Eθ) ⊆ Eθ.
Advancing to the case where E0 and E1 are operator spaces, if E0 ⊆ X and E1 ⊆ X ,
we then have Mn(E0) ⊆ Mn(X ) and Mn(E1) ⊆ Mn(X ). Thus (Mn(E0),Mn(E1))θ is
well-defined. We then define operator space norms on Eθ = (E0, E1)θ by
Mn(Eθ) = (Mn(E0),Mn(E1))θ,
i.e., for [xij ] ∈ Mn(Eθ),
‖ [xij ]‖Mn(Eθ) = ‖ [xij ]‖(Mn(E0),Mn(E1))θ.
We define E0 ⊕∞ E1 to be the direct sum of E0 and E1 with norm
‖(x0, x1)‖ = max‖x0‖E0, ‖x1‖E1
75
and E0 ⊕1 E1 as the direct sum with norm
‖(x0, x1)‖ = ‖x0‖E0+ ‖x1‖E1
.
Setting Mn(E0⊕∞E1) = Mn(E0)⊕∞ Mn(E1) provides E0⊕∞E1 with an operator space
structure. We can represent Mn(E0)⊕∞ Mn(E1) as matrices of the form
[x0 00 x1
]with
∥∥∥∥[
x0 00 x1
] ∥∥∥∥ = max‖x0‖Mn(E0), ‖x1‖Mn(E1).
Then Mn(E0∩E1) = Mn(E0)∩Mn(E1) can be represented by matrices of the form
[x 00 x
]
with ∥∥∥∥[
x 00 x
] ∥∥∥∥ = max‖x‖Mn(E0) , ‖x‖Mn(E1).
Thus E0 ∩E1 is an operator space as a subspace of an operator space.
Next, we observe that (E0 ⊕1 E1)′ = E′0 ⊕∞ E′1 (which is similar to (l12)
′ = l∞2 ),
so it is natural to give Mn(E0 ⊕1 E1) norms from CB(E†0 ⊕∞ E†1,Mn). This provides
E0 ⊕1 E1 with an operator space structure. Let ∆n = (x,−x) ⊆ E0 ⊕1 E1. Then
E0+E1 = (E0⊕1E1)/∆n is an operator space as a quotient. Note that for a+b ∈ E0+E1,
a + b = (a + x) + (b− x) where x ∈ E0 ∩E1. We then have that
‖x‖Mn(E0+E1) = inf‖(x0, x1)‖Mn(E0⊕1E1) : x = x0 + x1.
Thus we have the following.
Proposition ([45]) 5.1.1. Let E0 and E1 be operator spaces. Then the spaces E0 ∩E1,
E0 + E1 and Eθ = (E0, E1)θ, 0 ≤ θ ≤ 1, equipped with the above matrix structures are
operator spaces.
We now proceed to define the spaces Sp[E] and Sp[K;E] for 0 ≤ p ≤ 1. Noting that S∞
and S∞(K) are operator spaces, we begin by defining operator spaces S∞[E] and S∞[K; E]
by
S∞[E] = S∞∨⊗ E and S∞[K; E] = S∞(K)
∨⊗ E.
Building on concepts mentioned in Section 1.2, we recall that S1, the trace class oper-
ators on l2, is the dual of S∞, and with the appropriate matrix structure can be viewed
76
as the operator dual. We have a similar relationship between S1(K) and S∞(K). Then,
given an operator space E, we can define operator spaces S1[E] and S1[K; E] by
S1[E] = S1
∧⊗ E and S1[K; E] = S1(K)
∧⊗ E.
In line with Definition 1.2.1, we consider S1⊗E as a vector linear subspace of M∞(E).
For u ∈ S1 ⊗ E, we let [uij ] ∈ M∞(E) be the associated element to u. Then
‖u‖ = inf‖a‖2 ‖v‖S∞[E] ‖b‖2
where the infimum is taken over all representations of the form u = avb with a, b ∈ S2 and
v ∈ S∞[E].
We now concentrate on the case where the Hilbert space K is l2. For simplicity of
notation, we let C = (l2)c and R = (l2)r. By Theorem 1.3.25 ((1.3.20) and (1.3.21)) along
with the commutativity of “∧,”
S1
∧⊗ E ∼= R⊗h E ⊗h C
is a complete isometry. By equation (1.3.22) of the same theorem we get that
M∞(E†) ∼= (R⊗h E ⊗h C)†
is also a complete isometry. For this latter result, we note M∞CB(E, C ) = CB(E,M∞).
Therefore,
S1[E] ∼= R⊗h E ⊗h C and M∞(E†) ∼= (S1[E])†
are complete isometries. Also, from equations (1.3.20), (1.3.16) and (1.3.17) of Theo-
rem 1.3.25 and the commutativity of “∨,”
S∞∨⊗ E ∼= C
∨⊗R
∨⊗ E ∼= C
∨⊗ E
∨⊗R ∼= C ⊗h E ⊗h R
are complete isometries, so
S∞[E] ∼= C ⊗h E ⊗h R
is also a complete isometry.
77
Since we have completely contractive injections
S1[E] = S1
∧⊗E → S∞[E] = S∞
∨⊗ E → M∞(E),
(S∞[E], S1[E]) is a compatible couple of operator spaces. Then, for 1 < p < ∞, we define
Sp[E] = (S∞[E], S1[E])θ
where θ = 1/p. This defines an operator space structure on Sp[E].
We can find another realization of Sp[E] by considering the couple (R,C). We identify,
for purposes of interpolation, x ∈ R and y ∈ C if they correspond to the same element of
l2. This then gives meaning to R ∩C, C ∩R, R + C and C + R. For 0 < θ < 1, we define
R(θ) = (R,C)θ = (C, R)1−θ.
This also gives us
R(1− θ) = (R,C)1−θ = (C, R)θ.
For completeness, we define R(0) = R and R(1) = C.
Theorem ([44]) 5.1.2. Let E be an operator space, let 1 < p < ∞, and let θ = 1/p.
Then
Sp[E] ∼= R(1− θ)⊗h E ⊗h R(θ)
is a completely isometric isomorphism.
With this theorem in hand, to show that Sp[E] is matrix regular, it remains to show
that R(θ)∗ = R(1− θ). As a first step, we need the following lemma.
Lemma 5.1.3. Let H be a Hilbert space. Then (Hr + Hc)∗ = Hc + Hr.
Proof. We recall that (Hr)∗ = Hc and (Hc)
∗ = Hr. Then, for x = r + c ∈ Mn(Hr) +
Mn(Hc), x∗ = r∗ + c∗ ∈ Mn(Hc) + Mn(Hr). Since
‖x‖Mn(Hr)+Mn(Hc)= inf‖(r, c)‖Mn(Hr)⊕1Mn(Hc)
: x = r + c
= inf‖r‖Mn(Hr) + ‖c‖Mn(Hc): x = r + c
= inf‖r∗‖Mn(Hc)+ ‖c∗‖Mn(Hr) : x∗ = r∗ + c∗
= inf‖(r∗, c∗)‖Mn(Hc)⊕1Mn(Hr) : x∗ = r∗ + c∗
= inf‖(r′, c′)‖Mn(Hc)⊕1Mn(Hr) : x∗ = r′ + c′ = ‖x∗‖Mn(Hc)+Mn(Hr) ,
we have our result.
78
Lemma 5.1.4. Given 0 ≤ θ ≤ 1, (R(θ))∗ = R(1− θ).
Proof. Suppose
x ∈ R(θ) = (R,C)θ = (C,R)1−θ.
Suppose also that x = f(θ) where f : 0 ≤ Re z ≤ 1 → R+C is in F (R,C)θand 0 ≤ θ ≤ 1.
Since f is analytic on 0 < Re z < 1, for all w ∈ 0 < Re z < 1,
f ′(w) = limz→w
f(z) − f (w)
z −w.
Define f# : 0 ≤ Re z ≤ 1 → C + R by f#(z) = f(1− z)∗. For w ∈ 0 < Re z < 1,
(f#)′(w) = limz→w
f#(z)− f#(w)
z − w= lim
z→w
f (1− z)∗ − f (1− w)∗
z − w
= limz→w
[f(1− z) − f (1− w)
z − w
]∗= lim
z→w
(−
[f (1− z) − f (1− w)
(1− z) − (1 − w)
]∗)
= −f ′(1 − w)∗.
Thus f# is analytic on 0 ≤ Re z ≤ 1. Since
‖f(it)‖ = ‖f(1 − (1 − it))‖ =∥∥f#(1 + it)
∥∥ ,
‖f (1 + it)‖ = ‖f (1− (it))‖ =∥∥f#(it)
∥∥ ,
and f# is clearly continuous on 0 ≤ Re z ≤ 1, f# is in the set F (C,R)θ.
We also have that
(5.1.1)
‖f‖F (R,C)θ= maxsup ‖f (it)‖R , sup ‖f (1 + it)‖C
= maxsup∥∥f#(it)
∥∥C
, sup∥∥f#(1 + it)
∥∥R
=∥∥f#
∥∥F (C,R)θ
.
Since 0 ≤ θ ≤ 1, if x = f (θ) = g(θ) for f, g ∈ F (R,C)θ, then f#(1 − θ) = g#(1 − θ)
for f#, g# ∈ F (C,R)θ= F (R,C)1−θ
. Thus we can define an element x∗ ∈ R(1 − θ) by
x∗ = f#(1 − θ). Since x = f (θ) ∈ R + C, we get that x∗ ∈ C + R.
Suppose x = f(θ), y = g(θ), and α ∈ C . Then x+y = f (θ)+g(θ) = (f +g)(θ) and αx =
αf(θ) = (αf )(θ). This gives us x∗ = f#(1− θ), y∗ = g#(1− θ), (x+ y)∗ = (f + g)#(1− θ)
and (αx)∗ = (αf )#(1− θ). Thus
(x+y)∗ = (f +g)#(1−θ) = [(f + g)(θ)]∗
= f (θ)∗+g(θ)
∗= f#(1−θ)+g#(1−θ) = x∗+y∗
79
and
(αx)∗ = (αf )#(1− θ) = (αf)(θ)∗
= (αf(θ))∗ = αf (θ)∗
= αf#(1− θ) = αx∗.
We conclude that the operation “∗” is conjugate-linear.
Next, suppose that x = [xij] ∈ Mn(R,C)θ = (Mn(R),Mn(C))θ. Then x∗ =[x∗ji
]∈
Mn(C,R)θ = (Mn(C),Mn(R))θ. If x = [xij ] = f (θ) where f ∈ F (Mn(R),Mn(C))θ, the use
of the same argument as above gives us that f#(1− θ) = f (θ)∗ = x∗.
Finally, from (5.1.1) and the definition of interpolation norms, we also have that
‖x‖(Mn(R),Mn(C))θ= inf‖f‖F (Mn(R),Mn(C))θ
: f (θ) = x, f ∈ F (Mn(R),Mn(C))θ
= inf∥∥f#
∥∥F (Mn(C),Mn(R))θ
: f#(1− θ) = x∗, f# ∈ F (Mn(C),Mn(R))1−θ
= inf∥∥f#
∥∥F (Mn(C),Mn(R))θ
: f (θ) = x∗, f# ∈ F (Mn(C),Mn(R))θ
= ‖x∗‖(Mn(C),Mn(R))θ.
We have then established the sought result.
Since we then have for 0 ≤ p ≤ 1 and θ = 1/p that
Sp[E] ∼= R(θ)∗ ⊗h E ⊗h R(θ),
we have that Sp[E] is an involutive operator space if E is. Further, if E is a matrix ordered
operator space, then there are naturally defined cones given by
Mn(Sp[E])+ = Mn(R(θ)∗ ⊗h E ⊗h R(θ))+ = Pn(R(θ)∗ ⊗h E ⊗h R(θ))−‖·‖h .
Although we have limited this exposition to the case where the Hilbert space is l2, we
get similar results for any Hilbert space K. We then have reached our goal.
Theorem 5.1.5. Suppose that E is a matrix regular operator space and K is a Hilbert
space. Then, for 1 ≤ p ≤ ∞, Sp[E] and Sp[K;E] are matrix regular operator spaces.
Proof. This follows directly from Theorem 4.1.5 and Lemma 5.1.4.
80
5.2. Commutative Lp-spaces
We now turn to the commutative Lp-spaces, 1 ≤ p ≤ ∞. Focusing on the natural
operator space structures described by Pisier ([44]), we show that all such spaces are
matrix regular.
For this work, we assume that (Ω,Σ, µ) is a measure space. If X is a complex Banach
space, we denote by Lp(µ; X) the Lp-space of X-valued Bochner-integrable functions. We
use Lp(µ) to denote the Lp-space of complex valued functions.
When X is an involutive Banach space, given a function f : Ω → X , we define the
function f∗ : Ω → X by f∗(x) = f (x)∗. This provides the space of functions from Ω into X
with an involution. We say that a function f : Ω → X is self-adjoint if f∗(x) = f (x) a.e.(µ).
Further, if X is partially ordered, we say that the function f is positive if f (x) ∈ X+ a.e.(µ).
We recall the definition of a measurable function:
Definition 5.2.1. Suppose (Ω,Σ, µ) is a measure space and X is a Banach space. A
function f : Ω → X is called simple if there exist x1, . . . , xn ∈ X and E1, . . . , En ∈ Σ such
that f =n∑
i=1
xiχEi where χEi is the characteristic function of the set Ei.
A function f : Ω → X is called µ-measurable (or just measurable if µ is clear) if there
exists a sequence of simple functions (fn) with limn‖fn(x)− f (x)‖ = 0 a.e.(µ).
Proposition 5.2.2. Suppose (Ω, Σ, µ) is a measure space and X is an involutive Banach
space. Then f : Ω → X is µ-measurable if and only if f∗ : Ω → X is µ-measurable.
