existence of the c-envelope of an operator system · arveson introduced the c-envelope of a...

Existence of the C*-Envelope ofan Operator System

Adam Humeniuk

July 30, 2018

Master of Mathematics Research Paper

Department of Pure Mathematics

University of Waterloo

Abstract

In 1969, Arveson introduced the C*-envelope of an operator system

or operator algebra as a universal quotient amongst all C*-algebras it

generates. He left its existence as an open problem, expecting a proof

based on the existence of sufficiently many boundary representations.

In this paper, we present the diverse proofs of the C*-envelope’s ex-

istence that have been given in the four decades since. In full, we

present Hamana’s proof using the injective envelope, Dritschel and Mc-

Cullough’s dilation-theoretic argument, and Davidson and Kennedy’s

dilation-theoretic proof that enough boundary representations indeed

exist.

Contents

Contents

1 Introduction 1

2 Background 3

2.1 Operator Systems and Complete Positivity . . . . . . . . . . . 3

2.2 The Schwarz Inequality and Multiplicativity . . . . . . . . . . 5

2.3 Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 The Bounded Weak Topology . . . . . . . . . . . . . . . . . . 7

3 The C*-envelope and Boundary Representations 10

3.1 The C*-envelope . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Boundary Representations . . . . . . . . . . . . . . . . . . . . 11

3.3 The Commutative Case . . . . . . . . . . . . . . . . . . . . . . 16

4 The Injective Envelope 18

5 Maximal Dilations 25

6 Purity and Boundary Representations 32

7 Example: The Toeplitz-Cuntz System 39

7.1 Representations of the Toeplitz-Cuntz Algebra . . . . . . . . . 39

7.2 The C*-envelope of the Cuntz System . . . . . . . . . . . . . . 42

References 45

i

1. Introduction

1 Introduction

In 1969, William Arveson [1] established much of the study of operator systems

and non-selfadjoint operator algebras. These operator spaces exist concretely

as distinguished subspaces of C*-algebras. Arveson established an important

general question: To what extent does an operator space determine properties

of the C*-algebra it generates? Further, isomorphic copies of the same operator

system or algebra may exist within nonisomorphic C*-algebras. Which–if any,

properties do the various C*-algebras containing a given operator space share?

Arveson introduced the C*-envelope of a unital operator algebra or op-

erator system as one answer to this question. The C*-envelope is a universal

quotient amongst all C*-algebras generated by an operator system or algebra

X. That is, whenever A = C∗(X) is a C*-algebra generated by X (or an iso-

morphic copy), there is a unique ∗-homomorphism onto the C*-envelope that

preserves X. As a universal object in an appropriate category of C*-algebras

extending X, the C*-envelope is unique up to an isomorphism fixing X.

Arveson [1] was able to establish the existence of the C*-envelope for a class

of “admissible” operator spaces. The main hurdle in proving the existence

of the envelope in general was to show the existence of enough boundary

representations to completely norm all elements. These are representations

of an operator space that extend uniquely to irreducible representations of the

generated C*-algebra. Arveson left the existence of boundary representations,

and hence the C*-envelope, open.

The first proof that the C*-envelope exists would not appear for a full

decade. In 1979, Hamana [9] proved the existence of the injective envelope

of an operator system or algebra. This is the smallest injective C*-algebra

containing such an operator space. As a Corollary, Hamana showed that the

C*-subalgebra of the injective envelope that an operator space generates is

the C*-envelope. Though Hamana answered the question of the C*-envelope’s

existence, this didn’t prove that enough boundary representations exist. The

search for boundary representations would continue for decades.

Nearly two decades following, Muhly and Solel [10] gave an algebraic char-

acterization of boundary representations in terms of certain homological prop-

1

1. Introduction

erties. Using Hamana’s result, they established algebraic conditions for an

operator algebra to be completely normed by its boundary representations.

However, since Muhly and Solel’s result relied on Hamana’s previous proof,

this wasn’t a new proof of the existence of the C*-envelope.

In 2005, Dritschel and McCullough [8] gave an exciting new proof that the

C*-envelope exists. Their argument didn’t require the injective envelope, and

was instead purely dilation theoretic. The key tool was showing the existence

of maximal dilations of any representation of an operator space or algebra.

Maximally dilating a faithful representation yielded the C*-envelope.

With new inspiration from Dritschel and McCullough’s argument, Arveson

[2, 3] made new headway in the search for boundary representations. Using

maximal dilations, he showed in [3] that sufficiently many boundary represen-

tations existed in the case of a separable operator system. The general case was

close at hand, and in 2015, Davidson and Kennedy [6] finally established the

existence of enough boundary representations in general. Gratifyingly enough,

the key to their argument was present in Arveson’s original paper [1]: the use

of pure representations. By improving Dritschel and McCullough’s argument,

Davidson and Kennedy showed that a pure map can be dilated to a maximal

map while maintaining purity. The result is a boundary representation, and

there are enough pure maps to dilate to obtain sufficiently many boundary

representations.

The purpose of this Master’s research paper is to present the story of the

C*-envelope for operator systems. In a tour of the preceding arguments–which

span over four decades, we present the multiple proofs that the C*-envelope

exists. Following some preliminaries in Sections 2 and 3, we proceed chrono-

logically. In Section 4, we give Hamana’s proof that the injective envelope

exists, and show how it yields the C*-envelope. Section 5 details Dritschel and

McCullough’s proof that maximal dilations exist, and can be used to produce

the C*-envelope. Finally, we give Davidson and Kennedy’s recent dilation-

theoretic proof of sufficiency of boundary representations in Section 6.

Along the way, the reader is encouraged to note the pervasive importance

of completely bounded and completely positive maps. Most of our attention

will be dedicated to unital and completely positive (u.c.p.) maps. These are

2

2. Background

arguably the best notion of “morphism” between operator systems, and they

have fantastic extension and dilation properties. The two figureheads of these

properties are Arveson’s extension theorem and Stinespring’s dilation theorem,

respectively, which we’ll use repeatedly throughout.

2 Background

2.1 Operator Systems and Complete Positivity

We assume all basic facts and definitions surrounding concrete and abstract

C*-algebras as in [5]. All vector spaces in this paper are over C; and if V and

W are vector spaces, we write V ≤ W when V is a subspace of W . If H is

a Hilbert space, B(H) denotes the C*-algebra of bounded linear operators on

H. For a thorough treatment of the basic results and definitions in this section

we refer the reader to [11].

If A is a C*-algebra, Mn(A) denotes the ring of n×n matrices with entries

in A. Then Mn(A) is itself a C*-algebra with involution A = (aij)ni,j=1 7→

A∗ := (a∗ji)ni,j=1. The C*-norm is found by regarding Mn(A) as a concrete C*-

subalgebra of B(H⊕n) whenever A is represented faithfully as a C*-subalgebra

of B(H), for a Hilbert space H. When A = C we write Mn(C) =: Mn for

brevity. The C*-algebra Mn(A) is canonically isometrically ∗-isomorphic to

the tensor product A⊗Mn by the linear map that identifies a⊗ (λij)ni,j=1 with

(λija)ni,j=1 where a ∈ A and λij ∈ C.

Let A be a unital C*-algebra with identity 1. A subspace X ≤ A is called

an operator space. An operator system is a unital operator space that

is self-adjoint. A (unital) operator algebra is an operator space that is a

unital subalgebra. The closure of an operator space X (resp. operator system,

operator algebra) is again an operator space (resp. operator system, operator

algebra), and Mn(X) is an operator space (resp. operator system, operator

algebra) in Mn(A). If X is an operator space, then S = C1 + X + X∗ is the

smallest operator system containing it.

Recall that an element a ∈ A is positive if it is self-adjoint and has

nonnegative spectrum. It is of the form a = b∗b for some b ∈ A, and always

3

2. Background

acts by a positive operator in any ∗-representation A → B(H). An operator

system S is spanned by its positive elements, since if a = a∗ ∈ S is self-adjoint

then (‖a‖1 ± a)/2 ∈ S are positive elements whose difference is a. Since the

self-adjoints span, this is enough. Conversely any unital subspace spanned by

a set of positive elements is an operator system.

Let A and B be unital C*-algebras, S ≤ A an operator space, and ϕ : S →B a linear map. If S is an operator system we say ϕ is positive if whenever

a ∈ S+ := {b ∈ S | b ≥ 0} then ϕ(a) ≥ 0. A positive map is automatically

bounded with ‖ϕ‖ ≤ 2‖ϕ(1)‖ and self-adjoint meaning ϕ(a∗) = ϕ(a)∗ for

a ∈ S. For each n ≥ 0 there is an induced map ϕn : Mn(S) → Mn(B)

defined on A = (aij)ni,j=1 by ϕn(A) = (ϕ(aij))

ni,j=1. Alternatively identifying

Mn(S) ∼= S ⊗ Mn and Mn(B) ∼= B ⊗ Mn the map ϕn is simply ϕ ⊗ idMn

where idMn is the identity map. For any n the assignment ϕ 7→ ϕn is linear,

functorial, and norm-increasing. The map ϕ is n-positive if ϕn is positive,

and completely positive (c.p.) if it’s n-positive for all n ≥ 1. If ϕ is c.p.,

we write ϕ ≥ 0 and more generally if ϕ, ψ : S → B are linear we write ϕ ≥ ψ

when ϕ−ψ is c.p.. This gives a partial ordering. We say that ϕ is completely

bounded (c.b.) if

‖ϕ‖cb := sup{‖ϕn‖ | n ≥ 1} <∞,

completely contractive (c.c.) if ‖ϕ‖cb ≤ 1, and completely isometric if

ϕn is an isometry for every n. Also, ϕ is unital if ϕ(1) = 1. Since the identity

in Mn(S) is 1 ⊗ In where In ∈ Mn is the identity matrix this implies ϕn is

unital for all n. Finally if ϕ is an isomorphism and ϕ and ϕ−1 are both c.p.,

we call ϕ a complete order isomorphism.

If ϕ is 2-positive then ‖ϕ‖ = ‖ϕ(1)‖. If ϕ is completely positive then

‖ϕ‖cb = ‖ϕ(1)‖. Therefore a unital completely positive (u.c.p.) map is a

complete contraction. Conversely if S is a unital operator space and ϕ is

unital and completely contractive (u.c.c.) then ϕ extends to a unique u.c.p.

map ϕ : S + S∗ → B via ϕ(a + b∗) = ϕ(a) + ϕ(b)∗. In particular if S is

an operator system then ϕ = ϕ and a unital map ϕ : S → B is completely

positive if and only if it’s completely contractive. A completely positive (resp.

4

2. Background

completely bounded) ϕ : S → B is continuous and so extends uniquely to a

completely positive (resp. completely bounded) map ϕ : S → B on the closure

S. Conversely any completely positive ψ : S → B determines a c.p. map

S → B by restriction. Thus when considering completely positive maps we

may assume for convenience that the domain is norm-closed.

Let π : A → B be a ∗-homomorphism. Then π is positive and contractive.

Moreover πn is again a ∗-homomorphism for any n, so π is completely positive

and completely contractive. Further if π is injective then π is a complete isom-

etry. In this way the study of operator systems and u.c.p. maps generalizes

the study of unital C*-algebras and unital ∗-homomorphisms. Much as a ∗-representation A → B(H) on a Hilbert space H is a unital ∗-homomorphism,

we define a representation of an operator system S on H as a u.c.p. map

ϕ : S → B(H). Say that ϕ is faithful if it’s injective.

2.2 The Schwarz Inequality and Multiplicativity

Proposition 2.1 (Schwarz Inequality). [11] Let S be an operator system and

ϕ a u.c.p. map with domain S. For a ∈ S

ϕ(a)∗ϕ(a) ≤ ϕ(a∗a).

Proof. A self-adjoint operator matrix of the form

A =

(1 a

a∗ b

)

is positive if and only if b ≥ a∗a. Setting b = a∗a gives A ≥ 0 and since ϕ is

u.c.p.

ϕ2(A) =

(1 ϕ(a)

ϕ(a)∗ ϕ(a∗a)

)which implies ϕ(a)∗ϕ(a) ≤ ϕ(a∗a).

Failure of equality in the Schwarz inequality measures how much the u.c.p.

map ϕ fails to be multiplicative.

5

2. Background

Proposition 2.2. Let ϕ be a u.c.p. map on a C*-algebra A. Given a ∈ A,

ϕ(a)∗ϕ(a) = ϕ(a∗a) if and only if ϕ(ba) = ϕ(b)ϕ(a) for any b ∈ A. Similarly

ϕ(a)ϕ(a)∗ = ϕ(aa∗) if and only if ϕ(ab) = ϕ(a)ϕ(b) for every b ∈ A.

Proof. We prove only the first claim and the second follows immediately by

replacing a with a∗ and considering b∗ instead of b. If ϕ(ba) = ϕ(b)ϕ(a) always

we can set b = a∗. Conversely suppose ϕ(a∗a) = ϕ(a)∗ϕ(a). Let

A :=

(a b∗

0 0

).

The Schwarz inequality for ϕ2 gives

ϕ2(A∗A) =

(ϕ(a∗a) ϕ(ba)∗

ϕ(ba) ϕ(bb∗)

)≥

(ϕ(a)∗ϕ(a) ϕ(a)∗ϕ(b)∗

ϕ(b)ϕ(a) ϕ(b)ϕ(b)∗

)= ϕ2(A)∗ϕ2(A).

So

ϕ2(A∗A)− ϕ2(A)∗ϕ2(A) =

(0 (ϕ(ba)− ϕ(b)ϕ(a))∗

ϕ(ba)− ϕ(b)ϕ(a) ϕ(bb∗)− ϕ(b)ϕ(b)∗

)

is positive. This is only possible if the off-diagonal entry ϕ(ba) − ϕ(b)ϕ(a) is

zero.

We call the set {a ∈ A | ϕ(a∗a) = ϕ(a)∗ϕ(a) and ