John Wiley 8t Sons, Inc.. New York
Longman Scientific & Technical
Essex CM20 21E, England
and Associated companies throughout the world.
Copublished in the United Stales with
John Wiley & Sons Inc., 605 Third Avenue, New York, NY 10158
© Longman Group UK Limited 1992
All rights reserved; no part of this publication may be reproduced, stored in
a retrieval system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written
permission of the Publishers, or a licence permitting restricted copying in
the United Kingdom issued by the Copyright Licensing Agency Ltd,
90 Tottenham Court Road, London, WlP 9HE
First published 1992
(Subsidiary) 47D30, 46E30, 46842
British Library Cataloguing in Publication Data
A catalogue record for this book is
available from the British Library
Library of Congress Cataloging-in-Publication Data
Abramovich, Y. (Yuri)
Abramovich, E.L. Arenson, and A.K. Kitover.
p. cm. -- (Pitman research notes in mathematics series, ISSN
0269-3674)
1. Banach modules (Algebra) 2. Operator theory. 3. Banach
lattices. I. Arenson, E. L. (Eugene L.) II. Kitover, A. K.
(Arkady K.) III. Title. IV. Series.
QA326.A27 1992
by Biddles Ltd, Guildford and King’s Lynn
To our parents
COO-modules and d-homomorphisms
. Banach COO-modules and their relationship with Banach and vector lattices
. Comparison of operators taking values in a vector lattice or C(K)«module
. Some applications. A generalization of Bade’s Theorem.
Properties of d~homomorphisms
An analog of the factorization theorem of G. Lozanovsky
Special algebras of operators on X*. A dual Bade theorem
10. Operators conjugate to d-homomorphisms
1 1. Independence of d—homomorphisms
Part III. Spectral properties
14. Examples
B. Proofs of Lemmas 12.6.1 and 12.6.3
References
1. Introduction
The present work was motivated by our attempt to answer the following two basic
questions:
0 What are the properties, in particular the spectral properties, of (weighted)
composition operators in the spaces of measurable scalar and vector valued functions?
0 What classes of operators in Banach spaces are “similar” to composition opera-
tors and, at least partially, inherit their properties?
A list of works devoted to the same problems would be rather extensive. VVith—
out trying to present a complete bibliography, we mention here only several articles
(see for example, [AAK, AAV1-2, Abr1—3, AL, Art, Coel—Z, Har, HP, Karol—2, KK,
Kit1—3, Lat, MaCl—Z, MCS, Mon, MW, Mys, Pet, Sha, SS, ST, Wicll). There are
apparently two reasons for the continuing interest in composition operators. First, the
results of the investigation of composition operators find applications in the theory of
singular integral equations, dynamical systems, and differential equations with delayed
time (see, for example, [KK, Mys, 35]). Secondly, the properties of composition oper—
ators are closely related to the algebraic and order structures of linear spaces, and in
the investigation of these operators there arise very interesting and subtle problems,
which lie on the borderlines between the theories of Banach algebras, Banach lattices,
dynamical systems, and analytical functions.
Our work was inspired mostly by the second reason, but as the reader will see, many
of the results presented are useful for applications as well. Among the many intertwining
routes which arise in the investigation of composition and “similar” operators, we have
chosen a route (still far from being fully travelled) connected with the property of
preserving disjointness. The results obtained along this route can be divided into three
seemingly isolated (but in fact closely related) groups. The division of our work into
three parts corresponds to these three groups of results.
In the first part we deal with an operator from one Banach lattice into another and
find conditions to guarantee that this operator may be represented as a composition
operator. More precisely, we investigate the following problem: when is the obviously
necessary condition of preserving disjointness also sufficient? In solving this problem we
1
obtain a criterion for such a representation and also obtain several sufficient conditions
which can be easily verified. The results in this part essentially supplement and improve
the results in [Abr1, AAK].
The main focus of the second part of the work is a Banach space X and a subalg‘ebra
A of L(X) (the space of all continuous linear operators) which is closed in L(X) and
is isoxnetrically isomorphic to an algebra C(K) for some compact Hausdorff space K.
In typical situations X consists of vector valued functions and A is an algebra of scalar
multipliers. The spaces with such a structure (Banach GHQ—modules) are “locally”
similar to Banach lattices. More precisely, each closed principal A-submodule X(:r),
generated by an arbitrary 1: E X, has a natural structure of a Banach lattice. In terms
of these local lattices .X (1:), we are able to define several “global” concepts such as order
ideals, the center of X, disjointness of elements in X, d--hornomorphisms (= operators
preserving disjointncss), etc.
For these objects the following main results are obtained. (1) In Theorem 6.2 we
give a broad generalization of a famous theorem of Bade [DSZ, Ch.XVII.3]. Specifically,
we prove that each linear (a priori not assumed to be continuous) operator which
maps each C(K)-invariant subspace into itself belongs to the closure A of the algebra
A 2 C(K) in the strong operator topology of the space L(X), the algebra A is a
reflexive operator algebra, and A is characterized in order terms as the center of the
space X. (2) The class of continuous d—hornomorphisms is thoroughly studied and
we give a criterion for an operator T to belong to this class. (3) On the space X *,
the conjugate of a Banach C(K)—module X, we define an appropriate structure of a
C(K1 )-rnodule and in terms of this module investigate the local order properties of X *.
In the process, the following dual version of Bade’s theorem is obtained: the center
of the module X“ coincides with the space of operators for which each a(X",X)-closed
C(K)-invariant subspace of X* is invariant. The properties of operators conjugate to
d-homomorphisrns are also investigated.
It is worth pointing out that most of the above mentioned results have natural
analogues (and, as a rule, simpler proofs) for Banach lattices. One more comment
regarding Part II is in order. There is a substantial literature devoted to the extension
2
of Bade’s theory to the framework of locally convex spaces (see for example [DR DPR,
RS, Sc, W] and the references therein). Due to some natural “boundary conditions”
we do not consider this case in the present work.
The third and final part of the work is devoted to the investigation of the spectrum
of d—endomorphisms (i.e.. d-homomorphisms of a space into itself) of a Banach lattice
or a Banach COO-module. We show that the study of the spectral properties of
a d—endomorphism is closely related to the question of how the different powers of
this endomorphism interact among themselves. To demonstrate this we introduce a
new concept of independence of two arbitrary d—endomorphisms, a concept which in
the case of Dedekind complete Banach lattices coincides with the usual disjointncss of
operators in the Dedekind complete vector lattice of regular operators. The. main result
of this part is valid for a very large class of Banach lattices and C(K)—rnodules and
reads as follows: if the powers of a d—endomorphism T are pairwise independent, then
the spectrum 0(T) is rotation invariant. This theorem considerably improves all the
previously known results in this direction due to Arendt, Hart, Antonevich7 Lebedev,
Kitover, and others. The exact references will be given in the appropriate places.
The first and the third authors would like to point out that it was E. Arenson who
has inspired. our investigation of C(K)—modules and who has contributed many original
ideas.
The authors would like to express our deep gratitude to Marc Frantz and Cindy
Jones for their unfailing readiness to help correct and improve our English. Our special
thanks go to Mehmet Orhon for his interest and many valuable suggestions regarding
the manuscript and for his advice to include in Section 9 the description of the closure
of the central subalgebras in the weak* operator topology. Finally, the third author
acknowledges with sincere thanks the financial aid and hospitality of the Mathematical
Sciences Research Institute at Berkeley, where a part of the work was done.
2. Notations and abbreviations
For the convenience of the reader we start with a short section containing the basic
notation and terminology. All the terms, symbols or definitions not explained in this
section or in a corresponding place in the text are standard and may be found in [A131],
[KA], [Sch], [V111] or [Zaa].
The following notations and abbreviations will be used throughout the work.
C (resp. R) denotes the field of all complex (resp. real) numbers. Unless otherwise
stated explicitly, all linear spaces and vector lattices are considered over C.
C U {00} is the extended complex plane; we set A ~oo = 00 for each nonzero A 6 C.
R U {—00, 00} is the extended real line.
N=:{1, 2,. . .} is the set of all natural numbers.
VL stands for an (arbitrary Archimedean) vector lattice (2 Riesz space).
BL stands for a Banach lattice.
If X is a vector lattice, then Q(X) denotes its Stone space. Recall that Q(X) is
an extremally disconnected compact Hausdorff space.
For an arbitrary extremally disconnected compact Hausdorff space Q, the sym-
bol C00(Q) denotes the Dedekind complete vector lattice (and algebra) of all ex-
tended continuous functions on Q. More precisely, Cm(Q) consists of all continuous
mappings f : Q -—) C U {00}, such that f—1(oo) is a nowhere dense subset of Q.
For arbitrary f,g 6 (700(6)) and A E C there exist uniquely determined functions
ALRemef, lflif + g and f9 in 000(9)-
The real functions from 000(6)) will be occasionally treated as functions with values
in R. U {—00, 00}
X is the Dedekind completion of a vector lattice X. Whenever appropriate, X is
assumed to be embedded in X.
M(X) := M(X) is the universal completion of a vector lattice X. It is well known
that when a. unit 6 is fixed in M(X) the latter space can be identified with C00(Q(X))
in such a way that 6 becomes the constant function one.
A representation j of a vector lattice X is any order isomorphic embedding
j 2 X —+ Coo(Q(X)) which agrees with the canonical isomorphism between the Boolean
4
algebras of all bands in X and all clopen subsets of Q(X).
X2 is the characteristic function of a set E.
If X is a VL and 11,12 6 X, then the notation (Bldl'g says that the elements at]
and 3:2 are disjoint, i.e., |1z1| /\ |;I:2| = 0.
Ad is the disjoint complement of a subset A of an arbitrary vector lattice X, i.e.,
Ad 2: {1: E X :1da Va 6 A}. Accordingly, A“ = (Ad)d.
Xz is the smallest (order) ideal containing the element a: of a vector lattice X.
X(2:) is the smallest norm-closed ideal containing the element a: of a Banach
lattice X.
A linear operator T between two vector lattices X and Y is called a d-homo-
morphism or, equivalently, a disjointness preserving operator if a: 1d 12 in X implies
TibldTitz in Y.
dh(X,Y) denotes the collection of all regular d—homomorphisms from a vector
lattice X into a vector lattice Y. Accordingly, dh+(X, Y) = {T E dh(X, Y) : T Z 0} is
the subset of all lattice homomorphisms. Instead of dh(X,X) we simply write dh(X).
Z(X) is the center of the vector lattice X, i.e., the space of all linear operators
T : X —r X for which there exists A = /\(T) > 0 such that {Tel S Mel for all I E X.
We refer to [Wil] and [Wic2] for the standard facts about the centers. We will often
use the fact that for an arbitrary Banach lattice X, its center Z(X) is algebraically
isomorphic to the space C(K) of all continuous functions on an appropriate compact
Hausdorff space K. This implies, in particular, that Z(X) is a closed commutative
subalgebra of the algebra L(X) of all continuous linear operators on X .
Part I. Multiplicative representation
on vector lattices
Let X be an arbitrary VL and f E Z0?) (hence fa: E X for each a: 6 X).
Then there exists a unique function f E C(Q(X)) such that for each representation
j : X ——> C00(Q(X)) the following identity holds: jfa' 1- f-jx, at E X.
Allowing ourselves a bit of looseness we will identify f and f:
For an arbitrary subset A of X we denote by supp(A) the support set of A, i.e.,
the smallest clopen subset Q,4 of Q(X), such that XQAJ: = a: for each :5 6 A. For any
a: E X we put Q; 2: supp(1') := supp({a:}).
If Q0 is a clopen subset of Q and a: E Cm(Qo), then, without any special ex»
planations, we may (and will) consider as also as an element of the space C0002), i.e.,
whenever appropriate, we will identify 000(Q0) with the corresponding band X00 00°(Q)
in 000(Q).
3.1. Definition. Let X and Y be two vector lattices, let T : X —> Y be a
linear operator, and let E = ET = supp (TX). We say that T admits a multiplicative
representation (mm) if there exist
1) representations jl :X ——> Coo(Q(X)) and j2 1 Y —> CW(Q(Y)),
2) afunction e E Cm(Q(Y)), and
3) a continuous mapping so : E —r Q(X),
such that supp(e) = E and
(J'2Tx)(¢1) = 6(9) ' (j1$)(90(q)) (3-1)
for each x E X, and each q E Q(Y) satisfying 0 < |e(q)] < 00.
The last sentence needs some comments. Formally speaking, the function (jlx)o¢p
belongs to the space COO(E); however, in accordance with our agreement, it is treated
as an element of Cm(Q(Y)). The same should be said about the function e ~j151: 0 1,9.
6
Also note that the value (c - jm ogo)(q) of the latter function at a point 4 E E coincides
with 6(q) ~(j1$)(go(q)) at least at each point q of the dense open subset E0 == {q E E :
0 < |e(q)l < 00} and is defined by continuity at other points. Whenever there is no
danger of ambiguity we will shorten (3.1) as ng$ = e-J'grogo, or even as Ta, = e—roco
provided the representations j] and jg are fixed.
3.2. Definition. Let X,Y and T be as above. We say that T admits a local
XII, of the
multiplicative representation (l.m.r.) iffor each x’ E X... the restriction T
operator T to the ideal XI: generated by a’, admits a m.r.
In the case of real vector lattices it was shown in [AVKll that regular d—homomor-
phisms, and only they, admit a l.m.r. It was also shown in [Abrl] that if X and Y are
Banach lattices, then each continuous d—homomorphism is regular, and hence admits a
l.m.r. On the other hand, in [Abr2] there was constructed an example of a Dedekind
complete Banach lattice X and of a continuous d-homomorphism T : X —> ,X such
that T does not admit a m.r.
The purpose of this section is to generalize and[or improve the results in [Ahrl]
and [AAK] concerning the existence of a l.m.r. for regular d—homomorphisms, and
to investigate systematically the question of the existence of a m.r. Notice that each
operator admitting a l.m.r. is obviously an order bounded d—homoxnorphism.
Our first theorem shows that the problem of multiplicative representation for order
bounded d~homomorphisms (over C) can be reduced to that for lattice homomorphisms.
Consequently the choice of a scalar field becomes inessential, and this immediately
allows us to simplify many subsequent proofs by considering the real case only.
3.3. Theorem. Let X and Y be vector lattices, and T : X we Y a linear operator.
The following conditions are equivalent.
(1) T is an order bounded d-homomorphism.
(2) [all S |$2| implies that |T:r1| _<_ |T2:2|.
Each of these equivalent conditions implies that T is regular and, moreover, T admits
a unique factorization T = VT] (1%), where T1 : X —> Y is a. lattice homomorphism,
V E Z07) and {VI = XE” (The support set E = ET is introduced in Definition 3.1.)
Proof. Obviously the condition xldxz is equivalent to the following: [$1] g
|:::] + /\.’L'2I for all A E C. Hence, (2) :s (1).
To prove (1) => (2) we can assume (since T is order bounded) that X and Y
have strong units. This implies that there exist two compact Hausdorff spaces K1 and
K2, such that X and Y may be identified with dense sublattices of C(K1) and C(Kg),
respectively, that Xx, E X, Xx; 6 Y, and that T is continuous with respect to the usual
sup-norms.
Let t be an arbitrary point in K2 such that the linear functional 1 : .r H (T$)(t)
is nonzero on X. To establish (2) it is enough to show that [1(a)] S |l(:r2)| provided
[31] S [1'2]. Let p. be a regular Bore] measure in C(K1)* which represents 1 and let F
be the support of y. We are done if we verify that F is a singleton. Assume by way of
contradiction that F is not a singleton. Then there exist two positive functions In , u; E
C(Kl) such that u; duz and fum dn 7E 0 for m = 1,2. Now pick zm 6 X+ (m = 1,2)
that are close enough (in norm) to um and put .‘L‘m :: (2m —— ”2m — u,,,HXKl)+. It is
plain to see that (121,32 6 X and 3:1 d372, but (Tx1)(t) 96 O, (T$2)(t) ¢ 0 contrary to
the fact that T is a d—homomorphism. This establishes the equivalence (1) <=>(2).
Now we will proceed to prove the validity of the factorization (7t). Essentially,
we have already proved (using the fact that F is a singleton) that |T(11 + x2)l =
[Tell + lTIzl for all $1,:r2 E X+. This defines an additive operator 3: H ITxl on X+
that can be extended in a natural way to a linear operator T1 : X ——» Y.
Let us fix a representationj : Y ——r m(Q(Y)), and for each m E X+ introduce
a unimodular function V, = ij/IjTI] on Q71. Then the identity |jT(m1 + x2)| .—.—.
[ijl + leM, which is true for all $1,:r2 E X+, obviously implies that VI, : VI2
on QT,1 fl QT“, This and the density of the open subset U{QTI : a: 6 X4} of the
extremally disconnected compact space E allow us to “glue” together all the functions
Vt, i.e., there exists a unique function V E C(Q(Y)), such that [V] = XE and VlQT: =
V; for all ,r E X+, Thus, the factorization T = VT; is established, T1 being a lattice
homomorphism. This factorization implies immediately that T is a regular operator,
and therefore the proof of Theorem 3.3 is complete. I
Assume now that X and Y are Dedekind complete vector lattices and that a given
d-homomorphism T : X ——> Y admits a m.r. (3.1), and let ET :2 E, e, and (p be
as in Definition 3.1. Recall that a continuous mapping (,0 : E —> Q(X) generates in
a canonical way an algebraic homomorphism 'y = 7,, from C(Q(X)) to C'(Q(Y)) [as
always we identify C(E) with the corresponding band in C(Q(Y))] by the formula
7f=f°so-
It is immediate from the multiplicative representation (3.1) that for all f 6 C(Q(X ))
and for all a: E X the following identity holds:
T(f - r) = 7(f) - T($)- (3.2)
Since C(Q(X)) = Z(X) and C(Q(Y)) = Z(Y) we can reformulate (3.2) as follows. If
T admits a m.r., then there exists an algebraic homomorphism 'y : Z(X) -——> Z(Y) such
that
TU '1) = 7(f) ' T($) (f E Z(X),$ E X) (3-3)
It was an interesting and important observation due to Hart [Har], that (3.3) may
be true even in the absence of (3.1). More precisely, Hart proved that if X and Y are
Dedekind complete vector lattices and T E dh(X,Y), then there exists an algebraic
homomorphism ’YT : Z(X) —r Z(Y) such that (3.3) holds.’[ Note further that 711‘ in
turn generates in a standard way a continuous mapping 90,. 2 ET —> Q(X), and it is not
difficult to verify that whenever T admits a m.r. (3.1), the mapping (,0 in (3.1) must
coincide with SOT. An alternative and independent proof of a broad generalization of
Hart’s result will be given in Theorem 6.7; however, for our present purposes, we need a.
partial generalization of this result for arbitrary vector lattices X and Y (not assumed
Dedekind complete), for which the problem of the existence of (for is more difficult.
Let us agree (for brevity) on the following symbolism. For an arbitrary y E Y,
for an arbitrary clopen subset F of Q(Y) containing the support set Qy, and for an
1 Notice that in view of the existence of a local m.r. for TIXZ for each x G X, the
existence of 7T is not that surprising. In fact, 7T may by pieced together using these
local representations, which are sufficient to recapture 77', but which are not sufficient
to guarantee the existence of a (global) multiplicative representation.
arbitrary g E C(F), we denote by g ' y an element 91 -y 6 Y, where g; is an arbitrary
function in C(Q(Y)) = Z(Y) which coincides with g on F. Obviously, this function
g - y is independent of the choice of g1.
3.4. Theorem. Let X and Y be arbitrary vector lattices, T 6 dh(X,Y), and
E = ET. Then there exists a (not necessarily unique) mapping cp : E —> C(Q(X)), such
that
T(f;r) =fo<p-T:r (3.4)
whenever f E Z(X),:r E X and f1: 6 X. The set ‘I’T of ail such mappings coincides
with
{#95: S 6 (Niki), E3 = E and SIX = T}
Proof. To establish (3.4) we are going to use the factorization T : VT] obtained
in Theorem 3.3. Notice further that in View of a theorem due to Lipecki [Lip] and
Luxemburg—Sehep [LS] the d-homomorphism T1 from dh+(X, Y) can be extended to
a d-homornorphism S] E dh+(X,Y), in such a way that E3, : E. Letting 5' 2 V5,,
we obtain an extension of T which is a regular d—homomorphism between Dedekind
complete vector lattices X and Y. Consequently, the above mentioned theorem of Hart
may be applied to S, and there exists therefore a corresponding mapping (,0 == 905.
Obviously (,9 satisfies (3.4), since SIX = T.
Now, let (,9 be an arbitrary mapping satisfying (3.4). For each 2 E X there exist
f E Z(}i') and z E X+ such that z = f3. Let us define 52 := fogo . Ta: and show that
this expression determines a well—defined operator from X into Y.
To this end it is enough to show that if 23nd fmxm = 0 (*), where V E N, fm E
Z(X) and rm 6 X+, then 2;: fm 0 so - TIE", = 0. Let :1: :2 me. Hence, rm 2 gma:
for an appropriate gm 6 ZOE), and therefore (*) becomes (2 fmgm) - :c = 0. Since 30
satisfies (3.4) the previous equality implies that (megm) 0 go ' Trr 7: 0, and by the
same token (since em = gmar) it implies that 2 fm o Lp ~ Trm = 0. It is obvious that
s e dh(Ji',f/), SlX = T, E3 = E, and 995 = e l
10
The next corollary is just a reformulation, for further reference of our previous
discussion about relationship between (3.1) and (3.4).
3.5. Corollary. IfT admits a m.r. (3.1), then so 6 (PT.
The next result, which again is a reformulation in our notation of Proposition C
of [Abrl], will be needed later on.
3.6. Theorem. Let T G dh(X, Y) and T = VT} as in Theorem 3.3. Then for
each (,9 E ‘I’T the operator T admits a l.m.r. of the following special type. If for an
:1; E X+ we pick up representations j] : X —-» Coo(Q(X)) and j2 : Y —+ 00(Q(Y))
such that j1(.1:) = X01 and j2(|T:r|) 2 X0”, then the restriction ofT to X; admits the
following representation:
(J'2TZ)(q) = V(a)(jlz)(s-9(q)), 2 E Xx, q 6 QT:-
3.7. Remark. (1) The set <1), can be infinite. For example, let X = ”[0, 1], Y =
C and T1 = 3(0) (1 E X). Then ET consists of a single point go, and X = C(Q),
where Q = Q(X) is the corresponding Stone space. Each extension of T to an operator
5 E dh(X, C) is of the form 52 = 2(p) (z 6 X), where p is an appropriate point from
Q, for which 905(qo) = p. Clearly, there is an infinite number of such points p.
(2) If T is order continuous, then ‘15 consists of a unique mapping (,9 = cpT and this
mapping is open. Conversely, if there exists an open mapping cp E (1),. , then T is order
continuous. The proof consists of the verification of the following simple statements.
We will assume that T 2 0. Let 31,52 6 dh+(X,Y) such that 31 IX 2 5'ng = T and
E51 = E5, = ET. Then
(b) T is order continuous (:y S] is order continuous.
(c) If (p51 and (p52 are open, then 51 = 32.
(3) Later we will show that if X and Y are Banach lattices and T 6 dh(X, Y),
then operator T“ E dh(X”, Y”) and T" is order continuous.
Now we turn our attention to the problem of the existence of a Int.
11
3.8. Theorem. Let T 6 dh(X, Y), where X and Y are vector lattices. If there
exists an I e X such that the band {T$}‘“ generated by T2: coincides with (TX)‘“,
then T admits a m.r.
Proof. From what We have already proved it follows that, without loss of gener-
ality, we may assume that T Z 0, :1: Z 0, {1}“ = X, and {Tx}dd = Y. Let cp 6 <15.
Pick up representations j1,j2 such that jla: = 1 and jom = 1. Then (by Theorem
3.6) joz = (7'12) o (,0 for each 2 6 X,. We want to show that the same formula is
valid for each 2 E X. Evidently we can assume additionally that z 2 2:, since the
linear combinations of such 2’s generate all of X. Fix such a 2. Then a: = f2 for some
f E Z(X) and thus, by Theorem 3.4 , 1 =j1x 2 f -j12 and 1 = ngz = f0 (,9 -j2Tz.
This implies that jlz and ngz are invertible elements of the algebras Cw(Q(X)) and
COO(Q(Y)), respectively. Therefore, jlz = f‘1 and joz = f‘1 o (p = jlz o (p. I
3.9. Remark. In both examples (constructed in [Abrl], [Abr2]) of lattice homo-
morphisms that do not admit a m.r., the domain vector lattice X has weak units and
(TX)‘“ 2 Y, but for no weak unit It in X is its image Tu a weak unit in Y.
Using Theorem 3.8 we can infer the following criterion for the existence of a m.r.
3.10. Theorem. Let X and Y be vector lattices and T E dh(X, Y). The following
statements are equivalent.
(1) T admits a m.r.
(2) There exists an order dense ideal X1 in M(X) containing X and an operator
T1 6 dh(X1,M(l7)), such that the restriction T1|X = T and T1 satisfies the condition
of Theorem 3.8.
Proof. Obviously (2) => (1). Conversely, assume (1), i.e., that formula (3.1)
is valid for T. Without loss of generality we may assume that (TX )d‘i = Y, that
as usual X is embedded into COO(Q(X)), Y is embedded into COO(Q(Y)) and that
Ta: = e-a: 050 for all 1: E X. Now if we put X1 := X +C(Q(X)) and define an operator
T1 : X1 —> C00(Q(Y)) by the formula T12: = c - z o p, then we can easily see that X1
and T1 satisfy (2).
12
For Dedekind complete vector lattices the previous criterion can be reformulated in
“internal” terms. To do so we recall the following definition first introduced in [Abr3].
3.11. Definition. Let X be a Dedekind complete VL and L be a subset of X+.
We will call L laterally directed if for each 131,12 6 L there exists an .‘L‘ E L such
that both 1:; and 11:2 are components of 1:, that is,
Ia‘ — Iii A |$.| = 0 (i = 1,2).
It is easy to see that L is an increasing directed set (possibly with 3. largest element)
and for each 2:1 S on from L the element $1 is a component of 12.
3.12. Theorem. Let X and Y be Dedekind complete vector lattices and T belongs
to dh(X,Y). The following statements are equivalent.
(1) T admits a m.r.
(2) There is a laterally directed subset L of X+ such that ('TL)’id = (TX)dd.
Proof. (1) => (2) Let T admit a m.r. (3.1). Using this representation we can
define the following set
L ={:1: 6 X4. :jlr isacomponent of 1 2 X000}.
It is plain to see that L is laterally directed and (TL)'“ = (TX)‘“, i.e., condition
(2) holds.
Conversely, assume condition (2) Without loss of generality we can assume that
the operator T Z 0 and that (TX )d‘i = Y. Moreover, we can assume for simplicity that
L“ = X. Lateral directedness of L implies (see [Vul], Lemma V.2.1 and a footnote
to it) that in MO?) there exists are = sup L, and similarly in Md’) there exists
yo = sup TL. Obviously 2:0 and yo are weak units in MO?) and Md”) respectively.
Pick representations j; and jg such that jla'o = X0“) and jzyo 2 X0“). We will show
now that for each .2 E X we have joa: = (jla) o (p, where so = LpT. For simplicity of
notation we identify X with le and Y with ng. Then L C C(Q(X)) and L consists of
the components of the unit in C(Q(X)). For each 2 E L the element T2 is a component
of the unit in C(Q(Y)).
13
In View of Theorem 3.6, T2 = z o (,0 for each 2 E L. Let a: E X n C(Q(X)). Then
applying the equality (3.4) twice (first to 2 as to an element of Z02), and after that
similarly to :c) we have
zogo-Tx:T(zz}=:cogp'Tz=zongocp.
Since .2 is arbitrary and since (TL)'M = (TX)"Zd the above identity implies that Ta: =
x o (,9 for all an under consideration. To complete the proof let us introduce the vector
lattice X; z: X + C(Q(X)), define T1 : X1 ——+ Cm(Q(Y)) by the formula T1(:r + f) 2
T1 + f 0 go (a: 6 X,f E Cw(Q(X)), and then apply Theorem 3.10. I
3.13. Corollary. Let X and Y be vector lattices and T E dh(X, Y). If there exists
in X a subset M of pairwise disjoint elements such that (TM)‘M = (TX)'“, then T
admits a m.r.
Proof. It is enough to notice that the condition of the corollary remains valid
when we extend T to an operator T E thE’j’), and that the set M1 of finite sums
of absolute values of elements of M is laterally directed. After that one can apply the
previous theorem to T and A!1. I
3.14. Remark. A. V. Koldunov has constructed an interesting example of
Dedekind complete vector lattices X and Y, and an operator T E dh+(X,Y) such
that T admits a m.r., but for each subset M C X of pairwise disjoint elements,
(TMW r (TXW-
Using Theorem 3.12 we can describe a class of “good” vector lattices, for which
each regular d-homomorphism admits a m.r.
3.15. Theorem. Let X be a Dedekind complete BL. The following two conditions
are equivalent.
(1) For an arbitrary vector lattice Y any operator T E dh(X,Y) admits a m.r.
(2) There exists a laterally directed set L C X+ such that the norm closed order
ideal X(L), generated by L, coincides with X.
Proof. (1) => (2). We denote by 'H the collection of all norm-closed proper ideals
in X. For each H E 'H the canonical quotient mapping TH : X —> X/H is a lattice
14
homomorphism. Let us introduce Y := (EH67! EBX/Hflw, the direct foo—sum, and
define T : X —> Y by T1 = {Tuzhew (a E X). Obviously T G rth(X,Y), and, by
Theorem 3.12, condition (1) implies that there exists a laterally directed subset L E X+,
such that (TL)d‘i = Y. If X(L) 79 X, then the ideal H = X(L) would belong to ’H and
for each .1: E L we would have 12,55 = 0. But then (TLYM # Y. A contradiction. Hence
(1)=>(2).
Conversely, assume (2). Then X(L) = X for some laterally directed L E X r- Fix.
an arbitrary vector lattice Y' and an arbitrary T E dh(X, Y). We denote by PT a band
projection in l7 onto the band (TL)d and let
I,={a:EX: EFT$=O}.
Then obviously Ir is an ideal in X and L C II It is easy to verify that Ir is
closed in X in the relative—uniform topology (ru—topology), and consequently, Ir is
norm-closed ([Vul}, Thm.VII.2.1). Hence 1,. = X (since X(L) z X ), and this means
that (TL)'“ 2 (Ti/Y)“. Theorem 3.12 implies now that T admits a ma. I
3.16. Remark. In a similar way one can prove a generalized version of the
previous theorem for an arbitrary Dedekind complete vector lattice X, provided one
defines X(L) to be the smallest (rd—closed ideal generated by L.
The previous results can be applied to infer several useful and easily verified suf—
ficient conditions.
3.17. Theorem. Each of the following conditions implies that T E dh(X,Y)
admits a m.r.
(2) .X is a Banach lattice with order continuous norm.
(3) X is a separable Banach lattice.
(4) The Banach lattice X has a quasi-interior point 1:0, i.e., there is an element
so 6 X... such that X(ro) = X.
(5) Every subset of pairwise disjoint non—zero elements in Y is at most countable,
or equivalently, M(Y) satisfies the countable sup property.
15
Proof. Obviously (2) :> (1) and (3) => (4) The sufficiency of (1) follows from
Remark 3.7(2) and Corollary 3.13. (If (p = cpr is open, then the equality (T1M)dd :
(TX)‘M holds for each subset M of pairwise disjoint elements of X satisfying M“M = X.)
The sufficiency of (4) is proved in [Abr2] and it can also be immediately inferred from
Theorem 3.15. Indeed, each quasiuinterior point so 6 X+ is also a quasi—interior point
in X endowed with the natural norm Hzllx = inf{”z|[ : :r E X+, [2| S x} Setting
L = {10} we obtain X(L) = X.
To finish the proof of the theorem we need to show that condition (5) is also
suflicient. First we notice that this condition still holds if we replace T by its ex-
tension T E dh(X,i’). Thus, we can assume that X and Y are Dedekind complete
vector lattices. Second we notice that (5) implies that there is a countable sub—
set A : {arfifiil C X such that (Tr/1)“ = (TX)'”. Let us define by induction
21 = 1:1, . . . ,zn+1 = (Pn+l -- Pn)r,.+1, where Pn is the band projection on the band
generated by the element I111] V V [on]. Then the set M = {23:21 satisfies the
condition of Corollary 3.13, and thus T admits a m.r. I
Recall that in [Abr2] in an example of an operator T E dh(X,Y) that does not
admit a m.r., X is a nonseparable Orlicz space. Now, in view of Theorem 3.17(3), we
see that this assumption is indispensable. Moreover, the next theorem shows that for a
large class of Banach lattices, namely for the rearrangement invariant Banach function
spaces (see their definition in [LT]), this assumption is also necessary.
3.18. Theorem. Let X be a rearrangement invariant Banach function space on
[0,1]. The following conditions are equivalent.
(1) The function 1 is a quasi~interior element in X (or, equivalently, Loo[0, 1] is
norm dense in X).
(2) For an arbitrary vector lattice Y any operator T E dh(X,Y) admits a m.r.
Moreover, if X at Loo[0,1], then each of the previous conditions is equivalent to
the following one:
(3) X is separable.
Proof. The validity of (1) => (2) follows from Theorems 3.17(4) or 3.15. To
prove that (2) => (1) we first note that by Theorem 3.15 X(L) : X for some laterally
16
directed L C X+. Without loss of generality we can assume that for each a: E L each
of its components also belongs to L. Then there exists an a: E L and a number a > 0
such that the set F = supp (2:) satisfies the following two conditions: the measure of
F equals 1/2 and axF S a: S adxp Let us set X1 :2 XFX and L1 :2 XFL' The
condition X(L) = X obviously implies that X(L1) = X1. Since a: = sup L1, we have
X1(a:) = X; or, equivalently, X100.) = X1. Therefore Mr is a quasi-interior point in
X1‘ Let G = Q(X) \ F. The rearrangement invariance of X clearly implies that X0 is
a quasi—interior point of X2 =- XGX- This implies that 1 =2 Xp + x0 is a quasi—interior
point of X.
Finally, let X 9E Loo[0, 1]. We need to prove only the implication (1) => (3), since
(3) => (2) holds by the previous theorem. It is well known that the restriction of
the invariant norm [I - ”X to Loo[0, 1] is order continuous, and thus (Loom, 1], || - ”X)
is separable. But in view of (1) the space LOOK), 1] is dense in X. Therefore, X is
separable. I
3.19. Remark. Suppose T 6 dh(X, Y) admits a m.r. (3.1).
(1) If 3'; and Lp are fixed, then formula (3.1) remains valid for an arbitrary re-
placement of jg by any other representation j; (with a corresponding replacement of
the weight 6), and for an arbitrary replacement of e by an arbitrary function 6’ with
the same support as c (with a corresponding replacement of the representation jg).
In particular, (a) there exists a m.r. (31) for which I6] = Xm and (b) if X r: Y,
then we can replace jg by j} and thus T admits a m.r. with a single representation
j = j] = jg. This possibility is essential when one investigates the spectral properties
of d—homomorphisms.
(2) A replacement of the representation j; by an arbitrary representation is not
possible in general. However, if T is order continuous, then T admits a m.r. for
arbitrary representations jl and jg.
(3) If X and Y are two order dense ideals in a vector lattice W, then there exist
two representations j{ and jé of W, such that (for some mapping go and some function
e) formula (3.1) is valid for jl =ji|X and jg =jéll’.
(4) Let X be an arbitrary vector lattice and T E Z(X) be a central operator, Le,
17
szi S Afr} (*) for some /\ Z 0 and all a: 6 X. It is obvious that (*) implies that T is
order continuous, and consequently, in View of Theorem 3.17(1) and Remark (1} above,
T admits a m.r. of the form T1 = e - .1: o (,0. Two other immediate consequences of (*)
are that (p(q) r: q for all g 6 ET and that 6 belongs to C(Q) (rather than to Cm(Q)).
That is, T3? = e - m, and this means that T is a multiplication (by a. complex-valued
continuous function), not only on the appropriate compact space mentioned at the very
end of Section ‘2, but also on the Stone space Q(X). Another proof of this result may
be obtained by extending T E Z(X ) to an operator T E ZOE).
18
4. Banach GHQ-modules and their relationship with
Banach and vector lattices
Let K be a compact Hausdorff space. We say that a Banach space X is a Ba—
nach C(K)«module (or, shorter, a C(K)-module) if there is a bilinear mapping
(f, 1) l—-) f - a: from C(K) X .X into X satisfying the following conditions:
1'56 =8”, (fyl‘I =f'(9'T) and “1"!” S Hfllqmill'll
for each fig 6 C(K) and :1: E X. it
Usually, when this does not cause any ambiguity, we write simply fr instead of
f - :3. Numerous examples of GHQ—modules will be considered in Section 14.
One of our goals is to show that there is a very deep parallelism between the
properties of GHQ-modules on one hand and the properties of vector and Banach
lattices on the other. Since there are many examples of C(K)—modules which are not
Banach lattices, this parallelism allows us to develop for C(K)—modules a deep theory
reflecting the one existing for Banach lattices. Moreover, the relationships between
theories work both ways. Some special properties of C(K)~-modules allow us to look
differently at Banach lattices and to deepen their theory, too.
Recall that if X is a vector lattice, then for an arbitrary a: E X, the (order) interval]:
generated by a: is the following set
NE): {y€X:lyIS [wil-
1‘ It is worth mentioning that a more general case which assumes only the inequality
[If - :r“ S 7I|fllc(k)H:s|| for some 7 2 1 can be reduced to the case under consideration
with 7 = 1 by an equivalent renorming of X.
I This term is standard in the case of real vector lattices, and we preserve it for the
complex case, too.
19
The next definition establishes a natural bridge between Banach lattices and C(K)—
modules by postulating which objects in C(K)—modules should serve as a correct coun—
terpart of intervals in Banach lattices. We preserve for these objects the same symbol
A($), though occasionally (when some ambiguity is possible because of the different
algebras involved) we will use also symbol AC(K)(a) to express explicitly the algebra
generating this interval.
4.1. Definition. IfX is an arbitrary C(K)-module and a: E X, then
Ah) = clx{fz : f E C(K),IlflI31},
where “cl,“r ” denotes the norm closure in X.
In the next few definitions we are going to use this set—valued mapping a v—r A(:r)
to extend to C(K)—modules the main concepts which were previously known (and used)
in the context of vector and Banach lattices only.
4.2. Definition. A linear subspace Y of a GHQ-module X is said to be an
(order) ideal iffor each x E Y the whole interval A(:r) belongs to Y.
It is easy to see that each ideal of a C(K)-module is a C(K)—submodule, not
necessarily closed. The converse statement is not true in general; but each closed
C(K)—submodule is an ideal. It is also clear that the closure of each ideal is likewise
an ideal.
4.3. Definition. Let Y be an arbitrary ideal in a C(K)-module X. We denote
by Z0000”), or simply by Z(Y), the collection of all linear operators T : Y —r Y, such
that Ty E AA(y) for each y 6 Y, where A = /\(T) is a positive number (independent
of y), i.e.,
2m = {T : Y —> Y I (3A 2 mm e may 6 no)».
Obviously, Z(Y) is an algebra of operators on Y. We call this algebra the center of Y.
4.3.1. Remark. In Theorem 6.2, one of the main results of Section 6, we will
show that the assumption that /\ is independent of the element y is actually redundant,
provided T is an operator on the whole of a C(K)-module X.
20
4.3.2. Remark. It is obvious that for each a:’ E A(:r) we have A(a:’) g A(a:).
Therefore, for each T E Z(X) we have T1:' 6 AA(:I:') Q AA(a:), i.e., T(A($)) Q AA(:r).
4.4. Definition. Two elements 3:, y of a C(K)—module X are called disjoint (in
symbols: xdy), if MI + y) = A05) + A(y) and AW) 0 A(y) = {0},
4.4.1. Remark. The subsequent Lemma 4.8(4) shows that our definition of
disjointness in C(K)—rnodules is adequate for our pursuit of the parallelism between
Banach lattices and C(K)-modules. If X is a Banach lattice, then evidently zdy
(a) A($) fl A(y) = {0} and each of these implies that A(z + y) = Am) + A(y).
However, the last relation A(z + y) = A(.‘L‘) + A(y) alone does not imply that zdy,
because A(2:r) = A($) + A(a:) for each 2: E X.
It seems somewhat curious that if X is a C(K)—module then, in general, the
condition A(.1:)flA(y) = {0} does not imply that A($+y) = A($)+A(y). For example,
let X be the L°°[O,1]—module L2[0, 1] ®L2[0, 1], and let a: = (1,0) and y = (0,1). Then
Mac + y) 74 Mac) + A(y)-
We leave it as an exercise for the reader to prove that whenever f1 and f2 are
arbitrary disjoint functions in C(K), the elements fun and f2$2 are disjoint in the
C(K)-module X for arbitrary 1hr; E X.
4.5. Definition. Let T : X —» Y be a linear operator from a C(Kfl-module X
into a, C(K2)--module Y. We call operator T a d-homomorphitam (a d-endomorphism
provided X = Y) if (Blditz in X implies Txldng in Y.
An invertible operator T : X —v Y is called a d-isomorphism if T and T—1 are
d-homomorphisms.
We want to emphasize once again that for vector lattices the above definitions
define the conventional objects carrying the same names.
Observe that in view of Remark 4.3.2 and Lemma 4.8(4) each operator T from
Z(X ), where X is an arbitrary C(K)-module, is a d—endomorphism.
4.5.1. If T is a continuous d—endomorphism on a Banach lattice X and T is
invertible, then T”1 is likewise a d—endomorphism. However, it is curious to notice
21
that if X is a C(K)-module and T is an invertible continuous d—endomorphisrn on X,
then T“1 may not be a. d—endomorphism. A corresponding example will be given in
Remark 12.17.
4.5.2. Throughout the work, we will use the following notations, which are also
in complete agreement with their vector lattice counterparts. If G is a subset of a
C(K)—module X, then XG denotes the order ideal generated by G, and X(G) denotes
the closure of XG in X. In case G = {1'} is a singleton, we prefer a simpler notation
XI and X(.r), instead of X{,} and X({:c}), respectively. It is plain to see that the
centers Z(Xz) and Z(X(1)) are canonically isomorphic and, therefore, we will denote
them simply by 2(1)
If X is a Banach lattice, then the same identification between Z(XI) and Z(X(r))
will be always assumed and, again, the same symbol Z(:17) will be used to denote either
of them. However, in this case the objects Xz, X(x) and their respective centers should,
of course, be understood in the sense appropriate for Banach lattices, that is, as they
are defined in the introduction.
4.6. Lemma. Let :1: be an arbitrary element of a C(K)-module X. Let
X(.r)+ = clx{fr :0 _<_ f E C(K)}.
Then the space X(a:) ordered by the cone X(I)+ is a Banach lattice. Moreover, the
interval mapping :r’ H A($') on this Banach lattice X(z:) coincides with the mapping
23' H AC(K)(x’) generated in X(a:) by the structure of the GHQ-module. (That is why
it does not cause any ambiguity to use the same symbolfor both mappings.)
Proof. If X is a cyclic C(K)-module, which means that there exists an element
:50 E X such that cl{f:ro : f E C(K)} 2 X, then the first statement was proved
by Kaijser [Kai]. The case of general C(K)-modules can be routinely reduced to the
previous one. The second statement of the lemma follows easily from the definitions.
4.6.1. If X is an arbitrary Banach lattice and 1: E X, then the closed ideal
X($), generated by r, and with the order induced from X, is a Banach lattice, a
22
sublattice of X. On the other hand, since its center Z(x) can be identified with a
C(K)-space, where K is the maximal ideal space of the algebra Z(x), the space X(x) is
as well a C(K)-mod11le, and consequently, by Lemma 4.6, the structure of this C(K)-
module turns X(1:) into another Banach lattice. Though, in general, the orders of
these Banach lattices are difl'erent (they coincide i3 :17 e X+), the mappings a: P—+ A05)
generated by these orders coincide, since (as collections of elements) X(3:) X(IT!)
and Z(z) = Z(lzl).
From now on, when speaking about the order on X(7:) (be X a Banach lattice or
a GHQ—module), we will always understand (unless otherwise stated explicitly) that
X(I) is viewed as a Banach lattice in accordance with Lemma 4.6.
4.6.2. Let X be a COO—module. Y be an ideal in X and T E Z(Y). For each
1' E Y we let A(T, 3:) = inf{A > 0 : T: E AA(:I:)}. Since :5 is a quasi—interior point in
the Banach lattice X(:r) and TIX 6 Z(rr) we have “TIX I]: HT]X(:r)l] = A(T, :5).
Therefore "T” = sup ||T|X ||—_— sup /\(T, .2). This implies that all operators in Z(Y)
are continuous andEthat the set {T6 Z(Y): “T“ S 1}IS closedin the strong operator
topology. In particular, Z(Y)is a closed subalgebra of the algebra L(X).
Let us notice also that there obviously exists a natural algebraic homomorphism
of C(K) into Z(X) This homomorphism does not increase the norm and, in general,
is neither injective nor surjective.
The next technical definition plays an important role throughout the whole work.
4.7. Definition. Suppose I E X and 13" E X" are two fixed elements, where X is
a GHQ-module. Then we denote by a:*:! :r: the following functional from C(K)* (that
is, a regular Borel measure on K)
(Wu 1X1“) = $‘(fr), f 6 C(10-
In other words, we have a bilinear mapping (cc‘, 9:) H :5”: :1: with the values in the
AL-space C(K)*. It can be readily seen that |[:v"‘n 1:“ g ”1"“ - ”3:”.
23
4.8. Lemma. Let X be a C(K)-module and 3:,y E X. The following statements
are true.
1) A functional z“ 6 X‘ is orthogonal to X(:c) 4:} ft! 3: = 0.
2) A functional z“ E X" is nonnegati'oe on X($)+ <————> I‘U z 2 0 as an element of
C(K)‘.
3) y E A(:e) <=4> Heft] y|| S ||:r*cl all for each $* 6 X“ (E) |:c"‘u yl 3 [fun arl for each
3:" 6 X‘.
4) ady Q the elements a: and y belong to the vector lattice X(x+y) and are disjoint
in it <=> (:c‘n :e)d(a:*u y) in C(K)* for each 2* E X“.
Proof. Statements 1) and 2) follow easily from definitions. Let A, B and C
(respectively) denote three conditions which are claimed to be equivalent in 3). Im—
plication A => C follows from the definition of the set A(z); implication C :s B is
evident; and B => A follows by a standard separation theorem.
Let now D, E and F (respectively) denote the three conditions, the equivalence
of which is claimed in 4). The equivalence D 4:) E can be inferred easily from the
definition of disjointness in a C(K)—module. The equivalence E' 4:) F follows from 3).
I
4.9. Corollary. Let X be a Banach lattice or a C(K)-module. Then the graph
{($,y) :1 E X, y 6 A(a‘)} of the mapping :1“. —> A(:L') is closed in X X X.
Proof. For a Banach lattice the statement is obvious. For a C(K)—Inodule it
follows from Lemma 4.8(3). I
4.10. Corollary. Let Y be a C(K)-sub7nodule of a C(K)-module X, and let Y1
denote the ideal in X generated by Y. Assume also that U : Y —§ X is a linear operator
such that Uy E AA(y) for each y E Y, where /\ 2 0 is a fixed positive number. Then U
admits a unique extension V E Z(Y1).
Proof. Obviously U is continuous, and a fortiori, it has a unique continuous
extension V1 on X1 = chY. It is readily seen that X] is again a C(K)—submodule
24
of X, and, being closed, it is an ideal in X. Therefore Yl Q X1. By Corollary 4.9,
V] 6 Z(X1), and hence V1(Y1) C Y1. It remains to set V: = VllYl- I
The next definition introduces an object which is of crucial importance for what
will follow.
4.11. Definition. UK is a C(K)-module, then LIX denotes the band in C(K)*
generated by all measures :r‘Dx, where :c'“ E X‘, and I 6 X. Consequently Liv is a
Banach lattice, and moreover, an Ali-space.
4.11.1. Lemma. L; coincides with the closed subspace of C(K)" generated by
all measures 1"“:1 x, where to" E X‘, and :1: E .X.
Proof. It is enough to show that if V E C(K)* satisfies ll/I S |:r*u .7:| for some
:1," E X' and a: E X, then 1/ = y‘n a: for some 31" E X‘. Let p = x‘n 1. By the Radon-
Nikodym theorem 1/ = hp for some h E L°°(|/.t]) with “hue.o S 1. Take any sequence
{fa} in the unit ball of the space C(K) which converges to h in L1(|p|)-norm. For each
n E N we denote by y; a functional in X" defined as follows: y:(z) = 1*(fnz), z E X.
Clearly ”31;" S ”1*”. Let y“ be an arbitrary a(X*,X)-limit point of {y;}
Then for each f E C(K) we have
ffdu:/fhd(z*nx)=li'rrl/ffnd(1*nz)=1i'mz‘(fnfx)=
=1i;ny.t(fz) = mm) = / fd(y*n x),
whence V = y*n :r. I
4.12. Recall that a subset D of a vector lattice V is called solid if (|v| S [$|,a E
D,v E V) implies that v G D.
It is important to notice that in the course of the proof of Lemma 4.11.1 we showed
that the set D = {1*13 1: : ar“ 6 X',:r E X} of all “elementary” measures is a solid
subset of L}.
In addition we will need the following approximation result, whose simple proof is
omitted.
25
4.12.1. Lemma. Let D be a solid subset of a. Dedekind complete vector lattice
V. Then for each '0 in VD, the order ideal generated by D, there are pairwise disjoint
elements {rk}}:=l C D and numbers {Akfi‘zl C C such that v = EAkrk.
4.12.2. Remark. The assumption that V is Dedekind complete is essential.
4.12.3. Corollary. For each u 6 Li: and for each 6 > 0 there exists a fi-
nite collection of pairwise disjoint measures 9:20 at, 1:; 6 X*,:ck E X, and numbers
A1,... ,z\,. E C, such that Hp — EAkaD an.“ s e.
4.12.4. Corollary. For each u G Li? there exists a. sequence of pairwise disjoint
measures min $16, 1c: 6 X", M E X, such that ,u 6 {win rk}dd, i.e., the band generated
by this sequence contains [4.
Remark. Later on, in Theorem 8.1, we will be able to prove that in actuality
L1 = {f0 .1: : :c’“ E X*, :r E X}, which is a much stronger result than the preceding
two corollaries and Lemma 4.11.1.
4.13. Another structure on L} which can be introduced immediately is that of a
C(IQ—module. Indeed, for each f E C(K) and each measure [t E L}, the product f - u
is defined as the following functional on C(K)1
(f - #)(9) = < mfg > = Mfg) (s 6 C(10)-
It is plain to verify that f - u E L1 , and that under this definition L} becomes a
C(K)—module.
4.13.1. For further references note that the above introduced multiplication of
a function f E C(K) by a. measure u = fl: 1 E L}, z“ E X",:r E X, enjoys the
following useful property:
Indeed, for an arbitrary g E C(K) we have:
< f-(r*nat),y >=<a:*n:c,fg >=< m‘nx,gf >2
< 1*,(yf)($) >= $‘(g(f1‘)) = < $‘U fag >-
The first equality follows from the definition of the multiplication, the second from
26
commutativity in C(K), the third and the last from the definition of the operation D,
and the second to last follows from associativity. Since 9 is arbitrary, this proves (4.1).
Note also that 27*0 fa: = (f"‘2:*)u :1: because < (f*m*)cl w,g >= (f':r*)(ga:) = I‘(f(gr))
for arbitrary g E C(K).
4.13.2. Finally, denote by L“)? the center Z(L}) of the Banach lattice LIX. Since
Lk ia an AL—space, the algebra L? is an AM—space, which is isometric and order
isomorphic to the conjugate space (LIXY. Therefore, Li, may be viewed also as an
L3?--Inodu.le.
Using the above notation we can easily prove the following proposition.
4.14. Proposition. The mappings ,u -—+ A01), determined on LEX by the struc-
tures of a Banach lattice, of a C(K)-module, and of an L3?-module coincide. That is,
AOL) = A000(41): AWOL)-
4.15. Let A be an ideal in a GHQ—module X. Since A is a C(K)-—rnodule in its
own right we can consider the space L24 which is a band in C(K)*. It is important to
observe that L1, is actually contained in L1 , and therefore L; is a band in LIX. This
observation follows from the following simple fact. For arbitrary x“ E X* and a E A,
the measures 1*0 a E L} and (a:*|A)I:l a 6 L34 coincide.
27
5. Comparison of operators taking values
in a vector lattice or C(K)-module
Throughout this section X stands for a vector lattice or a C(K)—Inodule, n is a
natural number; E1,E2 . . .,En are linear spaces, and E = E1 >< E2 - -- X En denotes
their cross-product; T : E --» X is an n-Iinear operator; and finally, X1 = X7103) is the
ideal in X generated by the set THE).
5.1. Definition. We denote by the symbol Z 0 T the collection. of all n-linear
operators S : E —> X admitting a representation S = VT for some V E Z(X1).
Since V maps X1 into X], we can consider V as a mapping from X1 into X, and
therefore, the operator 3 is well—defined.
If E = X and T = idx, then Z 0 T = Z(X), and thus, the set Z 0 T of operators
may be viewed as a generalization of the center.
The next lemma follows easily from the previous definition, and its proof is omitted.
5.2. Lemma. Let S E Z 0T. The following statements are true.
(1) The factorization S = VT, with V 6 Z(X1), is unique.
(2) IfX is a Dedekind complete VL, then there exists a unique factorization S = WT
with W 6 Z((TE)‘”), or, what is essentially the same, W E Z(X) and W E 0 on
(Teri.
The goal of this section is to present several easily verifiable sufficient conditions
for S to belong to Z 0 T.
5.3. Definition. We say that an operator S : E —> X is dominated by T on
the elements of E iffor each e E E there is a constant c 2 0 such that Se 6 cA(Te).
In this case, we use the notation S -< T. If such a. constant c > 0 may be chosen
independently of e E E, then we say that S is uniformly dominated by T on the
elements of E and we use the notation S «4 T.
5.3.1. Examples.
(1) If S E Z 0 T, then clearly S -<-< T.
28
(2) Let E and X be vector lattices, and let S : E —v X be a regular d-homomorphism.
Then (by Theorem 3.3) there is a lattice homomorphism T : E —> X such that S' <~< T.
“
“ 4 ” and the standard order relation < ” for operators. Specifically, let X be a
vector lattice and S,T : X —r X two operators such that 0 S S' S T. It is easy to see
that the previous inequalities do not imply in general that S —< T, though of course,
SJ: 6 A(T$) for each a: 2 0.
In the next theorem we deal with the case when X is a vector lattice.
5.4. Theorem. Let either of the following two conditions be fulfilled.
(1) X is an (ru)-complete vector lattice and S -<-< T,
(2) X is a Banach lattice, E1, . . .,E,. are Banach spaces, T is continuous and S -< T.
Then S E Z 0 T.
Proof. We start with two special cases.
Case I. Let S and T be n-linear functionals, that is, dim(X) = 1. Using induction
on n, we will show that S -< T implies S E Z 0 T. The implication is trivially true if
n = 1. Let n > 1. It is convenient to treat .5' and T as linear operators defined on
the space E" and with values in the space H of all (n — 1)-linear functionals defined
on E1 x E2 X X En_1. To justify the induction, it is enough to verify the following
simple statement:
If for each e’ 6 E,1 there exists a number A(e’) such that 56’ = /\(e’)Te’, then
S = AoT for some number A0.
To prove this statement, we first note that for each 6’ E En and for eachscalar t,
we obviously have /\(te’) = /\(e').
Let e' and e" be two arbitrary elements from En. We want to show that A(e’) =
x\(e"). On one hand, we have 5(6' + e") = S(e') + 3(8") = A(e’)Te' + /\(e")Te", and
on the other, S'(e' + e”) = /\(e’ + e”)T(c’ + e") = A(e’ + e")T(e’) + A(e’ + e")T(e").
Assuming that Te' and Te” are linearly independent, we immediately obtain from
the previous identities that /\(e') = /\(e' + e") = /\(e").
Now let Te' and Te" be linearly dependent. Without loss of generality we may
29
assume that Te’ = Te” .7é 0. Consider now the element 6' - 6". Since T(e' -— e”) = 0,
we have 5(6' — e") = 0. That is, S(e') = 3(8”), and hence A(e')Te’ = A(e”)Te”. This
shows that /\(e’) = /\(e”).
Case II. Let X be a Dedekind complete VL. Without loss of generality, we can
assume that (T(E))‘“ = X and that X is embedded in 000(6)), where Q = Q(X) is
the Stone space of X. Then, as usual, we can identify Z(X) with C(Q).
First we show that 5' < T implies that there is a function V E 00°(Q) such that
Se = VTe for each 6 E E. To this end, introduce the following set Q0 := {q E Q :
(El 6 E E) (0 < (Te)(q) < 00)}. Case I implies that for each q 6 Q0 there is a unique
number Vo(q) such that (|(Te)(q)| < 00) => (Se)(q) = Vo(q)(Te)(q). The function V0
thus defined is obviously continuous on Q0, and therefore Va has a continuous extension
V G Cw(Q). Now, for each 6 6 E and for each q from a dense subset Q0, we have
(Se)(q) = V(q)(Te)(q). This implies that Se = VTe.
Assume that condition (1) of the theorem (i.e., that S -<-< T) holds. Then the
preceding proof shows immediately that the function V belongs to C(Q) (rather than
to COO(Q)), and hence S E Z 0 T.
Assume now that we have (2). To show that also in this case V 6 Z, it is enough
to verify that (2) => (1). It is readily seen that the operator 5' is continuous with
respect to each variable (this follows from the continuity of T, from the closed graph
theorem, and from the formula Sc = VTe, where e E E). For each positive number
a > 0 introduce the set EW 2 {e E E : [Se| _<_ aITeI}. Since Sis continuous with
respect to each variable, it is jointly continuous (see [Rud], Theorem 2.17) and hence
the set Em is closed in E. Obviously E = Ufi=1E(“"') for each sequence am T 00.
The Baire category theorem then implies that there is an a > 0 such that E(“) has a
nonempty interior. We will show that EM = E, and this will establish condition (1)
of the theorem.
If n = 1, then it is enough to show that E”) is a linear subspace of E = E1.
Obviously E(“) is closed under scalar multiplication. Let 61,62 6 3(a). The formula
Se = VTe (c E E) shows that e E E“) if and only if for each q E Q the following
implication holds ((Te)(q) 75 0 => IV(q)| S a). This clearly implies that 81+62 E E”).
30
Now let n > 1. For each i = 1,2,. . . ,n there is a nonempty open subset Ug C E.-
such that E(“) 3 U1 x U2 X . . . X U”. The above result for n = 1 implies that E(“) 3
E, x U2 >< . . . X Un, since 5' and T are linear with respect to the first variable. Applying
the same arguments to the second, third, etc variables, we obtain that E”) = E. This
completes the proof of the theorem in Case II.
General case. We may (and shall) assume that X is embedded in its Dedekind
completion X. Either of the conditions (1) or (2) of the theorem implies that there
exists a l7 E Z(X) such that Se = VTe for each e E E. Let Y = {.1: E X : Vr G X}.
If we show that X1 = XT(E) C Y (#), then letting V := le, we obviously have
V E Z(X1) and S = VT, that is, the conclusion of the theorem. To verify (#), it
will suflice to show that Y is an ideal in X (for, clearly, T(E) C Y, and therefore
Y contains X1, the ideal generated by T(E)). Assume that Ix'l S |:7:[, where :c’ E X,
and r E Y. We want to show that x' E Y. Since X is (ru)-complete, there exists an
f E Z(X:) such that a." = fr. Consider f as an operator from XI to X. Obviously,
it is uniformly dominated by the operator of the canonical embedding XI L—r X . By
Case II there is an f E Z(X) such that fy = fy for y 6 X1. Therefore, V3’ 2 Vfa: =
Vfa: = fVa: = sz‘ E X, since Va: 6. X,. This proves that Y is an ideal, and the
proof of the theorem is complete. I
Now we turn our attention to the case of C(K)—modules.
5.5. Theorem. Let X be a C(K)-module and let either of the following two
conditions be fulfilled.
(1) S << T.
(2) E1,E2, . . .,E,, are Banach spaces, T : E —-r X is continuous and 5' 4 T.
Then S 6 Z 0 T.
Proof. Introduce two (n + 1)—linear operators S1,T1 : X* x E ——i L} by the
following formulas: 51(1"‘,e) = a:*n.5'e, T1(:1:*, e) = x‘hTe.
Lemma 4.8(3) implies that if the operators S,T satisfy condition (k) of this theo-
rem, where k = 1 or 2, then the operators 31,51} satisfy condition (k) of Theorem 5.4.
Therefore, there exists W E L? = Z(L},() such that S; = WT; or (equivalently)
31
:r‘DSe = W - w‘DTc for 3* E X“ and e E E. Let Y denote the submodule generated
I
by T(E) in X. Pick an arbitrary element kaTek E Y, where 1']; E C(K) and ck 6 E.
k:]
Then, for each a." E X", we have z‘DkaSek = W - $*DkaTek. Hence (again by
k I:
Lemma 48(3)) kaSek E ”W” - A(kaTek). Thus we have defined an operator
[6 I:
kaTek H kaSek on Y, and it satisfies the condition of Corollary 4.10. Therefore
k I:
it can be extended to an operator V 6 Z(Xy), and this implies the desired formula
S = VT. I
Properties of d-homomorphisms
6.1. Definition. A Banach C(K)-module X is said to be an operator module if
X 7E {0} and the representation of C(K) in L(X) is isometric, i.e., for each function
f E C(K) the norm ||f||c(K) coincides with the norm of the operator :5 H fr in L(X).
In this case, we identify C(K) with a closed subalgebra of the algebra L(X).
6.1.1. For an arbitrary C(K)-module X let us introduce the set J = {f 6 C(K) :
fr = 0 Va 6 X}, which is evidently an ideal in C(K). Therefore C(K)/J = C(Kl) for
an appropriate compact Hausdorff space K1, and it is easy to verify that X becomes
an operator C(K1)-module. (This is a standard fact of the theory of commutative
Banach algebras; its generalization to non-commutative C*-algebras is given in [H02].)
In other words: (i) we can replace a C(K)-module X by a naturally associated operator
C(K1)—module, and (ii) a C(K)-module X is an operator module iff X is exact.
The next theorem is a generalization of a famous theorem of Bade about algebras
of spectral operators (see [D52], Chapter XVII.3). In some sense, our generalization
establishes a natural domain of applicability of the Bade Theorem.
6.2. Theorem. (Generalized Bade’s Theorem.) Let X be an operator C(K)-
module, and let T : X —) X be a linear operator (not a priori assumed to be continuous).
The following statements are equivalent.
(1) T E Z(X).
(2) T belongs to the closure of C(K) in the strong operator topology.
(3) T maps each ideal into itself.
(4) T maps each closed ideal (or, equivalently, each closed C(K)vinvariant subspace)
into itself.
Proof. (1) => (2). Let us introduce two bilinear operators 5'1, T1 : X“ x X —i Lk
by the following formulas: 51(1“,a') = :c‘uTa' and T1(a:", m) = a‘nr. By Lemma 4.8(3)
51 -<< T1, i.e., these operators satisfy condition (1) of Theorem 5.4. Hence, there is an
operator W E Z(fo) such that
1*UT2: = W~r*uz, a)“ E X", z E X.
33
This implies that whenever Zita-TI: = 0 for some finite collections {xi} C X* and
k
{mg} C X, then necessarily 21?:ka :2 0. In particular this implies that whenever
I:
Zr;(ka) = 0 for all f E C(K), then ZI:(T$k) : 0.
k 1:
Recall now (see [D81], Theorem V1.1.4) that for each linear continuous functional
6 on the space L(X) equipped with the strong operator topology T5, there exist some
k=n
r: E X‘ and .1"; E X, k = 1,2,...,n, such that {(5') = 2 32(51):) for all 5 6 L(X).
k=1
This allows us to reformulate the statement in the previous paragraph as follows: if
{(f) = 0 for each f E C(K), then {(T) = 0. Clearly this implies that T belongs to the
T,--closure of C(K)
(2) => (4) Since each operator from C(K) maps closed ideals into themselves,
the same is true for T, which belongs to the closure of C(K) in the strong operator
topology.
(4) => (3). It is enough to show that for each a: 6 X the restriction of T to
X(z) belongs to the algebra 2(3). It is plain to see that each band in the Banach
lattice X(9:) is a closed ideal in X; therefore, (4) implies that T maps bands of the
vector lattice X(1:) into themselves. As shown in [AVKl], the last property implies
that T|X(:L') C Z(x).
(3) => (1) Let S; = T and let T1 be the identity operator on X. Clearly (3)
implies that 51 and T1 satisfy condition (2) of Theorem 5.5. Therefore T 6 Z(X ) I
6.2.1. Remark. The assumption that X is an operator C(K)~module is not
essential for the validity of the previous theorem. Indeed, if X is an arbitrary C(K)—
module, then it can be replaced by the associated operator C(K1 )-module described in
6.1.1.
6.2.2. Remark. Under the assumption that T is a continuous operator, the
equivalence (2) <=> (4) in Theorem 6.2 was independently obtained by a different
method by D. Hadwin and M. Orhon [H01]. We take this opportunity to thank them
for bringing our attention to this and to several other related articles.
34
6.2.3. Comparison with the Bade theorem. To see that Theorem 6.2 is
indeed a generalization of Bade’s theorem we begin by recalling a bit of terminology
(see for example [H01] or [Gil]) that will be of use later on, too.
If S is a subset of L(X), where X is a Banach space, then the symbol Lats
denotes the collection of all closed subspaces of X which are invariant under each
8 E S. The symbol Alg LatS denotes the set of all operators in L(X) which leave
invariant each subspace in Lat 8. Clearly, the set Alg Lats is a Ts—closed subalgebra
of L(X) containing both 5 and the identity operator. The collection 5 is said to be
reflexive if Alg LatS coincides with the unital ‘r,-closed subalgebra generated by S.
In this notation Theorem 6.2 says (among other things) that Z(X ) = n—cl(C(K))
= Alg Lat C(K) for each operator C(K)—module.
The classical Bade theorem [D32], Theorem XVII.3.16, asserts that the “rs-closed
algebra generated by a complete Boolean algebra B of projections in a Banach space
X coincides with Alg Lat B. In view of the Stone representation theorem, 8 may
be identified with the set S of the characteristic functions of all clopen subsets of an
extremally disconnected compact Hausdorff space Q3, the Stone space of 3. Obviously
the linear span of S is norm-dense in C(QB), and consequently X is a C(Q3)-module.
Therefore, Theorem 6.2 is indeed a broad generalization of Bade’s theorem. Moreover,
we can show that if X is a C(K)—module, then C(K) is generated (in the way described
above) by a complete Boolean algebra of projections in X if and only if each ideal
X(x), a: e X, has order continuous norm. At this point it is also appropriate to
mention a paper by A. Veksler [Vek], where it is shown that each Banach space X with
a complete Boolean algebra of projections may be turned into a Banach lattice with
order continuous norm, provided that there exists a cyclic element in X.
It is well known that the center of an arbitrary Banach lattice is isometrically
isomorphic to a C(S)—space for an appropriate compact Hausdorff space (see [Wic2]
for a short proof of this result). Our next corollary extends this result to arbitrary
operator C(K)-modules.
6.3. Corollary. For an arbitrary operator C(K)-module X the algebra Z(X) is
35
isomorphic to C(Kl), where K1 is a compact Hausdorfi space. In particular, Z(X) is
commutative.
Proof. Being the closure of a C(K)-space, the center Z(X) is obviously a commu-
tative algebra. Let us verify that it is a uniform algebra, that is, that ”T2“ = 1 for each
T E Z(X) with “T" = 1. Fix an arbitrary e > 0 and a normed element a: E X with
”Tr“ > 1 — 6. Consider now the restriction of T to the closed ideal X(.2) Obviously
T|x(1) 6 Z(X('r)) and the latter is isomorphic to some C(K,)—space, since X(.’L‘) is a
Banach lattice. Therefore ”TZIXQ)“ = HT]X(z)“2 > (1 -— e)2, whence ”T2” = 1.
Clearly the natural involution f —> ion C(K) satisfies the condition ”sz 2 ”fr”,
since the elements f1- and ftt of the Banach lattice X(1:) have the same modulus. The
equivalence (1) 4:) (2) of the previous theorem implies that this involution may be
extended to Z(X) and, therefore, it generates in Z(X) a structure of a commutative
C*—algebra. But each commutative C*—algebra with a unit coincides with the space of
continuous functions on some compact Hausdorff space [Dug],[Tak]. I
The next proposition supplements Theorem 6.2 by describing a relationship be-
tween the unit balls in C(K) and Z(X). In the case when X is a Hilbert space
this result may be inferred from a result due to I. Kaplansky on approximation of
self-adjoint elements of the rs-closure of self-adjoint algebras of operators (see [Arv]7
Theorem 1.2.2).
6.4. Proposition. Let X be an operator GHQ-module. Then the unit ball of the
algebra Z(X) coincides with the closure in the strong operator topology of the unit ball
of the algebra C(K).
Proof. Let T E Z(X) and ”T” S 1. Fix an arbitrary positive integer n and
consider the space X", the direct sum of n copies of X with the El-norm, i.e., ”if” =
illxk“, where f = (ark) and 1;; E X. Letting f - (x1,---,a:n) 1: (fx1,-~-,f$n),
.klee C(K), we equip X" with the structure of a GHQ-module. It is plain to see
that L} = Liv": and that fix? 2 izinrk for arbitrary a?" =(1c'f,--~,z;)€(X")*
k=l
and f = ($1, - - - ,1") E X". The operator T generates in a natural way an operator
T : X" —> X", by the formula T5 3: (Tan, - - ~ , T3"). Introduce also the following two
36
bilinear operators 31(5“, 5) = PETE and T1(.i’*,5:') = FEE. It is easy to verify that
the operators $1,T1 : (X")* X Xn —» L} satisfy the conditions of Theorem 5.4, and
hence there is a W 6 L7, ”W” s 1 such that rare = W - (arms) for all r e (X")*
and f E X". By Lemma 4.8(3) we have TE 6 13(5) for each 51' E X". Since n and
f = ($1, - - 31'“) E X" are arbitrary, the conclusion of Proposition 6.4 follows from the
definition of the set 13(5). I
6.4.1. Below we will present an important corollary to our previous proposition.
Let X be an arbitrary Banach lattice and let Z be its center. As we know, Z :2 C(K)
for some compact Hausdorff space K, and consequently X may be considered as a
GHQ-module. Now, being a C(K)—module, X has its own center ZC(;‘r)(X), and the
next result shows that (as one might expect) both these centers coincide.
6.4.2. Corollary. ZC(K)(X) = C(K), i.e., the centers of an arbitrary Banach
lattice X and of the corresponding C(K)-mod'u.le coincide.
Proof. By Theorem 6.2 the center ZC(K)(X) is the closure of C(K) in the strong
operator topology, and therefore, in view of Proposition 6.4, our claim will follow if
we show that the unit ball of C(K) is closed in L(X) in the strong operator topology.
Since C(K) is the center of the Banach lattice X, for each T E C(K) the condition
“T” S 1 is equivalent to the condition [Tr] S la] for all :5 6 X. Clearly this implies
that the unit ball of C(K) is closed in the strong operator topology. I
6.5. Now for a while we turn our attention to d-homomorphisms between C(K)-
modules. Recall that for vector lattices X and Y the symbol dh(X, Y) denotes the set of
all regular d—homomorphisms from X into Y, and dh+(X, Y) = {T E d(X, Y) : T Z 0}
is the set of all positive d-homornorphisms, i.e., lattice homomorphisms. If X is a
C(K1 )—module and Y is a C(K2)-module, then the same symbol dh(X, Y) will be used
to denote the set of all continuous d—homomorphisms from the C(Kfl—module X into
the C(K2 )-module Y.
The following result is an analog for GHQ-modules of Theorem 3.3.
6.6. Theorem. Let T be a continuous linear operator from a C(K1)~module X
into a C(Kfl-module Y. The following two conditions are equivalent:
37
(2) 9:1 6 A(a:2) :=> Tan 6 Ami-’62).
Proof. (1) -—-> (2). Let I] E A(a-2). Fix y” E Y“ and consider an operator
U : X(a:2) ——v Li, defined by the following formula: Ua‘ = y‘DTI, a: E X(:1:2). By
Lemma 4.8(4), the operator U is a continuous d-homomorphism from X(a:2) into Lt.
But each continuous d~homomorphism between Banach lattices is regular. (For real
Banach lattices this is proved in [Abrl], and for complex valued lattices it can be
reduced to the previous case by decomposing the operator into real and imaginary
parts.) Since $1 6 A($2), Theorem 3.3 implies that U33; 6 A(U.’e2), or, in other words,
Iy‘DTIII S ly‘DTm]. A second application of Lemma 4.8(3) allows us to conclude
that Txl E A(Trg).
(2) => (1). Let $1,“ 6 X and and“. Then, by Lemma 4.8(4), mldrg in the
vector lattice X(r1 + $2), and (2) implies that the restriction of T to X(9:1 + :52) takes
values in the vector lattice Y(T(a‘1 +12» and satisfies the condition (2) of Theorem 3.3.
By that theorem, T is a d-homomorphism between the vector lattices X(3:1 + $2) and
Y(T($1 + 222)), and hence TaldTrrz. I
The next result is a broad generalization, promised in Section 3, of a theorem due
to Hart [Bar]. This theorem will be generalized in two ways: on the one hand we
extend it to C(K)—modules, and on the other, in the case of vector lattices, we weaken
the assumptions on the spaces in question.
6.7. Theorem. Let X be a vector lattice and Y be an (rd—complete vector lattice,
or else let X be a C(K1)-m0dule and Y be a C(K2)umadule. Let T E dh(X,Y) and
let Y1 denote the ideal in Y generated by TX. Then there exists a unique algebraic
homomorphism 7 : Z(X) —+ Z(Yl) sending the unit of Z(X) into the unit of Z(Yl)
and such that
T(f-r) =7(f)-T$ (37 G X, f E Z(X))-
Proof. Fix any f E Z(X) and define an operator 5 : X —v Y by letting SJ: : Tfa‘
for x E X. By Theorem 3.3 or 6.6 (depending on the case in question), there exists
38
a number c 2 0 such that Tfrr E cA(Tx) for all a: e X, i.e., 5 <4 T. Therefore, by
Theorem 5.4 or 5.5 respectively, we can conclude that S (E Z 0 T, and thus, in View of
Lemma 5.2, there is a unique element 7(f) of the algebra Z(Yl) such that S = 7(f)T.
Obviously, the mapping f —» 7(f) is an algebraic homomorphism sending the unit of
Z(X) into the unit of Z(Yl). I
Remark. In the above cited work [Har], the space Y was assumed to be a.
Dedekind complete vector lattice. Under this assumption, the algebras Z(Yl) and
Z((TX)‘”) are clearly canonically isomorphic, and thererfore, Theorem 6.7 implies
immediately Hart’s theorem.
6.8. As one more illustration of the applicability of the results obtained in Sec--
tion 5, we are going to present a generalization of a known theorem of S. Kutateladze
[Kut] (see also [AB 1], Theorem 8.16). We precede it with some discussion.
6.8.1. Recall that if X and Y are vector lattices, then the space of all order
bounded (linear) operators from X into Y is denoted by Lb(X, Y).
Let Y denote, as usual, the Dedekind completion of Y, and J be the canonical
embedding of Y into li’. Then Lb(X, Y) is a Dedekind complete vector lattice and for
each T E Lb(X,Y) there exists an operator |JT| : X -—+ 1?. If |JT|(X) C Y, then
obviously there exists a module [TI : X —-> Y, and IJTI = JITI. Finally, for each
T E Lb(X,Y), let Ab(T) = {S E Lb(X,Y) : IJSI S IJTI}. If Y is Dedekind complete,
then clearly Ab(T) = A(T) = {5' E Lb(X, Y) : [3| 3 IT”.
6.8.2. For the next lemma we need the following result, which follows directly
from Theorem 3.3 and which (in the case of real vector lattices) is a reformulation of
a theorem due to Meyer [Mey]. If T E dh(X,Y), where X,Y are vector lattices, then
there exists an operator [TI : X _, Y', IT] 6 dh+(X,Y), and [Ta] = |T|(|a:[) for each
a: E X.
6.8.3. Lemma. Let T E dh(X, Y), where X,Y are vector lattices. Then an
operator 5 E Ab(T) if and only if (ISxI S ITrI Vrr E X). This implies, in particular,
that S E dh(X, Y).
39
Proof. By the definition of the modulus of an operator, the condition S E A5(T)
is equivalent to the condition lJSzI S |JT|(|J:|), x 6 X. By 6.8.2 the last condition
is equivalent to the condition [.1le S ITa‘I, a: E X. Thus, if 11d n, then clearly
|J5a1|A|J3z2| S [Tall/\[Txgl = 0, and therefore, 5 6 dh(X, Y). A second application
of 6.8.2 gives the existence of |S| and the equality IJS'xI = [31L This implies the desired
inequality [Sal S IT?! for all a: E X. I
6.8.4. Recall that the Kutateladze theorem states that if T E dh+(X,Y) and
0 S S S T, where X and Y are real vector lattices with Y Dedekind complete, then
S 2 RT for some T1 e Z(Y) In our notation it means that
T e dh+(X, Y) => A(T) c Z 0 T.
Now we are in a position to present its generalization, which includes all (m)-
complete vector lattices and, in particular, all Banach lattices.
6.9. Theorem. Let T E Lb(X,Y), where X and Y are vector lattices. The
following statements are true.
(1) IfY is (ru)-campletc and T E dh(X,Y), then Ab(T) C Z 0 T.
(2) If X is (ru)—camplete and Ab(Tl-Xz) C Z 0 (T|Xz) for each a E X, then the
operator T E dh(X, Y).
(3) If X is Dedekind complete and Ab(T) C Z 0 T, then T E clh(X,Y).
Proof. The first statement follows from Lemma 6.8.3 and Theorem 5.4. Now let
the conditions of Statement (2) be satisfied. Pick up 2:1, $2 6 X with [:51] S Izzl. Since
X is (rd-complete, 3:1 = fag for some f E Z(X,2) with If] S 1. It is plain to see
that the operator :1." H T(fa") defined on XI2 belongs to Ab(TlXI2), and therefore
there exists 9 = 7(f) €- Z(YT(X:2)) such that Tfa' = ng’ for each x’ E X“. The
correspondence f H —y( f) clearly generates an algebraic homomorphism, and thus the
condition If] S 1 implies lg] = l7(f)| S 1. This shows that l$1l S 11:2] implies that
ITxll S lTl‘zl. By Theorem 3.3 we can conclude that T 6 dh(X,Y),that is, the
conclusion of Statement (2) is proved.
Finally, let the conditions of Statement (3) hold. Take a: 6 X and S E Ab(T|X,).
In view of (2), to prove (3) it is enough to prove that 5 E Z 0 (TlXx). Let P denote
40
the band projection from X onto the band {1}“. Clearly, the operator 1’ H 3(P1'),
z' E X, belongs to ANT), and therefore it belongs to Z 0 T. This immediately implies
that S E Z0(T|XI). I
41
7. The center of the conjugate space
In this section, we continue the investigation of the parallelism between C(K)-
modules and Banach lattices.
Let X be a Banach lattice. Then X“ is likewise a Banach lattice with its natural
mapping 3:" H A(.1:"). Recall that X" is always Dedekind complete. Let K1 : Q(X")
be its Stone space. Then the center Z(X *) of the Banach lattice X" is (isomorphic
to) C(K1), and therefore X“ becomes an operator C(K1)~module. The structure of
this C(K1 )-module generates its own mapping 1“ H A(:I:*), which obviously coincides
with the initial one. As it follows from Corollary 6.4.2 the center of the C(K1)~module
X" coincides with C(Kl), that is, with the center of the Banach lattice X“. In short,
Zc<K1)(X‘) = C(K1)(= Z(X‘))-
7.1. In this section, a mapping x" H A(a:“) and an algebra Z(X*) will be defined
for an arbitrary C(K)-module X. Also, for an arbitrary Banach lattice X , we will
define a bilinear mapping; (x*,:1:) H 1* u :r, and the spaces Lk and L3?; and, as we
will show, the order and algebraic properties of these objects (for Banach lattices and
C(K)modules) are absolutely similar.
7.2. It is plain to see that if X is an operator C(K)-module, then X * is likewise
a C(K)—module, provided one defines the action on X" in a natural way as
f I F H f'99,
where f* is the conjugate of f E C(K) C L(X). Consequently one can talk about
the center of this module. Therefore, the reader may be tempted to ask why we then
need a special definition for the center of the conjugate space. The point is that the
above suggested definition, though natural and sometimes useful (see for instance 9.14.3
and Proposition 9.8) does not allow us to preserve the desired parallelism between the
theory of C(K)—modules and that of Banach lattices.
We begin with a formal definition of a new mapping (temporarily denoted by)
I' i—> A(.r*) on X* using only the fact that X * is the conjugate of a space X carrying
the mapping a: I—+ A(a:). As soon as we show in Theorem 7.6 that in the case of a. C(K)-
42
module X this new mapping on X * coincides with the interval mapping z* H A(:c")
on the C(K1 )—module X * defined there, we will use only the latter notation.
7.3. Definition. Let X be an arbitrary Banach lattice or a C(K)-module and
1:" 6 X‘. Then let
A<x')=={y*e X‘zw‘ens sup Iw‘(y)l Vase-X}. yemr)
To justify the usage of the triangular symbol A(a*), we start by verifying that
for any Banach lattice X the just introduced mapping .73“ H A(a:*) coincides with the
canonical one 3* H A(a‘*) existing on the Banach lattice X *.
7.4. Theorem. Let X be an arbitrary Banach lattice. Then Ate“) = A(:::"‘) for
each 1" E X‘.
Proof. Take y" e X". The inclusion y“ 6 A(:c“), that is, the inequality ly‘l S ia'|
is equivalent to the condition that for each 1: E X we have |y*(rc)| S lx*|(]x|) =
3:13 )iz"(y)], and this is precisely the statement that y" E A(a‘*). I
116