Proof. If f is µ-measurable, then there exists a sequence of simple functions (fn) with
limn‖fn(x)− f(x)‖ = 0 a.e.(µ). Clearly, each f∗n is also simple. Since X is an involutive
Banach space,
limn‖f∗n(x)− f∗(x)‖ = lim
n‖fn(x)∗ − f (x)∗‖ = lim
n‖fn(x)− f(x)‖
for all x ∈ Ω. Then limn‖f∗n(x)− f∗(x)‖ = 0 a.e.(µ), so f∗ is µ-measurable. The reverse
implication is similar.
We now define the natural operator space structures for Lp(µ), 1 ≤ p ≤ ∞. For p = ∞,
L∞(µ) is a C∗-algebra and thus possesses the usual C∗-algebra operator space structure.
For p = 1, L1(µ) possesses a natural operator space structure induced by the operator dual
81
space L∞(µ)†. This gives an operator space structure on L1(µ) such that L1(µ)† ∼= L∞(µ)
is a complete isometry ([4, 5, 6, 25]).
For 1 < p < ∞, we use complex interpolation as described in Section 5.1 to define the
natural operator space structures. Thus the norm for Mn(Lp(µ)) is the norm of the space
(Mn(L∞(µ)), Mn(L1(µ)))θ where θ = 1p.
Suppose f = [fij ] ∈ Mn(Lp(µ)). Then we view f as f : Ω → Mn with f(x) = [fij(x)].
It follows directly from the definition that f = [fij ] is measurable in Mn(Lp(µ)) if and
only if each of the component functions fij is measurable in Lp(µ). We define f∗ =[f∗ji
],
i.e., f∗(x) = [fij(x)]∗. We say that f ∈ Mn(Lp(µ)) is self-adjoint if f (x) = f∗(x) a.e.(µ)
and that f is positive if f (x) ∈ M+n a.e.(µ).
We will have need for the following result which intertwines complex interpolation with
the Haagerup tensor product.
Theorem ([45]) 5.2.3. Let (E0, E1) and (F0, F1) be two compatible couples of operator
spaces. Then the couple
(E0 ⊗h F0, E1 ⊗h F1)
is a compatible couple of operator spaces and we have a complete isometry
(E0 ⊗h F0, E1 ⊗h F1)θ∼= Eθ ⊗h Fθ
where
Eθ = (E0, E1)θ and Fθ = (F0, F1)θ.
Picking up on our presentation of Pisier’s work ([44]) in Section 5.1, but restricting
ourselves to the finite dimensional case, we let Cn = (C n)c and Rn = (C n)r. For eij a
matrix unit of Mn, the map
J : Cn ⊗h E ⊗h Rn → Sn
∨⊗E ⊆ Mn(E)
J(ei1 ⊗ x⊗ e1j) = eij ⊗ x
is a completely isometric embedding. This allows us to identify Cn ⊗h E ⊗h Rn with the
image of J . We also have a completely isometric map
k : Rn ⊗h E ⊗h Cn → Mn(E)
k(e1i ⊗ x⊗ ej1) = eij ⊗ x.
82
These two maps are compatible with the “identification” of ei1 and e1i (resp. e1j and ej1)
needed to define the interpolation space (Rn, Cn)θ.
Viewing Cn and Rn as subsets of Mn where all entries except those in the first column
and first row, respectively, are 0, let ξ1, . . . , ξn be the orthonormal basis of Rn(1−θ) corre-
sponding to ei1 and e1i, and let η1, . . . , ηn be the orthonormal basis of Rn(θ) corresponding
to e1j and ej1. Then the mapping
Jp : Rn(1− θ)⊗h E ⊗h Rn(θ) → Mn(E)
Jp(ξi ⊗ x⊗ ηj) = eij ⊗ x
is the “natural” inclusion mapping used in Theorem 5.1.2 to identify Rn(1−θ)⊗hE⊗hRn(θ)
with the subset Snp [E] of Mn(E). But since R(1 − θ) = R(θ)∗ by Lemma 5.1.4, we also
have ξi = η∗i for i = 1, . . . , n.
The following theorem of Pisier allows us to “compute” the norm in the space Snp [E].
Theorem ([44]) 5.2.4. Let E be an operator space and 1 ≤ p < ∞. Let u ∈ Sp[E] (resp.
u ∈ Snp [E]) and let [uij ] ∈ M∞(E) (resp. [uij ] ∈ Mn(E)) be the corresponding matrix with
uij ∈ E. Then ‖u‖Sp[E] (resp. ‖u‖Snp [E]) is equal to
inf‖a‖S2p‖v‖M∞(E) ‖b‖S2p
where the infimum runs over all representations of the form
[uij] = avb
with a, b ∈ S2p and v ∈ S∞[E] ⊆ M∞(E) (resp. with a, b ∈ Sn2p and v ∈ Mn(E)).
For the case where E is an involutive operator space and u is self-adjoint, we can build
on the previous theorem to get the following:
Theorem 5.2.5. Let E be an involutive operator space and 1 ≤ p < ∞. Let u ∈ Snp [E]sa
and let [uij] ∈ Mn(E)sa be the corresponding matrix with uij ∈ E. Then
‖u‖Snp [E] = inf‖c‖2Sn
2p‖v‖Mn(E)
where the infimum runs over all representations of the form
[uij ] = c∗vc
83
with c ∈ Sn2p and v ∈ Mn(E)sa.
Proof. From Theorem 5.1.2, Snp [E] ∼= Rn(1 − θ) ⊗h E ⊗h Rn(θ). Using this with the
definition of the Haagerup tensor product and a technique of Itoh’s ([33]),
‖u‖Snp [E] = inf
‖ [ a1 . . . an ] ‖ ‖ [wij ] ‖Mn(E)
∥∥∥∥∥∥
b1...
bn
∥∥∥∥∥∥
with [ a1 . . . an ] ∈ M1,n(Rn(1− θ)) and
b1...
bn
∈ Mn,1(Rn(θ)), where the infimum runs
over all representations of u of the form
u =
n∑
i,j=1
ai ⊗ wij ⊗ bj .
From the definition, the infimum would run over all such representations where the indices
i and j would run from 1 to an arbitrary p ∈ N . Suppose u =∑p
i,j=1 αi ⊗ zij ⊗ βj . We
may assume that the subset β1, . . . , βkk≤p is a maximal linearly independent subset of
β1, . . . , βp. Then there exists L1 ∈ Mp,k such that β =
β1...
βp
= L1
β1...
βk
. Let U1 |L1|
be the polar decomposition of L1. Then U1 ∈ Mp,k is a partial isometry and |L1| ∈ Mk
with ‖ |L1| ‖ = ‖L1‖. Then b =
b1...bk
= |L1|
β1...
βk
and ‖b‖ = ‖β‖. Working similary
with the row [α1 . . . αp ], we end up with u = [ a1 . . . αk ] ¯ U∗2 zU1 ¯
b1...bk
and
‖a‖ ‖U∗2 zU1‖ ‖b‖ ≤ ‖α‖ ‖z‖ ‖b‖. Since Rn(θ) and Rn(1 − θ) are n-dimensional spaces,
k ≤ n, and if k < n, we just fill in with 0’s.
Then given ε > 0, there exist a = [ a1 . . . an ] ∈ M1,n(Rn(1 − θ)), [wij ] ∈ Mn(E),
and b =
b1...
bn
∈ Mn,1(Rn(θ)) such that u = a¯ [wij ]¯ b and
‖u‖Snp [E] ≤ ‖ [ a1 . . . an ] ‖‖ [wij ] ‖Mn(E)
∥∥∥∥∥∥
b1...
bn
∥∥∥∥∥∥< ‖u‖Sn
p [E] + ε.
84
Then, for all λ > 0,
u =1
2(u + u∗)
=1
2(λa¯ [wij ]¯ λ−1b + λ−1b∗ ¯ [wij ]
∗ ¯ λa∗)
=1√2
[λa λ−1b∗ ]¯[
0 [wij ][wij]
∗0
]¯
1√2
[λa∗
λ−1b
].
We get a representation of the form d∗ ¯ y ¯ d with y =
[0 [wij]
[wij ]∗
0
]∈ M2n(E)sa and
d = 1√2
[λa∗
λ−1b
]∈ M2n,1(Rn(θ)).
For the norm of u, since
∥∥∥∥[
0 [wij][wij]
∗0
] ∥∥∥∥ = ‖ [wij] ‖ ,
∥∥∥∥1√2
[λa∗
λ−1b
] ∥∥∥∥2
≤ 1
2(λ2 ‖a‖2 + λ−2 ‖b‖2), and
minλ>0
1
2(λ2 ‖a‖2 + λ−2 ‖b‖2) = ‖a‖‖b‖ ,
there exists λ0 > 0 such that
‖u‖Snp [E] ≤
∥∥∥∥1√2
[λa∗
λ−1b
] ∥∥∥∥2 ∥∥∥∥
[0 [wij ]
[wij ]∗
0
] ∥∥∥∥ ≤ ‖a‖ ‖ [wij ] ‖ ‖b‖ < ‖u‖Snp [E] + ε,
the leftmost inequality coming from the definition of the Haagerup tensor norm. This then
establishes that
‖u‖Snp [E] = inf‖d‖2 ‖y‖,
or to change notation,
‖u‖Snp [E] = inf
∥∥∥2n∑
i=1
d∗i ⊗ e1i
∥∥∥Rn(1−θ)
∨⊗R2n
‖y‖M2n(E)
∥∥∥2n∑
j=1
dj ⊗ ej1
∥∥∥Rn(θ)
∨⊗C2n
where the infimum runs over all representations of u of the form
u =2n∑
i,j=1
d∗i ⊗ yij ⊗ dj
with y ∈ (M2n)sa.
85
Using the facts that
R∨⊗R ∼= C
∨⊗ C ∼= S2
C∨⊗R ∼= R
∨⊗ C ∼= S∞
(Mm,n, Sm,n2 )2/p
∼= Sm,np
and adjusting the first two to finite dimensional cases, we note that
Rn(1− θ)∨⊗Rm
∼= (Rn, Cn)1−θ ⊗h (Rm, Rm)1−θ
∼= (Rn ⊗h Rm, Cn ⊗h Rm)1−θ∼= (Cn
∨⊗Rm, Rn
∨⊗Rm)θ
∼= (Mn,m, Sn,m2 )θ
∼= Sn,m2p
and
Rn(θ)∨⊗ Cm
∼= Cm
∨⊗Rn(θ) ∼= (Cm, Cm)θ ⊗h (Rn, Cn)θ
∼= (Cm ⊗h Rn, Cm ⊗h Cn)θ∼= (Cm
∨⊗Rn, Cm
∨⊗ Cn)θ
∼= (Mm,n, Sm,n2 )θ
∼= Sm,n2p
are isometries when θ = 1p .
This implies that∥∥∥
2n∑i=1
d∗i ⊗ e1i
∥∥∥Rn(1−θ)
∨⊗R2n
= ‖D∗‖Sn,2n2p
where D∗ is the matrix with
d∗1, . . . , d∗2n as its columns and∥∥∥
2n∑j=1
dj ⊗ ej1
∥∥∥Rn(θ)
∨⊗C2n
= ‖D‖S2n,n2p
where D is the matrix
with d1, . . . , d2n as its rows. Recalling the correspondence between u and Jp(u), as ex-
plained before Theorem 5.2.4, and the fact that ξi = η∗i for i = 1, . . . , n, let d∗i =∑
di(k)η∗k
and dj =∑
dj(l)ηl. Then D = [dj(l)] ∈ S2n,n2p , so D has a polar decomposition D = s |D|
where s ∈ M2n,n is a partial isometry and |D| ∈ Sn2p with ‖ |D| ‖Sn
2p= ‖D‖S2n,n
2p.
Let c = |D| and label its rows c1, . . . , cn. Let v = s∗ys and note that ‖v‖Mn(E) ≤
‖y‖M2n(E). Then, as above,
c∗i =∑
ci(k)η∗k, cj =∑
cj(l)ηl,
∥∥∥n∑
i=1
c∗i ⊗ e1i
∥∥∥Rn(1−θ)
∨⊗Rn
= ‖c∗‖Sn2p
,
∥∥∥n∑
j=1
cj ⊗ ej1
∥∥∥Rn(θ)
∨⊗Cn
= ‖c‖Sn2p
,
86
and
‖u‖Snp [E] = inf‖c‖2Sn
2p‖v‖Mn(E)
= inf∥∥∥
n∑
i=1
c∗i ⊗ e1i
∥∥∥Rn(1−θ)
∨⊗Rn
‖v‖Mn(E)
∥∥∥n∑
j=1
cj ⊗ ej1
∥∥∥Rn(θ)
∨⊗Cn
where the infimum runs over all representations of u of the form
u =n∑
i,j=1
c∗i ⊗ vij ⊗ cj
with v ∈ Mn(E)sa. Then we have
[uij ] = Jp(∑
i,j
c∗i ⊗ vij ⊗ cj)
=∑
i,j,k,l
c∗i (k)cj(l)ekl ⊗ vij =∑
i,j,k,l
c∗i (k)cj(l)ekieijejl ⊗ vij
=(∑
i,k
c∗i (k)eki ⊗ idE
)(∑
i,j
eij ⊗ vij
)(∑
j,l
ejlcj(l)⊗ idE
).
Hence, since c∗ =∑i,k
c∗i (k)eki and c =∑j,l
ejlcj(l), we obtain our result.
Lemma ([44]) 5.2.6. Let E be an operator space and 1 ≤ p ≤ ∞. If a ∈ S2q, b ∈ S2q
and if 1t = 1
p + 1q ≤ 1, then
‖axb‖St[E] ≤ ‖a‖S2q‖x‖Sp[E] ‖b‖S2q
.
We will need the following useful fact.
Lemma ([44]) 5.2.7. Let F be an operator space, n ≥ 1, and let [yij ] ∈ Mn(F ). Then
for all 1 ≤ p ≤ ∞,
‖ [yij] ‖Mn(F ) = sup‖a [yij] b‖Snp [F ] : ‖a‖Sn
2p, ‖b‖Sn
2p≤ 1.
Again, we build on the above lemma to prove a corollary for the special case where F
is an involutive operator space and [yij ] is self-adjoint.
87
Corollary 5.2.8. Let F be an involutive operator space, n ≥ 1, and let [yij] ∈ Mn(F )sa.
Then, for 1 ≤ p ≤ ∞,
‖ [yij ] ‖Mn(F ) = sup‖c [yij ] c∗‖Sn
p [F ] : ‖c‖Sn2p≤ 1.
Proof. Since Mn(F )′ = Sn1 [F ′], we have that
‖ [yij ] ‖Mn(F ) = sup〈y, ξ〉 : ‖ξ‖Sn1 [F ′] ≤ 1.
But since y is self-adjoint, we may further restrict ourselves to ξ ∈ Sn1 [F ′]sa. Hence by
Theorem 5.2.5,
‖ [yij] ‖Mn(F ) = sup〈y, c∗zc〉 : ‖c‖Sn2≤ 1, ‖z‖Mn(F ′) ≤ 1.
This yields (note 〈y, z〉 =∑ij
〈yij, zji〉, hence 〈y, azb〉 = 〈bya, z〉)
‖ [yij] ‖Mn(F ) = sup‖cyc∗‖Sn1 [F ] : ‖c‖Sn
2≤ 1.
This proves the theorem for p = 1. The general case follows from Lemma 5.2.6 and the
fact that any c ∈ Sn2 with ‖c‖Sn
2≤ 1 can be written as c = c′c′′ such that c′ ∈ Sn
2p′ with
‖c′‖Sn2p′≤ 1 and c′′ ∈ Sn
2p with ‖c′′‖Sn2p≤ 1 where p and p′ are conjugate exponents. Then,
from the last identity, we have
‖ [yij] ‖Mn(F ) ≤ sup∥∥c′′yc′′
∗∥∥Sn
p [F ]: ‖c′′‖Sn
2p≤ 1
since
‖cyc∗‖Sn1 [F ] =
∥∥c′c′′yc′′∗c′∗∥∥
Sn1 [F ]
≤ ‖c′‖Sn2p′
∥∥c′′yc′′∗∥∥
Snp [F ]
‖c′‖∗Sn2p′≤
∥∥c′′yc′′∗∥∥
Snp [F ]
.
The converse inequality follows from Lemma 5.2.6 since
∥∥c′′yc′′∗∥∥
Snp [F ]
≤ ‖c′′‖Sn2p‖y‖∞
∥∥c′′∗∥∥
Sn2p≤ ‖y‖∞ .
We can now state our main tools for proving that the commutative Lp-spaces are matrix
regular.
88
Theorem ([44]) 5.2.9. Let E be an operator space, (Ω,Σ, µ) a measure space, and
1 ≤ p < ∞.
(1) Let a = [aij ] ∈ Mn ⊗ Lp(µ;E). Then
‖ [aij] ‖Mn(Lp(µ;E)) = sup‖αaβ‖Snp [Lp(µ;E)] : ‖α‖Sn
2p≤ 1, ‖β‖Sn
2p≤ 1.
(2) The spaces Lp(µ; Sp) and Sp[Lp(µ)] are completely isometric. More generally, the
spaces Lp(µ;Sp[E]) and Sp[Lp(µ; E)] are completely isometric.
Corollary 5.2.10. Let E be an involutive operator space, (Ω,Σ, µ) a measure space, and
1 ≤ p < ∞. Let a = [aij ] ∈ (Mn ⊗ Lp(µ;E))sa. Then
‖ [aij ] ‖Mn(Lp(µ;E)) = sup‖α∗aα‖Snp [Lp(µ;E)] : ‖α‖Sn
2p≤ 1.
Proof. This is an immediate application of Corollary 5.2.8.
We now proceed to the main results of this section.
Proposition 5.2.11. Let (Ω, Σ, µ) be a measure space. Then Lp(µ) is a matrix ordered
operator space.
Proof. Since Snp is a matrix ordered operator space, for α, β ∈ Sn
2p, f ∈ Mn(Lp(µ)), and
x ∈ Ω, ‖(αf∗β)(x)‖Snp
= ‖(β∗fα∗)(x)‖Snp. Then ‖αf∗β‖Lp(µ;Sn
p ) = ‖β∗fα∗‖Lp(µ;Snp ). Thus
‖f∗‖Mn(Lp(µ)) = sup‖αf∗β‖Snp [Lp(µ)] : ‖α‖Sn
2p≤ 1, ‖β‖Sn
2p≤ 1
= sup‖αf∗β‖Lp(µ;Snp ) : ‖α‖Sn
2p≤ 1, ‖β‖Sn
2p≤ 1
= sup‖β∗fα∗‖Lp(µ;Snp ) : ‖α∗‖Sn
2p≤ 1, ‖β∗‖Sn
2p≤ 1
= sup‖β∗fα∗‖Snp [Lp(µ)] : ‖α∗‖Sn
2p≤ 1, ‖β∗‖Sn
2p≤ 1 = ‖f‖Mn(Lp(µ)) ,
so Lp(µ) is an involutive operator space.
It remains to show condition (3) of Definition 2.3.4. Suppose f ∈ Mm(Lp(µ))+. Then
f (x) ∈ M+m a.e.(µ), so, for γ ∈ Mm,n, γ∗f (x)γ ∈ M+
n a.e.(µ) since C is matrix ordered.
But then γ∗fγ ∈ Mn(Lp(µ))+, so we conclude that Lp(µ) is matrix ordered.
If (Ω,Σ, µ) is a measure space and f : Ω → Mn is measurable, then so is |f | : Ω → Mn
where |f | is defined by |f | (x) = |f (x)| with |f(x)| = (f(x)∗f(x))1/2 since involutions,
products, and roots of measurable functions are measurable.
89
Theorem 5.2.12. Let (Ω,Σ, µ) be a measure space. Then Lp(µ) is a matrix regular
operator space for 1 ≤ p ≤ ∞.
Proof. In this proof, we will use Theorem 5.2.9 (1) to compute norms in Mn(Lp(µ)) and
the identity in Theorem 5.2.9 (2) to compute norms in Sp[Lp(µ)].
Suppose −f ≤ g ≤ f in Mn(Lp(µ))sa. Then −f(x) ≤ g(x) ≤ f (x) a.e.(µ) in Mn, so for
all α ∈ Mn, −α∗f (x)α ≤ α∗g(x)α ≤ α∗f(x)α a.e.(µ). Since the orderings in Mn and Snp
are the same and Snp is regular, this gives us that
‖α∗g(x)α‖Snp≤ ‖α∗f (x)α‖Sn
pa.e.(µ),
which yields that
‖α∗gα‖Lp(µ;Snp ) ≤ ‖α
∗fα‖Lp(µ;Snp ) .
Since this also means that
‖α∗gα‖Snp [Lp(µ)] ≤ ‖α
∗fα‖Snp [Lp(µ)] ,
we conclude by using Corollary 5.2.10 that
‖g‖Mn(Lp(µ)) ≤ ‖f‖Mn(Lp(µ)) .
Now suppose that g ∈ Mn(Lp(µ))sa with ‖g‖Mn(Lp(µ)) < 1. For all x ∈ Ω, we have
− |g(x)| ≤ g(x) ≤ |g(x)| in Mn, so − |g| ≤ g ≤ |g| in Mn(Lp(µ)).
For all x ∈ Ω, we have by polar decomposition that g(x) = |g(x)| ux = ux |g(x)| where
ux is a unitary, thus giving that |g(x)| = u∗xg(x) = g(x)u∗x. We also have for all x ∈ Ω that
∥∥α |g(x)|β∥∥
Snp
=(tr(β∗g(x)uxα∗αu∗xg(x)β)
p2
) 1p
=(tr(γ)
p2
) 1p
and
∥∥αg(x)β∥∥
Snp
=(tr(β∗g(x)α∗uxu∗xαg(x)β)
p2
) 1p
=(tr(δ)
p2
) 1p
.
Since γ, δ ∈ M+n , σMn(γ) ∈ [0,∞) and σMn(δ) ∈ [0,∞). Now (idR )
p2 is continuous on
σMn(γ) ∪ σMn(δ), and on the same set there exists a sequence of polynomials qn such
that∥∥(idR )
p2 − qn
∥∥∞ → 0. Then
∥∥fp2 − qn(f )
∥∥ → 0 in C∗(γ, idMn) and C∗(δ, idMn), the
C∗-subalgebras of Mn generated by γ and δ along with the identity. Since the trace is
continuous on Mn, we then have that
∥∥∥tr fp2 − tr qn
∥∥∥∞→ 0 on C∗(γ, idMn) and C∗(δ, idMn).
90
But since the trace has the property that tr(xy) = tr(yx), we have for all n ∈ N that
tr qn(γ) = tr qn(δ).
Then
∣∣∣tr fp2 (γ)− tr qn(δ)
∣∣∣ ≤∣∣∣tr f
p2 (γ) − tr qn(γ)
∣∣∣ +∣∣∣tr qn(γ) − tr qn(δ)
∣∣∣ +∣∣∣tr qn(δ) − tr f
p2 (δ)
∣∣∣ → 0
as n →∞. Thus, for all x ∈ Ω,∥∥α |g(x)| β
∥∥Sn
p=
∥∥αg(x)β∥∥
Snp, resulting in
∥∥α |g|β∥∥
Snp
=∥∥α |g|β
∥∥Lp(µ;Sn
p )=
∥∥αgβ∥∥
Lp(µ;Snp )
=∥∥αgβ
∥∥Sn
p.
Then ‖ |g| ‖Mn(Lp(µ)) = ‖g‖Mn(Lp(µ)) < 1, proving that Lp(µ) is a matrix regular operator
space.
91
6. An Application
As an application of matrix regularity, we present here a representation theorem for cer-
tain completely bounded, self-adjoint linear maps involving the Haagerup tensor product.
With this theorem, we can then prove the Christensen-Sinclair multilinear representation
theorem as a corollary. The advantage to this approach for the proof of the Christensen-
Sinclair multilinear representation theorem is that it replaces Wittstock’s complicated con-
cept of matricial sublinearity ([53]) with the more accessible concept of matrix regularity.
In order to prove our representation theorem, we will need to call on some results regard-
ing completely bounded and completely positive maps. Although several of these results
were originally proven by means of Wittstock’s matricial sublinearity, all of them have now
been provided with proofs that do not rely on this concept. Good sources for such proofs
include Paulsen’s book ([42]) and a forthcoming monograph on operator spaces by Effros
and Ruan ([29]). The first result we call on is Stinespring’s representation theorem for
completely positive maps
Theorem (Stinespring [50]) 6.1. Let A be a C∗-algebra and let ϕ : A → B(H)
be a completely positive map. Then there exists a Hilbert space K, a ∗-homomorphism
π : A → B(K), and a bounded operator V : H → K with ‖ϕ‖cb = ‖ϕ‖ = ‖V ‖2 such that
ϕ(a) = V ∗π(a)V.
If A is a unital C∗-algebra, we may take π to be a unital ∗-representation. If ϕ is unital,
then V is an isometry.
We next present the operator space version of the Arveson-Wittstock Hahn-Banach
Extension Theorem.
Theorem (Arveson-Wittstock Hahn-Banach [1, 54, 29]) 6.2. Given operator
spaces V and W with V ⊆ W , we have that any completely bounded map ϕ : V → B(H)
has a linear extension Φ : W → B(H) satisfying ‖Φ‖cb = ‖ϕ‖cb.
The following corollary will prove useful since we will be working with self-adjoint com-
pletely bounded maps.
92
Corollary 6.3. Given involutive operator spaces V and W with V ⊆ W , we have that
any self-adjoint completely bounded map ϕ : V → B(H) has a self-adjoint completely
bounded linear extension Φ : W → B(H) satisfying ‖Φ‖cb = ‖ϕ‖cb.
Proof. Suppose ϕ : V → B(H) is self-adjoint and completely bounded. By the Arveson-
Wittstock Hahn-Banach Theorem, ϕ has an extension ψ to W such that ‖ ψ ‖cb = ‖ϕ‖cb.
Then, by Proposition 2.3.2, ‖ ψ ∗‖cb = ‖ϕ‖cb also. Let Φ = 12 ( ψ + ψ ∗). Φ is clearly self-
adjoint and completely bounded with ‖Φ‖cb ≤ ‖ ψ ‖cb. But Φ|V = ϕ since ψ |V = ψ ∗|V = ϕ.
Thus
‖ϕ‖cb = ‖Φ|V ‖cb ≤ ‖Φ‖cb ≤ ‖ψ ‖cb = ‖ϕ‖cb ,
and so ‖Φ‖cb = ‖ϕ‖cb.
The following is a generalization of Stinespring’s theorem.
Theorem ([53, 42]) 6.4. Let A be a C∗-algebra with unit and let ϕ : A → B(H)
be a completely bounded map. Then there exists a Hilbert space K, a ∗-homomorphism
π : A → B(K), and bounded operators Vi : H → K, i = 1, 2, with ‖ϕ‖cb = ‖V1‖ ‖V2‖ such
that
ϕ(a) = V ∗1 π(a)V2
for all a ∈ A. Moreover, if ‖ϕ‖cb = 1, then V1 and V2 may be taken to be isometries and
π to be unital.
Theorem (Wittstock Decomposition [53, 42]) 6.5. Let A be a unital C∗-algebra,
and let ϕ : A → B(H) be completely bounded. Then there exists a completely positive map
ψ : A → B(H) with ‖ ψ ‖cb ≤ ‖ϕ‖cb such that ψ ±Re (ϕ) and ψ ± Im (ϕ) are all completely
positive.
Using matrix regularity, we can say even more in the self-adjoint case.
Corollary 6.6. Let A be a unital C∗-algebra, and let ϕ : A → B(H) be completely
bounded and self-adjoint. Then there exists a completely positive map ψ : A → B(H)
with ‖ ψ ‖cb = ‖ϕ‖cb such that ψ ± ϕ are completely positive. Letting ϕ+ = 12(ψ + ϕ) and
ϕ− = 12(ψ − ϕ), we also have ϕ = ϕ+ − ϕ− and ‖ϕ‖cb = ‖ϕ+ + ϕ−‖cb.
93
Proof. If ϕ is self-adjoint, then Re (ϕ) = ϕ and Im (ϕ) = 0 in Theorem 6.5. Since A,
as a C∗-algebra, is matrix regular, so is CB(A,B(H)). Then ψ ± ϕ ∈ CB(A,B(H))+,
so ‖ϕ‖cb ≤ ‖ ψ ‖cb also, yielding ‖ϕ‖cb = ‖ψ ‖cb. For the last statement, notice that ψ =
ϕ+ + ϕ−.
Theorem ([11, 29]) 6.7. Given operator spaces V1, . . . , Vn, a linear mapping
ϕ : V1 ⊗h · · · ⊗h Vn → B(Hn,H0)
is completely bounded if and only if there exist Hilbert spaces Hs and completely bounded
mappings ψ s : Vs → B(Hs, Hs−1), s = 1, . . . , n, such that
ϕ(v1 ⊗ · · · ⊗ vn) = ψ 1(v1) · · · ψ n(vn).
Furthermore, we can choose ψ s (s = 1, . . . , n) such that
‖ϕ‖cb = ‖ ψ 1‖cb · · · ‖ψ n‖cb .
Theorem 6.8. Let B be a C∗-algebra, X an operator space, and
ϕ : X∗ ⊗h B ⊗h X → B(H)
a completely bounded self-adjoint linear map. Then there is a Hilbert space K, a ∗-
representation π of B on K which is unital if B is, a completely bounded linear map
V : X → B(H,K) with ‖ϕ‖cb = ‖V ‖2cb, and a bounded linear map S ∈ B(K)sa with
‖S‖ = 1 such that
ϕ(y∗ ⊗ v ⊗ x) = V (y)∗Sπ(b)V (x).
If ϕ is also completely positive, then S = idK .
Proof. (1) Let B be a C∗-algebra and suppose
ψ : X∗ ⊗h B ⊗h X → B(H)
is completely bounded and completely positive. By Theorem 1.3.25 (1.3.22),
CB(X∗ ⊗h B ⊗h X,B(H)) ∼= (Hr ⊗h X∗ ⊗h B ⊗h X ⊗h Hc)†
94
completely isometrically, the correspondence between ˜ψ ∈ (Hr ⊗h X∗ ⊗h B ⊗h X ⊗h Hc)†
and ψ given by
˜ψ (η ⊗ y∗ ⊗ b⊗ x⊗ ξ) = 〈 ψ (y∗ ⊗ b⊗ x)ξ | η〉H .
We then have that
‖ ψ ‖cb = ‖ ψ ‖ = ‖ ˜ψ ‖ = ‖ ˜ψ ‖cb,
and by Theorem 3.2.3, ˜ψ is completely positive.
Let Y = X ⊗h Hc, so that Y ∗ = Hr ⊗h X∗ also. Without loss of generality, we assume
that ‖ ψ ‖cb = 1. On B ⊗ Y , define
⟨[ a1 . . . αn ]¯
x1...
xn
, [ b1 . . . βn ]¯
y1...
yn
⟩
ψ
=
〈a¯ x, b¯ y〉 ψ = ˜ψ (y∗ ¯ [b∗i aj ]¯ x) = ˜ψ (y∗ ¯ b∗a¯ x)
where
b∗a =
b∗1 0 . . . 0...
.... . .
...b∗n 0 . . . 0
a1 . . . an
0 . . . 0. . . . . . . . . . .0 . . . 0
.
We have that
〈a¯ x, a¯ x〉 ψ = ˜ψ (x∗ ¯ a∗a¯ x) ≥ 0
since ˜ψ is positive, so 〈·, ·〉 ψ is a positive semidefinite sesquilinear form. Let
N = u ∈ B ⊗ Y : 〈u, u〉 ψ = 0.
Then K = [B ⊗ Y/N ]−
is a Hilbert space with
〈u + N | v + N 〉K = 〈u, v〉 ψ
as its inner product.
For b ∈ B, define π(b) : B ⊗ Y → B ⊗ Y by π(b)(d⊗ y) = bd⊗ y. Then
〈π(b)(d ¯ y), π(b)(d¯ y)〉ψ =
〈[ bd1 . . . bdn ]¯ y, [ bd1 . . . bdn ]¯ y〉ψ =
˜ψ (y∗ ¯ [d∗i b∗bdj ]¯ y) ≤
‖b‖2 ˜ψ (y∗ ¯ d∗d¯ y) = ‖b‖2 〈d ¯ y, d¯ y〉 ψ
95
since[d∗i b∗bdj ] ≤ ‖b∗b‖ [d∗i dj ] and ˜ψ is completely positive.
We conclude that π(b) leaves N invariant and thus defines a linear transformation on
B ⊗ Y/N , still denoted by π(b). We also have that π(b) is bounded with ‖π(b)‖ ≤ ‖b‖.
Then π(b) extends to a bounded linear operator on K, still denoted by π(b). We get that
π : B → B(K)
is a unital ∗-homomorphism with
π(b)(d ⊗ y + N ) = bd⊗ y + N.
We note that if B is a unital C∗-algebra, then π is a unital ∗-representation.
If B is not a unital C∗-algebra, let µλλ∈Λ be the canonical approximate identity for
B (the set of all positive elements of B of norm less than 1, directed by “<”). Then
(6.1)
limλ〈b ⊗ x⊗ ξ + N | µλ ⊗ y ⊗ η + N 〉K = lim
λ
˜ψ (η ⊗ y∗ ⊗ µλb⊗ x⊗ ξ)
= ˜ψ (η ⊗ y∗ ⊗ b⊗ x⊗ ξ)
= 〈ψ (y∗ ⊗ b⊗ x)ξ | η〉H .
Then we can define W : Y → K by W (y) = w*-limµλ ⊗ y + N , the convergence being in
the point-weak* topology. Since, for all µλ,
‖µλ ⊗ y + N‖2 = 〈µλ ⊗ y + N | µλ ⊗ y + N〉K = ˜ψ (y∗ ⊗ µ2λ ⊗ y) ≤ ‖y‖2 ,
‖W‖ ≤ 1.
However, if B is unital, we can define W : Y → K by W (y) = 1⊗ y + N . Since
‖Wy‖2 = 〈Wy | Wy〉K = 〈1⊗ y, 1⊗ y〉 ψ = ˜ψ (y∗ ⊗ 1⊗ y) ≤ ‖y‖2 ,
‖W‖ ≤ 1.
Next, define V : X → B(H,K) by V (x)ξ = W (x ⊗ ξ). Proceeding only with the case
where B is nonunital, since the unital case is similar, we have that V is linear, and for
x = [xij] ∈ Mn(X) and h =
h1...
hn
∈ Hn, we get that
‖Vn([xij ])h‖2 = 〈Vn([xij ])h | Vn([xij ])h〉Kn =∑
i,j,k
〈V (xij)hj | V (xik)hk〉K
96
=∑
i,j,k
〈W (xij ⊗ hj) | W (xik ⊗ hk)〉K
=∑
i,j,k
⟨w*-limλ µλ ⊗ xij ⊗ hj + N | w*-limγ µγ ⊗ xik ⊗ hk + N
⟩K
= w*-limλ w*-limγ
∑
i,j,k
〈µλ ⊗ xij ⊗ hj + N | µγ ⊗ xik ⊗ hk + N 〉K
= w*-limλ w*-limγ
∑
i,j,k
˜ψ (hk ⊗ x∗ik ⊗ µγµλ ⊗ xij ⊗ hj)
= w*-limλ w*-limγ˜ψ (h∗ ¯ x∗ ¯
µγµλ
. . .
µγµλ
¯ x¯ h) ≤ ‖h‖2 ‖x‖2 ,
so ‖Vn‖ ≤ 1. Thus V is completely bounded with ‖V ‖cb ≤ 1.
We then have, using (6.1), for y∗ ∈ X∗, b ∈ B, x ∈ X, and ξ, η ∈ H, that
〈 ψ (y∗ ⊗ b⊗ x)ξ | η〉H = limλ〈b⊗ x⊗ ξ + N | µλ ⊗ y ⊗ η + N〉 ψ
= limλ〈π(b)V (x)ξ | π(µλ)V (y)η〉K
= limλ〈V (y)∗π(µλb)V (x)ξ | η〉H
= 〈V (y)∗π(b)V (x)ξ | η〉H ,
so
ψ (y∗ ⊗ b⊗ x) = V (y)∗π(b)V (x).
Since ‖ ψ ‖ = 1 and V and π are completely contractive, we must have that ‖V ‖ = 1,
giving us also that ‖V ‖cb = 1. Thus the completely positive case is finished.
(2) We now assume that
ϕ : X∗ ⊗h B ⊗h X → B(H)
is completely bounded and self-adjoint. By Corollary 6.3, we may assume that B is a
unital C∗-algebra. We also assume ‖ϕ‖cb = 1. Since B is a C∗-algebra, it is matrix
regular. Then X∗ ⊗h B ⊗h X is matrix regular by Theorem 4.1.5, from which it follows
that CB(X∗ ⊗h B ⊗h X, B(H)) = (Hr ⊗h X∗ ⊗h B ⊗h X ⊗h Hc)† is matrix regular as
an operator dual space by Theorem 3.2.4. Thus, by Theorem 2.2.9, the intersection of
the self-adjoint elements of CB(X∗ ⊗h B ⊗h X,B(H)) with its closed unit ball is solid.
That means that there exists ψ ∈ CB(X∗ ⊗h B ⊗h X,B(H))+, ‖ ψ ‖cb = 1, such that
97
ψ ±ϕ ∈ CB(X∗⊗h B⊗h X,B(H))+. We take ψ exactly as in the unital version of part (1)
of this proof.
Define Fϕ : K ×K → C by
Fϕ(b ¯ y ¯ η + N, d ¯ x¯ ξ + N) = 〈ϕn(x∗ ¯ d∗b¯ y)η | ξ〉Hn
where b, d ∈ M1,n(B), x, y ∈ Mn(X), and ξ, η ∈ Mn,1(H).
Fϕ is sesquilinear, and since ϕ is self-adjoint,
Fϕ(b¯ y ¯ η + N, d¯ x¯ ξ + N) = 〈ϕn(x∗ ¯ d∗b¯ y)η | ξ〉Hn =
〈ϕ∗n(x∗ ¯ d∗b ¯ y)η | ξ〉Hn = 〈ϕn(y∗ ¯ b∗d ¯ x)∗η | ξ〉Hn =
〈ϕn(y∗ ¯ b∗d¯ x)ξ | η〉Hn = Fϕ(d¯ x¯ ξ + N, b¯ y ¯ η + N ),
so Fϕ is Hermitian.
Then
|Fϕ(b¯ y ¯ ξ + N, b¯ y ¯ η + N )| = |〈ϕn(y∗ ¯ b∗b¯ y)η | η〉Hn|
≤ 〈ψ n(y∗ ¯ b∗b¯ y)η | η〉Hn .
But since
‖b¯ y ¯ η + N‖2 = 〈b¯ y ¯ η + N | b¯ y ¯ η + N 〉K
= 〈b¯ y ¯ η, b¯ y ¯ η〉ψ = ˜ψ (η ⊗ y∗ ⊗ b∗b⊗ y ⊗ η)
= 〈 ψ n(y∗ ⊗ b∗b ⊗ y)η | η〉Hn ,
‖Fϕ‖ ≤ 1. Then there exists S ∈ B(K)sa, ‖S‖ = ‖Fϕ‖, such that
Fϕ(b⊗ y ⊗ η + N, d ⊗ x⊗ ξ + N) = 〈S(b⊗ y ⊗ η + N ) | d⊗ x⊗ ξ + N 〉K .
Thus
〈ϕ(x∗ ⊗ b⊗ y)η | ξ〉H = Fϕ(b⊗ y ⊗ η + N, 1⊗ x⊗ ξ + N )
= 〈S(b⊗ y ⊗ η + N ) | 1⊗ x⊗ ξ + N 〉K
= 〈Sπ(b)V (y)η | V (x)ξ〉K
= 〈V (x)∗Sπ(b)V (y)η | ξ〉H ,
98
so
ϕ(x∗ ⊗ b⊗ y) = V (x)∗Sπ(b)V (y).
Since ‖ϕ‖ = 1 and V , π and S are contractive, ‖S‖ = 1, and thus we have completed the
completely bounded self-adjoint case.
We now turn to the Christensen-Sinclair multilinear representation theorem. However,
since we are going to prove a linearized version of this theorem using the Haagerup tensor
product, we first need to provide the definitions used by Christensen and Sinclair and then
point out the equivalence of the multilinear and linear versions.
Let A and B be C∗-algebras and let V be a k-linear operator from Ak into B. We say
that V is bounded if
‖V ‖ = sup‖V (a1, . . . , ak)‖ : ‖a1‖ ≤ 1, . . . , ‖ak‖ ≤ 1 < ∞.
The k-linear operator Vn : Mn(A)k → Mn(B) is defined by
Vn(A 1, . . . ,A k) =
[∑
r,s,... ,t
V (a1ir , a2rs , . . . , aktj)
]
i,j
for all A l =[alij
]∈ Mn(A) (1 ≤ l ≤ k) and all n ∈ N . Note that this definition of Vn is
related directly to matrix multiplication.
The k-linear operator V is said to be completely bounded with completely bounded
norm ‖V ‖cb if ‖V ‖cb = sup‖Vn‖ : n ∈ N < ∞. For notation,
CB(Ak, B) = V : Ak → B : V is k-linear and completely bounded.
The k-linear operator V ∗ : Ak → B is defined by V ∗(a1, . . . , ak) = V (a∗k, . . . , a∗1)∗ for
all a1, . . . , ak ∈ A. We say that V is symmetric (self-adjoint when k = 1) if V = V ∗. For
every V ∈ CB(Ak, B), we have that
V =V + V ∗
2+ i
V − V ∗
2i,
i.e., V is a linear combination of symmetric operators.
A k-linear operator V : Ak → B is completely positive if Vn(A 1, . . . ,A k) ≥ 0 for every
(A 1, . . . ,A k) = (A ∗k, . . . , A ∗1) ∈ Mn(A)k with A m ≥ 0 if k is odd, where m =[
k+12
](with
[·] being the greatest integer function), and all n ∈ N . Note that there are completely
positive multilinear operators that are not completely bounded. For notation,
CB(Ak, B)+ = V ∈ CB(Ak, B) : V is completely positive.
99
Lemma ([11]) 6.9. Given the k-linear operator V : Ak → B:
(1) For all n ∈ N , (V ∗)n = (Vn)∗.
(2) If V is symmetric, then V is completely symmetric in that V ∗n = Vn for all n ∈ N .
(3) ‖V ∗‖cb = ‖V ‖cb.
We now establish the relationship between the multilinear maps and their corresponding
linearized maps.
Definition 6.10. Let E1, . . . , Ek be subspaces of C∗-algebras A1, . . . , Ak, respectively.
If V : E1× · · · ×Ek → B(H) is a multilinear map, we define the corresponding linear map
ϕ : E1 ⊗h · · · ⊗h Ek → B(H) by
V (a1, . . . , ak) = ϕ(a1 ⊗ · · · ⊗ ak).
Note. We then have
Vn(A 1, . . . ,A k) = ϕn(A 1 ¯ · · · ¯A k)
where A j ∈ Mn(Ej).
Proposition ([11]) 6.11. Let E1, . . . , Ek be subspaces of C∗-algebras A1, . . . , Ak, re-
spectively. Let V : E1× · · ·×Ek → B(H) be a multilinear map and ϕ : E1⊗h · · · ⊗h Ek →
B(H) be the associated linear map. Then V is completely bounded if and only if ϕ is
completely bounded and ‖V ‖cb = ‖ϕ‖cb.
We now need to show that the cones CB(Ak, B(H))+ are in an appropriate one-to-one
correspondence with the cones CB(X⊗h A⊗h X,B(H))+ where X = A⊗h · · · ⊗h A︸ ︷︷ ︸m−1 copies
, m =
[k+12
], for k odd, and with the cones CB(X ⊗h X,B(H))+ = CB(X ⊗h C ⊗h X,B(H))+
where X = A ⊗h · · · ⊗h A︸ ︷︷ ︸m copies
, m =[
k+12
], for k even.
Definition 6.12. Given a C∗-algebra A , we define CB(A⊗h · · · ⊗h A︸ ︷︷ ︸k copies
, B(H))⊕ for k ∈ N
by ϕ ∈ CB(A⊗h · · · ⊗h A︸ ︷︷ ︸k copies
, B(H))⊕ if and only if V ∈ CB(Ak, B(H))+, where ϕ and V
are associated maps.
100
Proposition 6.13. Let A be a C∗-algebra and let k ∈ N . For k odd,
CB(A⊗h · · · ⊗h A︸ ︷︷ ︸k copies
, B(H))⊕ = CB(X ⊗h A ⊗h X,B(H))+
where X = A ⊗h · · · ⊗h A︸ ︷︷ ︸m−1 copies
and m =[
k+12
]. For k even,
CB(A ⊗h · · · ⊗h A︸ ︷︷ ︸k copies
, B(H))⊕ = CB(X ⊗h C ⊗h X,B(H))+
where X = A ⊗h · · · ⊗h A︸ ︷︷ ︸m copies
and m =[
k+12
].
Proof. We only need consider the case where k is odd. Let P = CB(X⊗h A⊗h X,B(H))+
and Q = CB(A ⊗h · · · ⊗h A︸ ︷︷ ︸k copies
, B(H))⊕.
Suppose ϕ ∈ Q. Then the associated map V to ϕ is in CB(Ak, B(H))+. Suppose
u = (a∗1 ¯ · · · ¯ a∗m−1)¯ am ¯ (am−1 ¯ · · · ¯ a1) ∈ Pn(X ⊗h A⊗h X). Let p be the largest
number from among all the dimensions of a1, . . . , am. Create matrices b1, . . . , bm ∈ Mp(A)
by embedding ai into the upper left corner of bi, i = 1, . . . ,m, and then filling in with 0’s.
Then w = (b∗1 ¯ · · · ¯ b∗m−1)¯ bm ¯ (bm−1 ¯ · · · ¯ b1) ∈ Pp(X ⊗h A⊗h X). We have that
Vp(b∗1, . . . , b∗m−1, bm, bm−1, . . . , b1) ≥ 0, so ϕp(w) ≥ 0. Then
ϕn(u) =
[In 0n,p−n
0p−n,n 0p−n,p−n
]ϕp(w)
[In 0n,p−n
0p−n,n 0p−n,p−n
]≥ 0.
We conclude that ϕ ∈ P .
Now suppose ϕ ∈ P . Suppose (a1, . . . , am−1, am, a∗m−1, . . . , a∗1) ∈ Mn(Ak). Then
u = a1 ¯ · · · ¯ am−1 ¯ am ¯ a∗m−1 ¯ · · · ¯ a∗1 ∈ Pn(X ⊗h A ⊗h X), so ϕn(u) ≥ 0. Then
for the associated map V to ϕ, Vn(a1, . . . , am−1, am, a∗m−1, . . . , a∗1) ≥ 0. We conclude that
V ∈ CB(Ak, B(H))+, so ϕ ∈ Q. Thus P = Q.
Thus we have that the needed compatibility between the orderings in the multilinear
and linear approaches. We proceed to the proof of the Christensen-Sinclair theorem.
Theorem 6.14. Let A be a C∗-algebra, let H be a Hilbert space, and let ϕ be a completely
bounded, self-adjoint linear map from A ⊗h · · · ⊗h A︸ ︷︷ ︸k copies
into B(H). Let m =[
k+12
].
(1) k odd. There are ∗ representations θ1, . . . , θm−1, ψ 1, ψ 2 of A on Hilbert spaces
H1, . . . ,Hm−1,K1,K2, linear operators Vj ∈ B(Hj ,Hj+1) for 0 ≤ j ≤ m − 2
101
where H0 = H with
‖V0‖ ‖V1‖ · · · ‖Vm−2‖ = ‖ϕ‖1/2cb
and W1 ∈ B(Hm−1,K1) and W2 ∈ B(Hm−1,K2) with ‖W ∗1 W1 + W ∗
2 W2‖ = 1 such
that
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V
∗1 θ2(a2) · · · θm−1(am−1)
× W ∗1 ψ 1(am)W1 −W ∗
2 ψ 2(am)W2
× θm−1(am+1) · · · V1θ1(ak)V0
for all a1, . . . , ak ∈ A. If in addition ϕ is completely positive, then W2 = 0.
(2) k even. There are ∗ representations θ1, . . . , θm of A on Hilbert spaces H1, . . . ,Hm,
linear operators Vj ∈ B(Hj ,Hj+1) for 0 ≤ j ≤ m − 1 where H0 = H, and W ∈
B(Hm−1)sa with ‖W‖ = 1 and
‖V0‖ ‖V1‖ · · · ‖Vm−2‖ = ‖ϕ‖1/2cb
such that
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V∗1 · · ·V ∗
m−1θm(am)W
× θm(am+1)Vm−1 · · ·V1θ1(ak)V0
for all a1, . . . , ak ∈ A. If in addition ϕ is completely positive, then W ≥ 0.
Proof. We assume that ‖ϕ‖cb = 1. Suppose k is odd. For convenience sake, we will refer
to the middlemost copy of A (copy m) as B. Let X = A⊗h · · · ⊗h A︸ ︷︷ ︸m−1 copies
. Noting X = X∗,
we apply Theorem 6.8 to ϕ : X∗ ⊗h B ⊗h X → B(H) to get a Hilbert space K, a ∗-
representation π of B on K, a completely bounded linear map V : X → B(H,K) with
‖ϕ‖cb = ‖V ‖2cb, and a bounded linear map S ∈ B(K)sa with ‖S‖ = 1 such that
ϕ(x∗ ⊗ v ⊗ y) = V (x)∗Sπ(b)V (y).
If ϕ is also completely positive, then S = idK . Let H = L0, K = Lm−1. By Theorem 6.7,
there exist Hilbert spaces L1, . . . , Lm−2 and, for s = 1, . . . ,m − 1, completely bounded
mappings γs : A → B(Ls−1, Ls) such that
V (b1 ⊗ · · · ⊗ bm−1) = γm−1(b1) · · · γ1(bm−1)
102
and
‖V ‖cb = ‖γ1‖cb · · · ‖γm−1‖cb .
Without loss of generality, let ‖γs‖cb = 1 for s = 1, . . . ,m− 1.
By the Arveson-Wittstock Hahn-Banach theorem (Theorem 6.2), we may extend each γs
to A, the standard C∗-algebraic unitisation of A, the completely bounded norm remaining
the same.
We may also assume that γs : A → B(Ls−1 ⊕ Ls), s = 1, . . . , m − 1, i.e., γs(a) ∈
B(Ls−1, Ls) corresponds to
[0 0
γs(a) 0
]∈ B(Ls−1 ⊕ Ls).
By applying Theorem 6.4 to γs for s = 1, . . . ,m − 1, there exist Hilbert spaces Hs,
unital ∗-homomorphisms θs : A → B(Hs), isometries Ws,1 : Ls−1 ⊕ Ls → B(Hs) and
Ws,2 : Ls−1 ⊕ Ls → B(Hs) with γs(a) = W ∗s,1θs(a)Ws,2 and ‖γs‖cb = ‖Ws,1‖ ‖Ws,2‖ = 1.
But γs(A)Ls−1 ⊆ Ls. Let Vs,1 = Ws,1|Ls and Vs,2 = Ws,2|Ls−1 . Vs,1 and Vs,2 are then
isometries, and for all a ∈ A, γs(a) = V ∗s,1θs(a)Vs,2. Also, ‖γs‖cb = ‖Vs,1‖ ‖Vs,2‖ = 1. We
then have
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗12θ1(a1)V11V
∗22θ2(a2)V21 · · ·V ∗
m−1,2θm−1(am−1)Vm−1,1
× π(am)
× V ∗m−1,1θm−1(am+1)Vm−1,2 · · · V ∗21θ2(ak−1)V22V∗11θ1(ak)V12.
Let V0 = V1,2 and Vs = Vs+1,2V∗s,1 for s = 1, . . . ,m − 2. Then ‖Vs‖ = 1 for s =
1, . . . ,m− 2.
Now suppose that ϕ is completely positive. Since π : B → B(K), as a ∗-homomorphism,
is completely positive and completely contractive, there exists a Hilbert space K1, a
∗-representation ψ 1 of B on K1, and Vm : K → K1 such that π(b) = V ∗m ψ 1(b)Vm and
‖π‖cb = ‖Vm‖2. Let W1 = VmV ∗m−1,1. With
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V
∗1 θ2(a2) · · · θm−1(am−1)
×W ∗1 ψ 1(am)W1
× θm−1(am+1) · · · V1θ1(ak)V0,
we have ‖W1‖ = 1 since ‖ϕ‖ = 1 and each of the operators to the right of the “=” sign are
contractive. Then ‖W ∗1 W1‖ = ‖W1‖2 = 1. This gives us our representation for the odd,
completely positive case.
103
For the even, completely positive case, let B = C . With index adjustment for the even
case, we have
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V∗1 · · ·V ∗
m−1θm(am)
× Vm,1π(1)V ∗m,1
× θm(am+1)Vm−1 · · ·V1θ1(ak)V0.
Noting that π(1) = id |K, let W = Vm,1V∗m,1. We have that W ≥ 0 and ‖W‖ = 1. We then
have the representation for this case also.
Next, assume that we are in the case where k is odd and ϕ is self-adjoint. By Corol-
lary 6.3, we may now assume that B is unital. Based on the above, we have
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V
∗1 θ2(a2) · · · θm−1(am−1)
× Vm−1,1Sπ(am)V ∗m−1,1
× θm−1(am+1) · · · V1θ1(ak)V0.
The map
Z : Vm−1,1Sπ(·)V ∗m−1,1 : B → B(K)
is completely bounded and self-adjoint with ‖Z‖cb ≤ 1. By Corollary 6.6, there exists
T ∈ CB(B,B(K))+ such that ‖T‖cb = ‖Z‖cb and T ± Z ∈ CB(B,B(K))+. Letting
Z+ = 12(T +Z) and Z− = 1
2(T−Z), we have that Z = Z+−Z− and ‖Z‖cb = ‖Z+ + Z−‖cb.
Since Z+ and Z− are completely positive, by Stinespring’s theorem (Theorem 6.1)
there exist Hilbert spaces K1 and K2, unital ∗-representations ψ 1 and ψ 2 of B on K1
and K2, respectively, and bounded linear maps W1 : K → K1 and W2 : K → K2 with
‖Z+‖cb = ‖W1‖2 and ‖Z−‖cb = ‖W2‖2 such that
Z+(b) = W ∗1 ψ 1(b)W1 and Z−(b) = W ∗
2 ψ 1(b)W2.
This gives us that
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V
∗1 θ2(a2) · · · θm−1(am−1)
× W ∗1 ψ 1(am)W1 −W ∗
2 ψ 2(am)W2
× θm−1(am+1) · · · V1θ1(ak)V0.
104
Then
‖W ∗1 W1 + W ∗
2 W2‖ = sup‖(W ∗1 id |K1W1 + W ∗
2 id |K2W2)k‖ : k ∈ K, ‖k‖ ≤ 1
= sup‖(W ∗1 ψ 1(1)W1 + W ∗
2 ψ 2(1)W2)k‖ : k ∈ K, ‖k‖ ≤ 1
= sup∥∥(Z+(1) + Z−(1))k
∥∥ : k ∈ K, ‖k‖ ≤ 1
= sup‖T (1)k‖ : k ∈ K, ‖k‖ ≤ 1 = ‖T (1)‖ = ‖T‖ = ‖T‖cb = ‖Z‖cb ,
since for completely positive maps γ between unital C∗-algebras, ‖γ‖cb = ‖γ‖ = ‖γ(1)‖.
Using the same argument as previously, ‖Z‖ = 1, so ‖Z‖cb = ‖W ∗1 W1 + W ∗
2 W2‖ = 1.
Thus we now have the representation for the odd, self-adjoint case.
For the even, self-adjoint case, again let B = C . We have
ϕ(a1 ⊗ · · · ⊗ ak) = V ∗0 θ1(a1)V∗1 · · ·V ∗
m−1θm(am)
× Vm,1Sπ(1)V ∗m,1
× θm(am+1)Vm−1 · · ·V1θ1(ak)V0.
Again noting that π(1) = id |K , let W = Vm,1SV ∗m,1. Then W is self-adjoint with ‖W‖ = 1,
so we have the representation in this final case also.
105
Appendix: Ordered Vector Spaces
In this appendix, we provide a development leading up to Theorem A.45, of which
Theorem 2.1.9 in Section 2.1 is a special case. We do this since Theorem 2.1.9 is so crucial
to our work. Proofs of the other statements in Section 2.1 are also included as part of this
development.
The material in this appendix is a distillation from several sources, but primarily comes
from the books of Wong and Ng [56] and Asimow and Ellis [2].
Notation. Throughout this section, E denotes a real vector space, ‖·‖ denotes a norm on
E, τ denotes a topology on E, and P denotes a cone (not necessarily proper) on E which
gives E a partial ordering. All spaces are assumed to be Hausdorff.
Definition A.1. Let E be a real vector space.
(1) B ⊆ E is symmetric if −B = B.
(2) B ⊆ E is circled if λB ⊆ B whenever |λ| ≤ 1.
(3) Let A ⊆ E and B ⊆ E. A absorbs B if there exists λ ≥ 0 such that B ⊆ µA for all
µ with |µ| ≥ λ. A is absorbing if it absorbs every finite subset of E.
(4) If A ⊆ E is absorbing, the functional pA defined by pA(x) = infλ > 0 : x ∈ λA
for all x ∈ E is called the Minkowski functional of A.
Theorem (Bonsall) A.2. Let (E, P ) be a real ordered vector space and p a sublinear
functional on E, and suppose q is a superlinear functional on P such that
q(u) < p(u) for all u ∈ P.
Then there exists a linear functional f on E such that
f (x) ≤ p(x) for all x ∈ E, and
q(u) ≤ f (u) for all u ∈ P.
Proof. For all x ∈ E, define r(x) = infp(x + u)− q(u) : u ∈ P . Since
p(u) ≤ p(x + u) + p(−x),
106
it follows that r(x) ≥ −p(−x), and hence that r is finite on E. Thus r is a sublinear
functional on E, r(x) ≤ p(x) for all x ∈ E, and r(−u) ≤ −q(u) for all u ∈ P . By the
Hahn-Banach theorem, there exists a linear functional f on E such that f (x) ≤ r(x) for
all x ∈ E, and so f (x) ≤ p(x) for all x ∈ E. Since f (−u) ≤ r(−u) ≤ −q(u) for all u ∈ P ,
we conclude that f(u) ≥ q(u) for all u ∈ P .
Definition A.3. Let (E,P ) be an ordered real vector space.
(1) The order interval [x, y] = z ∈ E : x ≤ z ≤ y for x ≤ y in (E,P ).
(2) A ⊆ (E,P ) is order convex if [a1, a2] ⊆ A whenever a1, a2 ∈ A and a1 ≤ a2.
(3) A ⊆ (E,P ) is absolutely order convex if [−a, a] ⊆ A whenever a ∈ A and a ∈ P .
(4) A ⊆ (E, P ) is absolutely dominated if for every a ∈ A there exists x ∈ A such that
−x ≤ a ≤ x.
(5) A ⊆ (E,P ) is solid if it is both absolutely order convex and absolutely dominated.
Definition A.4. Let E and F be real vector spaces and let f0 be a bilinear functional
on E × F such that
(1) for all x 6= 0, there exists y ∈ Y such that f0(x, y) 6= 0, and
(2) for all y 6= 0, there exists x ∈ X such that f0(x, y) 6= 0.
Then E and F are said to be in duality (with respect to f0) and 〈E,F 〉 is said to form a
dual system. We denote f0(x, y) = 〈x, y〉.
Examples.
(1) If Ed is the algebraic dual of E, the bilinear function 〈x, x∗〉 = x∗(x) defines the
duality 〈E,Ed〉.
(2) If E is a topological vector space with topological dual E′, then the duality 〈E,Ed〉
induces a duality 〈E,E′〉.
Let E and F be endowed with the weakest topologies for which the functionals 〈·, y〉
and 〈x, ·〉 are continuous on E and F for all x ∈ E and for all y ∈ F . We refer to these
topologies as σ(E,F ) and σ(F,E). Then E and F are locally convex linear topological
spaces with each being the (dual) space of continuous linear functionals over the other.
107
Theorem (Separation Theorem) A.5. If A and B are disjoint convex subsets of a
real vector space E with A compact and B closed, then there exists a linear functional f
such that
max〈a, f〉 : a ∈ A < inf〈b, f 〉 : b ∈ B.
Proof. Consequence of Hahn-Banach theorem.
Definition A.6. If E is a real vector space and A ⊆ E, the polar of A, taken in Ed, is
Aπ and is defined by
Aπ = f ∈ Ed : f(a) ≤ 1 for all a ∈ A.
Definition A.7. If (E,P ) is an ordered real vector space, the dual cone in Ed is
P d = f ∈ Ed : f (x) ≥ 0 for every x ∈ P.
Theorem A.8. Suppose A and B are subsets of a real vector space E.
(1) If B ⊆ A, then Aπ ⊆ Bπ.
(2) If λ 6= 0, (λA)π = λ−1Aπ.
(3) (A ∪ B)π = Aπ ∩ Bπ.
(4) Aπ is convex and is closed in the σ(Ed, E) topology.
(5) (Bipolar Theorem) Aππ = co(A ∪ 0), where the closure is taken in the σ(E,Ed)
topology.
(6) Aπππ = Aπ.
Proof.
(1), (2), and (3) follow from the definition.
(4) Convexity is immediate. The equality Aπ =⋂
a∈A
f ∈ Ed : 〈a, f〉 ≤ 1, together
with the σ(Ed, E) continuity of each functional 〈a, ·〉 implies that Aπ is closed.
(5) From part 4, Aππ is σ(E,Ed)-closed and convex. Clearly, 0 ∈ Aππ and A ⊆ Aππ.
Letting A = co(A∪ 0), A ⊆ Aππ. To show equality, we need that a0 6∈ A implies
a0 6∈ Aππ. Suppose a0 6∈ A. By the Separation Theorem, there exists α ∈ R and
f ∈ Ed, f σ(E,Ed)-continuous, such that f (a0) < α < f (a) for all a ∈ A. Since
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0 ∈ A, 0 = 〈0, f 〉 > α. Let g = 1αf . Then g(a) < 1 < g(a0) for all a ∈ A, so
g ∈ (A)π ⊆ Aπ. Thus a 6∈ Aππ.
(6) Follows from part 5.
Definition A.9. For any subset V of an ordered real vector space (E, P ), define
S(V ) =⋃[−v, v] : v ∈ V ∩ P.
Proposition A.10. Let V be a subset of an ordered real vector space (E,P ).
(1) V is absolutely dominated ⇐⇒ S(V ) ⊇ V .
(2) V is absolutely order convex ⇐⇒ S(V ) ⊆ V .
(3) V is solid ⇐⇒ S(V ) = V .
Proof. Follows from the definitions.
Lemma A.11. Let V be a subset of an ordered real vector space (E,P ). Then S(V π) ⊆
(S(V ))π. Consequently, if V is absolutely dominated, then V π is absolutely order convex
in Ed.
Proof. Let f ∈ S(V π). Then there exists 0 ≤ g ∈ V π such that −g ≤ f ≤ g. Let x ∈ S(V )
and suppose −v ≤ x ≤ v for some v ∈ V . Then
0 ≤ (g − f)(v + x) = g(v) + g(x)− f (v)− f (x)
and
0 ≤ (g + f )(v − x) = g(v)− g(x) + f(v)− f(x),
so 0 ≤ 2g(v)− 2f(x). Since v ∈ V and g ∈ V π, f (x) ≤ g(x) ≤ 1. Thus f(x) ≤ 1 for all
x ∈ S(V ), so f ∈ (S(V ))π.
If V is absolutely dominated, then V ⊆S(V), so S(V π) ⊆ (S(V ))π ⊆ V π. Thus V π is
absolutely order convex.
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Theorem (Jameson) A.12. Suppose V is a convex and absorbing subset of an ordered
real vector space (E,P ). Then (S(V ))π = S(V π). Thus, if V is absolutely order convex,
then V π is absolutely dominated, and if V is solid, then V π is solid.
Proof. In view of Lemma A.11, we need only show (S(V ))π ⊆ S(V π). Let g ∈ S(V ))π.
For all x ∈ P , define q(x) = supg(y) : y ∈ E,−x ≤ y ≤ x. Since g ∈ (S(V ))π and V
is absorbing, q is a well-defined superlinear functional on P . Let pV be the Minkowski
functional of V . Now, if −x ≤ y ≤ x in V , g(y) ≤ 1, and so q(v) ≤ pV (v) for all v ∈ P .
Since pV is sublinear, by Theorem A.2 there exists f ∈ Ed such that f(y) ≤ p(y) for
all y ∈ E and q(v) ≤ f(v) for all v ∈ P . Notice −q(v) ≤ g(v) ≤ q(v) for all v ∈ P , so
−f (v) ≤ g(v) ≤ f (v) for all v ∈ P . Hence −f ≤ g ≤ f . Notice also f(y) ≤ p(y) ≤ 1 for all
y ∈ V , so f ∈ V π. Then g ∈ S(V π), so (S(V ))π ⊆ S(V π).
If V is absolutely order convex, then S(V ) ⊆ V , so V π ≤ (S(V ))π ≤ S(V π), making V π
absolutely dominated. Using Lemma A.11 also, we then get that V solid implies that V π
is solid.
Definition A.13. Suppose (E, τ) is a real topological vector space. The polar of A,
taken in E′, is
A = f ∈ E′ : f(a) ≤ 1 for all a ∈ A, i.e., A = Aπ ∩E ′.
Definition A.14. Suppose (E,P, τ) is an ordered real topological vector space. The
dual cone P ′ in E′ is P ′ = P d ∩ E′.
Proposition A.15. Suppose (E,P ) is an ordered real vector space.
(1) P π = −P d.
(2) (A + P )π = Aπ ∩ Pπ.
Proof.
(1) Pπ = f ∈ Ed : 〈p, f 〉 ≤ 1 for all p ∈ P = f ∈ Ed : 〈p, f 〉 ≤ 0 for all p ∈ P =
−f ∈ Ed : 〈p, f〉 ≥ 0 for all p ∈ P = −f ∈ Ed : 〈p, f 〉 ≥ 0 for all p ∈ P =
−P d.
(2) (A + P )π = f ∈ Ed : 〈a + p, f 〉 ≤ 1 for all p ∈ P and for all a ∈ A =
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f ∈ Ed : 〈a, f〉+ 〈p, f〉 ≤ 1 for all p ∈ P and for all a ∈ A =
f ∈ Ed : 〈a, f〉 ≤ 1 and 〈p, f〉 ≤ 1 for all p ∈ P and for all a ∈ A = Aπ∩P π.
Corollary A.16. Suppose (E,P, τ) is an ordered real topological vector space.
(1) P = −P ′. Thus P ′ is closed in the σ(E′, E) topology.
(2) (A + P ) = A ∩ P whenever A contains the origin.
Proof.
(1) P = P π ∩ E = (−P d) ∩ E′ = −P ′. The closure of P ′ follows from the above
definition and Theorem A.8.
(2) (A + P ) = (A + P )π ∩E′ = (Aπ ∩ Pπ) ∩E′ = A ∩ P .
Proposition A.17. Let (E, P ) be an ordered real vector space and let τ be a (Hausdorff)
locally convex topology on E. Then P is a τ -closed set if and only if the following condition
is satisfied: If x0 ∈ E and f (x0) ≥ 0 for all f ∈ P ′, then x0 ∈ P .
Proof. [=⇒] Suppose P is closed. If x0 6∈ P , by Theorem A.5 there exists f ∈ E′ such
that f(x0) < inff(x) : x ∈ P . In particular, for x ∈ P , f (x) > α−1f (x0) for all α > 0,
hence f(x) ≥ 0. Then f ∈ P ′ and f(x0) < 0. Thus necessity is proved.
[⇐=] For f ∈ P ′, let Ef = x : f(x) ≥ 0. Each Ef is closed for f ∈ P ′. Given the
condition, then P =⋂
f∈P ′Ef , so P is closed.
Corollary A.18. Let (E, P, ‖·‖) be an ordered real Banach space. Then P = P ′′ ∩E.
Proposition A.19. Let E be a real locally convex space, and let S and T be convex
subsets containing 0. If S and T are closed, or neighborhoods of 0, then (S ∩ T ) is the
σ(E′, E)-closed convex hull co(S ∪ T ) of S ∪ T .
Proof. In either case, S ∩ T = S ∩ T . In the second case, let x ∈ S ∩ T . Since S is convex
and contains 0 as an interior point, λx ∈ S for all 0 ≤ λ < 1. Similarly, λx ∈ T . Letting
λ → 1 in λx ∈ S ∩ T , we have x ∈ S ∩ T . Thus S ∩ T ⊆ S ∩ T and so S ∩ T = S ∩ T since
the opposite inclusion is obvious.
By the bipolar theorem, and noting Theorem A.8 holds for as well as π,
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co(S ∩ T ) = (S ∩ T ) = (S ∩ T ) = (S ∩ T ).
Since S ∩ T = S ∩ T and the polar in E′ of a set is the same as the polar of its closure,
(S ∩ T ) = (S ∩ T ) = (S ∩ T ) = co(S ∩ T ) = co(S ∪ T ).
Note. If S and T are neighborhoods of 0, then S and T are equicontinuous subsets of
E′, and thus compact in the σ(E′, E) topology [49, p.84]. Then co(S∪T ) = co(S∪T ).
Definition A.20. Let (E,P ) be an ordered real vector space.
(1) A ⊆ (E, P ) is decomposable if for all a ∈ A there exists a1, a2 ∈ A ∩ P such that
a = λ1a1 − λ2a2 for some λ1, λ2 ≥ 0 with λ1 + λ2 = 1.
(2) The decomposable kernel of A, the largest decomposable set in A, is
D(A) = co(−(A ∩ P ) ∪ (A ∩ P )).
(3) The order convex hull of B, the smallest order convex set in E containing B, is
F (B) = (B + P ) ∩ (B − P ).
Theorem A.21. Let V be a symmetric, convex neighborhood of 0 in an ordered real
topological vector space (E, P, τ). Then V π = V and
(1) (F (V ))π = D(V π) (i.e., (F (V )) = D(V )),
(2) (D(V ))π = F (V π) = F (V ),
where F (V ) denotes the order convex hull of V in (Ed, P d), that is,
F (V ) = (V + P d) ∩ (V − P d).
Proof. That V π = V follows from the hypotheses on V .
(1) V + P and V − P are convex neighborhoods of 0. Then, using Propositions A.16
and A.19, (F (V )) = ((V + P ) ∩ (V − P )) = co((V + P ) ∪ (V − P )) =
co((V ∩ P ) ∪ (V ∩ (−P ))) = co(−(V ∩ P ′) ∪ (V ∩ P ′)) = D(V ).
(2) Let f ∈ (D(V ))π. Then f (x) ≤ pV (x) for all x ∈ P , where pV denotes the
Minkowski functional of V . By Theorem A.2 (Bonsall), there exists a linear func-
tional g on E such that f (x) ≤ g(x) and g(y) ≤ pV (y) for all x ∈ P and all y ∈ E.
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Then f ≤ g and g ∈ V . Similarly, considering −f instead of f , one can find
h ∈ V such that h ≤ f . Then f ∈ F (V ), and hence (D(V ))π ⊆ F (V ).
Conversely, let f ′ ∈ F (V ), x ∈ D(V ), and suppose x = λ1x1 − λ2x2 and
h′ ≤ f ′ ≤ g′ where λ1, λ2 ≥ 0, λ1 + λ2 = 1, x1, x2 ∈ V ∩ P , and h′, g′ ∈ V . To
complete the proof, we must show that f ′(x) ≤ 1. Note that f ′(x1) ≤ g′(x1) ≤ 1
and f ′(−x2) ≤ h′(−x2) ≤ 1 (since V is symmetric, −x2 ∈ V ). Hence
f ′(x) = λ1f′(x1) + λ2f
′(−x2) ≤ λ1 + λ2 = 1.
Definition A.22. If p is a positively homogeneous, sub-additive functional on a real
normed space (E, ‖·‖) such that∞∑
n=1p(xn) < ∞ implies both that x =
∞∑n=1
xn exists and
p(x) ≤∞∑
n=1p(xn), then p is called a gauge. It is the Minkowski functional of the set
B = x ∈ E : p(x) ≤ 1.
If p has the weaker property that p(x) ≤∞∑
n=1p(xn) whenever x =
∞∑n=1
xn, then p is called
a pre-gauge.
We say the convex set B (containing 0) is a gauge set, or a pre-gauge set, if pB has the
corresponding property.
Note. It may happen that B is a proper subset of x : pB(x) ≤ 1.
Proposition A.23. Let A and B be closed convex sets containing 0 in the real normed
space (E, ‖·‖).
(1) A and B are pre-gauge sets.
(2) If B is complete and bounded, then B is a gauge set.
(3) If B is a gauge set, the A + B and co(A ∪ B) are pre-gauge sets.
Proof.
(1) Note first that pA is lower semi-continuous if and only if A is closed, since
x ∈ E : pA(x) ≤ α = αA for 0 < α < ∞ and x ∈ E : pA(x) = 0 =⋂
α>0αA.
We recall pA is lower semi-continuous if for every α ∈ R , x : pA(x) > α is open.
Thus, if x = lim sn, where sn =n∑
i=1
xi, then
pA(x) ≤ lim inf pA(sn) ≤ lim infn∑
i=1
pA(xi) =∞∑
n=1pA(xn).
(2) If B is complete and bounded, then, for some M > 0, ‖·‖ ≤ MpB ; thus∑
pB(xn) < ∞ implies∑‖xn‖ < ∞ and hence x =
∑xn exists.
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(3) Let C = co(A ∪B) and let x =∞∑
n=1xn with
∞∑n=1
pC(xn) < ∞. Let αn be such that
0 ≤ pC(xn) < αn < pC(xn) + ε/2n.
Then xn ∈ αn co(A ∪ B), so that
xn = αnλnan + αn(1− λn)bn = yn + zn
where
pA(yn) + pB(zn) = αnλnpA(an) + αn(1− λn)pB(bn) ≤ αn.
In particular,∑
pB(zn) < ∞, so that z =∑
zn with pB(z) ≤∑
pB(zn). It follows
that y =∑
yn =∑
(xn − zn) = x− z exists and hence pA(y) <∑
pA(yn) < ∞.
Thus, if pA(y) < λ and pB(z) < µ, then
x = (λ + µ)
[λ
λ + µ· λ
µ+
µ
λ + µ· z
µ
]∈ (λ + µ) co(A ∪B).
Hence pC(x) ≤ λ + µ. Since λ and µ are arbitrary,
pC(x) ≤ pA(y) + pB(z) ≤∑
[pA(yn) + pB(zn)] ≤∑
an ≤∑
pC(xn) + ε.
The argument for A + B is similar.
Definition A.24. Let A be a subset and B a gauge set of a real normed space E.
(1) d(A,B)(x) = infr ≥ 0 : x ∈ A + rB gives a distance from a point x to a set A
via a gauge set B. We will abbreviate this by dB , or just d, if the sets A and B
are understood.
(2) The closure of d is d(A,B)(x) = infr ≥ 0 : x ∈ (A + rB)−.
Theorem (Gauge Lemma) A.25. Let A, B, D be convex sets containing 0 in the real
normed space E and satisfying the following hypotheses:
(1) D + 〈B〉 ⊆ D, where 〈B〉 is the linear subspace spanned by B,
(2) d(A,B) ≤ αd(A,E1) on D for some α > 0,
(3) any one of
(a) A is closed and B is a gauge set,
(b) A is compact and B is a pre-gauge set,
(c) A is complete and B is a bounded pre-gauge set.
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Then d(A,B) = d(A,B) on D.
Proof. Denote d(A, E1) by δ. We have d ≤ d by definition. Let x0 ∈ D with d < r0 < r <
∞. Choose a sequence of positive numbers with r1 > r0 and∞∑
n=1
rn < r. Since d(x0) < r0,
we have x0 = a1 + b1 + c1 where a1 ∈ A, pB(b1) < r1, and ‖c1‖ < r2/α. Then by (1),
x0 − b1 ∈ D, and x0− b1 = a1− c implies δ(x0 − b1) < r2/α. Hence x0 − b1 = a2 + b2 + c2
where a2 ∈ A, pB(b2) < r2, and ‖c2‖ < r3/α.
Continuing by induction we obtain sequences (an), (bn), and (xn) such that (an) ∈ A,
pB(bn) < rn, ‖cn‖ < rn+1/α, and x0 = an + (b1 + b2 + · · ·+ bn) + cn.
Thus, if (3)(a) holds, there exists b =∑
bn with pB(b) ≤∑
pB(bn) < r. Hence (an)
converges to a ∈ A and so x0 ∈ A + rB.
If (b) holds, then a subsequence (am) converges in A and so∑
bn = b exists with
pB(b) < r.
In case (c) we have ‖an+1 − an‖ = ‖bn=1 + cn+1 − cn‖ ≤ Mrn+1 + 1α(rn+1 + rn), where
M is a bound for B. Thus (an) is Cauchy, so that (an) converges to a ∈ A and consequently
b =∑
bn exists with pB(b) < r.
Definition A.26. Let B and D be convex sets containing 0 in the real normed space E.
We say B is D-regular if pB = pB on D. We say B is regular if B is E-regular.
Corollary A.27. Let D be a closed convex cone in the real normed space (E, ‖·‖), and
let B be a pre-gauge set with ±B ⊆ D. If D1 ⊆ αB for some α ≥ 0, then B is D-regular
and D1 ⊆ α′B for all α′ > α.
Proof. Apply the Gauge Lemma with A replaced by 0. Then d, d, δ are pB, pB , and ‖·‖.
Thus D1 ⊆ αB says d ≤ αδ, so that the conclusion follows.
Definition A.28. Suppose (E, P ) is an ordred real vector space. We say P ∈ E is
generating if E = P − P .
Definition A.29. Suppose (E,P, ‖·‖) is an ordered real normed space.
(1) P ⊆ (E,P, ‖·‖) is α-generating if α > 0 such that for all x ∈ E, there exist
x1, x2 ∈ P with x = x1 − x2 and ‖x1‖+ ‖x2‖ ≤ α ‖x‖.
(2) (E, P, ‖·‖) is approximately α-generated if P is α′-generating for all α′ > α.
(3) P ⊆ (E,P, ‖·‖) is α-normal if x ≤ y ≤ z in (E,P ) implies ‖y‖ ≤ αmax‖x‖ , ‖z‖.
115
Note. α ≥ 1 in both definitions (1) and (3).
Proposition A.30. Suppose (E,P, ‖·‖) is an ordered real normed space.
(1) P is α-generating if and only if Σ ⊆ αD(Σ) where Σ is the closed unit ball of E.
(2) P is α-normal if and only if F (Σ) ⊆ αΣ.
Proof. These follow directly from the definitions.
Theorem (Grosberg-Krein) A.31. Let (E,P, ‖·‖) be an ordered real normed space
and α ≥ 0. Then (E′, P ′, ‖·‖) is α-generating if and only if (E,P, ‖·‖) is α-normal.
Proof. Let Σ, Σ′ denote the closed unit balls in E and E′ respectively. By Theorem A.21,
and since Σπ = Σ = Σ′,
(F (Σ))π = D(Σπ) = D(E′).
Then (E, ‖·‖ , P ) is α-normal ⇐⇒ F (Σ) ⊆ αΣ ⇐⇒ (F (Σ))π ⊇ (αΣ)π = 1αΣπ ⇐⇒
αD(Σ′) ⊇ Σ′ ⇐⇒ (E′, ‖·‖ , P ′) is a-generating.
Theorem (Ando-Ellis) A.32. Let (E, P, ‖·‖) be an ordered real Banach space. Then
(E′, P ′, ‖·‖) is α-normal if and only if (E,P, ‖·‖) is approximately α-generated.
Proof. From Proposition A.16, (Σ′+P ′) = (Σ′)∩ (P ′) = −(Σ∩P ). Note that (Σ′+P ′)
is σ(E′, E)-closed.
In view of the bipolar theorem, we have that
(E′, ‖·‖ , P ′) is α-normal⇐⇒ F (Σ′) ⊆ αΣ′ ⇐⇒ (Σ′+P ′)∩(Σ′−P ′) ⊆ αΣ′ ⇐⇒ (Σ′+P ′)∩
(Σ′ − P ′) ⊇ 1αΣ ⇐⇒ co(Σ′ + P ′) ∪ (Σ′ − P ′) ⊇ 1
αΣ ⇐⇒ co−(Σ ∩ P ) ∪ (Σ ∩ P ⊇1αΣ ⇐⇒ D(Σ) ⊇ 1
αΣ ⇐⇒ Σ ⊆ αD(Σ). But B = D(Σ) is a regular pre-gauge set
by Corollary A.27, so Σ ⊆ αD(Σ) ⇐⇒ Σ ⊆ α′D(Σ) for all α′ > α ⇐⇒ (E, ‖·‖ , P ) is
approximately α-generated.
Definition A.33. Suppose (E,P ) is an ordered real vector space.
(1) A ⊆ (E,P ) is o-convex if it is both convex and order convex.
(2) A topology τ on (E,P ) is locally order convex if it admits a neighborhood base at
0 of order convex sets.
(3) A topology τ on (E,P ) is locally o-convex if it is both locally convex and locally
order convex.
116
Proposition A.34. Let (E,P, ‖·‖) be an ordered real normed space. If there exists α ≥ 1
such that P is α-normal in (E, ‖·‖), then the ‖·‖-topology is locally o-convex.
Proof. P α-normal implies F (Σ) ⊆ αΣ. Then the family of sets of the form
F (rΣ) : r > 0 = rF (Σ) : r > 0 is a neighborhood base at 0 of o-convex sets with
respect to the norm topology.
Definition A.35. The ordered real topological vector space (E,P, τ) has the open de-
composition property if each V ∩ P − V ∩ P is a τ -neighborhood of 0 whenever V is a
τ -neighborhood of 0. We say P gives an open decomposition in (E, τ).
Suppose we are given an ordred real topological vector space (E,P, τ) with E = P −P .
Let U be a neighborhood base at 0 of circled sets. For all U ∈ U , U ∩P −U ∩P is circled
and absorbing in E. Then U ∩ P − U ∩ P : U ∈ U uniquely determines a topology τD
on E, called the vector topology with the open decomposition property associated with τ .
If τ is locally convex, so is τD, and τD will be called the locally decomposable topology
associated with τ .
Proposition A.36. Suppose (E,P, τ ) is an ordered real topological vector space with
E = P − P . τD is the greatest lower bound of all vector topologies which are finer than τ
and have the open decomposition property, and τ gives an open decomposition if and only
if τ = τD.
Proof. Follows from the definitions.
Again, suppose we are given an ordred real topological vector space (E,P, τ ). Let U
be a neighborhood base at 0 for τ . Let F (U ) = F (V ) : V ∈ U . There exists a unique
locally order convex topology τF such that F (U ) is a neighborhood base at 0 for τF . τF
is called the locally order convex topology associated with τ . If τ is locally convex, τF is
a locally o-convex topology, called the locally o-convex topology associated with τ .
Proposition A.37. Suppose (E,P, τ ) is an ordered real topological vector space with
E = P − P . τF is the least upper bound of all locally order convex topologies coarser than
τ .
Proof. Follows from the definitions.
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Definition A.38. Let (E,P ) be an ordered real vector space, and let τ be a vector
topology on (E,P ) with a neighborhood base at 0 of circled sets.
(1) S(U ) = S(V ) : V ∈ U
(2) Let τS denote the vector topology for which S(U ) is a neighborhood base at 0.
Theorem A.39. Let (E,P, τ) be an ordered real topological vector space such that E =
P − P . Then
(1) τS is locally order convex and has the open decomposition property.
(2) τF D = τS = τDF
Proof.
(1) Note that if V + V ⊆ V , then we have that S(V ) + S(V ) ⊆ S(V ) and 2S(V ) ⊆
S(V ) ∩ P − S(V ) ∩ P ; hence τS has the open decomposition property. Now S(U )
is a neighborhood base at 0 for τS where U is a neighborhood base of circled sets
for τ at 0. Then S(U ) consists of circled sets. Since they are also order convex, τS
is also locally order convex.
(2) Since F (V ) ⊇ S(V ), τF ≤ τS (i.e., τS is finer than τF ). Since V ∩P −V ∩P ⊆ S(V )
whenever V + V ⊆ V (recall the V are circled here), τS ≤ τD. Thus τF ≤ τS ≤ τD.
From (1), τFD ≤ τS ≤ τDF . It remains to show τDF ≤ τF D.
Take a τDF -neighborhood of 0 in E, say F (V ∩P −V ∩P ), where V is a circled
τ -neighborhood of 0. Then, suppose x = y − z ∈ F (V ) ∩ P − F (V ) ∩ P with
y, z ∈ F (V ) ∩ P . Then there exist u, w ∈ V such that 0 ≤ y ≤ u and 0 ≤ z ≤ v.
In particular, u,−v ∈ V ∩ P − V ∩ P since 0 ∈ V ∩ P , and x = y − z ∈ [−v, u] ⊆
F (V ∩P−V ∩P ). Thus F (V )∩P−F (V )∩P ⊆ F (V ∩P−V ∩P ), so F (V ∩P−V ∩P )
is a τFD-neighborhood of 0, hence τF D ≥ τDF .
Corollary A.40. Let (E, P, τ) be an ordered real topological vector space. If τ is locally
order convex and P gives an open decomposition, then τ = τS .
Proof. From the hypotheses, τ = τF = τD. Hence τFD = τD = τ , so by Theorem A.39(2)
τ = τS .
118
Definition A.41. Let (E,P ) be an ordred real vector space.
(1) A locally convex topology τ on (E,P ) is a locally solid topology if τ admits a
neighborhood base at 0 consisting of solid sets.
(2) A norm ‖·‖ on (E,P ) is called absolute-monotone if −u ≤ x ≤ u in E implies
‖x‖ ≤ ‖u‖.
Since V is solid if and only if V = S(V ), a locally convex topology on (E,P ) admits
a neighborhood base of solid sets if and only if τ = τS . In fact, since S(V ) is circled
and convex whenever V is circled and convex, τ is locally solid if and only if it admits a
neighborhood base at 0 of circled convex and solid sets.
Definition A.42. A norm ‖·‖ on an ordered real vector space (E,P ) is called a regular
(or Riesz) norm if it satisfies the following two conditions:
(1) ‖·‖ is absolute-monotone: −u ≤ x ≤ u in E implies ‖x‖ ≤ ‖u‖.
(2) For each x ∈ E with ‖x‖ < 1 there exists u ∈ E with ‖u‖ < 1 such that −u ≤ x ≤ u.
Proposition A.43. Let U be the open unit ball of an ordred real normed space (E,P, ‖·‖).
Then ‖·‖ is a regular norm if and only if U is solid.
Proof. Follows from the definitions.
Proposition A.44. Let (E,P, ‖·‖) be an ordered real normed space with (E′, P ′, ‖·‖) the
dual real Banach space with induced cone. Then Σ′ solid, where Σ′ is the closed unit ball
of E′, implies P ′ is 2-normal and 3-generating.
Proof. Suppose x ≤ y ≤ z in E ′. Let t = max‖x‖ , ‖z‖. WLOG, t 6= 0, sox
t≤
y
t≤
z
tand
x
t,z
t∈ Σ′. Since Σ′ is absolutely dominated, there exist u1, u2 ∈ Σ′ ∩ P ′ such that
−u1 ≤x
t≤ u1 and −u2 ≤
z
t≤ u2. Then −1
2(u1 + u2) ≤
y
2t≤ 1
2(u1 + u2), so
y
2t∈ Σ′
since Σ′ is absolutely order convex. Thus ‖y‖ ≤ 2t, and so P ′ is 2-normal.
Given x ∈ E′, x 6= 0,x
‖x‖∈ Σ′. Since Σ′ is absolutely dominated, there exists b ∈ Σ′∩P ′
such that −b ≤ x
‖x‖≤ b, so −‖x‖ b ≤ x ≤ ‖x‖ b. Then x = ‖x‖ b − (‖x‖ b − x) and
‖ ‖x‖ b‖+ ‖ (‖x‖ b − x)‖ ≤ 3 ‖x‖. Thus P ′ is 3-generating.
119
Theorem (Davies) A.45. Let (E, P, ‖·‖) be an ordered real normed space, and let U
and Σ denote its open and closed unit balls. Let (E′, P ′, ‖·‖) be the real Banach dual space
with dual cone P ′, and let U ′ and Σ′ denote its open and closed unit balls. Consider:
(1) ‖·‖ is a regular norm in (E, P );
(2) U is solid in (E, P );
(3) ‖·‖ is a regular norm in (E′, P ′);
(4) U ′ is solid in (E′, P ′);
(5) Σ′ is solid in (E′, P ′);
Then (1) ⇐⇒ (2) =⇒ (3) ⇐⇒ (4) ⇐⇒ (5).
Further, if (E, ‖·‖) is a Banach space and P is closed, then (3) =⇒ (1), so (1)—(5) are
mutually equivalent.
Proof. (1) ⇐⇒ (2) and (3) ⇐⇒ (4) is Proposition A.43. (5) =⇒ (3) also follows easily
from the definitions.
(3) =⇒ (5): Since ‖·‖ is absolutely monotone on (E′, P ′), Σ′ is certainly absolutely
order convex. Now let f ∈ Σ′, f 6= 0. For all n ∈ N , there exists gn ∈ E′ with ‖gn‖ < 1
such that −gn ≤f
(‖f‖+ 1n)
≤ gn. Let hn = (‖f‖ + 1n)gn. Then ‖hn‖ < 1 + 1
n for all
n ∈ N . By Alaoglu’s Theorem, (hn) has a σ(E′, E)-cluster point, say h. Then ‖h‖ ≤ 1
and −h ≤ f ≤ h. Thus Σ′ is absolutely dominated, and thus solid.
(2) =⇒ (5): Since U is solid, U = S(U). By Theorem A.12, we then have Uπ =
(S(U))π = S(Uπ). Since Uπ = U = Σ′, Σ′ = S(Σ′), so Σ′ is solid by Proposition A.10.
We now assume (E, ‖·‖) is a Banach space and P is closed.
(3) =⇒ (1): Since ‖·‖ is a regular norm on (E′, P ′), (2) =⇒ (3) gives that ‖·‖ on (E′′, P ′′)
is a regular norm. In particular, ‖·‖ is absolutely monotone on E′′, so the norm in E must
be absolutely monotone also since (E, ‖·‖) is isometrically isomorphic to a subspace of
(E′′, ‖·‖). Thus we still only need to show U is absolutely dominated.
Let τ be the norm topology on E. From (3) =⇒ (5), Σ′ is solid, so by Proposition A.44
P ′ is 2-normal and 3-generating. From Theorems A.31 and A.32, (E, P, ‖·‖) is 3-normal
and α-generating for every α > 2. By Proposition A.34, τ is locally o-convex. Since τ is
α-generating for α > 2, E = P − P . Let A be a τ -neighborhood of 0. Then there exists
β > 0 such that βΣ ⊆ A. Also, τ α-generating for α > 2 implies 1αΣ ⊆ Σ ∩ P − Σ ∩ P .
120
Then
β
αΣ = β(
1
αΣ) ⊆ β(Σ ∩ P − Σ ∩ P ) = βΣ ∩ P − βΣ ∩ P ⊆ A ∩ P −A ∩ P,
so A ∩ P − A ∩ P is a τ -neighborhood of 0. Thus (E,P, τ ) has the open decomposition
property. By Corollary A.40, τ = τS, hence S(λU) : λ > 0 = λS(U) : λ > 0 is a
neighborhood base at 0 in (E, τ ). Notice then that αS(U) ⊆ βS(U) whenever 0 < α < β.
On the other hand, since S(Σ′) = Σ′ (by (5)), from Theorem A.12 (s(U))π = S(Uπ) =
S(U) = S(Σ′) = Σ′, so that (S(U))ππ = Σ. From the bipolar theorem, Σ ⊆ S(U). Hence
Σ ⊆ S(U) ⊆ (1+ε)S(U) for all ε > 0. Then U ⊆ S(U), i.e., U is absolutely dominated.
121
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Vita
Walter J. Schreiner was born in St. Paul, Minnesota on April 1, 1945. After graduatingfrom high school, he spent the 1963–64 academic year as a Novice with the ChristianBrothers, a Catholic Religious Order, concomitantly taking religious education courses asa student at St. Mary’s College of Minnesota. He has been a member of the ChristianBrothers continuously since August, 1968.
During the years 1964–69, Walter was a student at the College of St. Thomas in St.Paul, Minnesota, majoring in Mathematics. He became a member of Delta Epsilon Sigma,the national Catholic honor fraternity, in 1966, received a Bachelor of Arts degree, Magnacum Laude, in 1968, and received a Master of Arts in Teaching degree in 1969.
Walter attended the University of Notre Dame during the summers of 1970–73, receivinga Master of Science degree in Mathematics in 1973. He also attended the University ofMinnesota in 1977–78 as a graduate student in Educational Administration, and has beena graduate student in Mathematics at the University of Illinois at Urbana-Champaign sinceAugust, 1989.
During his Senior year in college (1967–68), Walter taught seventh and eighth gradeMathematics at St. Kevin’s School in Minneapolis Minnesota. From 1968–70, he wasa Mathematics teacher, coach, and Assistant Athletic Director at Grace High School inFridley, Minnesota.
For the next 15 years, 1970–85, Walter was a member of the faculty of Cretin HighSchool in St. Paul, Minnesota. He taught at least one class in each of those years, was abaseball and soccer coach, served as Athletic Director from 1971–77, and was Academic As-sistant Principal from 1977–82. Walter moved to De La Salle High School in Minneapolis,Minnesota, in 1985, where he served until 1989 as Vice Principal for Academics, Calculusteacher, and baseball coach. During the years 1984–88, he was also a member of the St.Paul-Minneapolis Archdiocesan Board of Education.
Finally, throughout his time as a graduate student, Walter has been a Teaching Assistantat the University of Illinois.
125
MATRIX REGULAR ORDERS ON OPERATOR SPACES
Walter James Schreiner, Ph.D.Department of Mathematics
University of Illinois at Urbana-Champaign, 1995Zhong-Jin Ruan, Advisor
In this thesis, the concept of the regular (or Riesz) norm on ordered real Banach spaces is
generalized to matrix ordered complex operator spaces in a way that respects the matricial
structure of the operator space. A norm on an ordered real Banach space E is regular if:
(1) −x ≤ y ≤ x implies that ‖y‖ ≤ ‖x‖; and (2) ‖y‖ < 1 implies the existence of x ∈ E
such that ‖x‖ < 1 and −x ≤ y ≤ x. A matrix ordered operator space is called matrix
regular if, at each matrix level, the restriction of the norm to the self-adjoint elements is
a regular norm. In such a space, elements at each matrix level can be written as linear
combinations of four positive elements.
After providing the necessary background material on operator spaces, especially with
respect to the Haagerup (⊗h), operator projective (∧⊗), and operator injective (
∨⊗) tensor
products, the concept of the matrix ordered operator space is made specific in such a
way as to be a natural generalization of ordered real and complex Banach spaces. For
the case where V is a matrix ordered operator space, a natural cone is defined on the
operator space X∗⊗h V ⊗h X so as to make it a matrix ordered operator space. Exploiting
the advantages gained by taking X to be the column Hilbert space Hc, an equivalence is
established between the matrix regularity of a space and that of its operator dual.
Beginning with the fact that all C∗-algebras, and in fact all operator systems, are matrix
regular, it is shown that all operator spaces of the form X∗ ⊗h V ⊗h X and CB(V,B(H))
are matrix regular whenever V is. Reverse implications are also shown in some cases. The
replacement of B(H) by an injective von Neumann algebra R is explored, leading to more
general results and some extra results regarding R′-module projective tensor products.
Complex interpolation is used to define operator space structures on the Schatten class
spaces Sp and the commutative Lp-spaces. These spaces are then shown to be matrix
regular. Some generalized Schatten class spaces are also shown to be matrix regular.
Finally, as an application, an alternative proof is presented for the Christensen-Sinclair
Multilinear Representation Theorem that depends on matrix regularity rather than on
Wittstock’s complicated concept of matricial sublinearity.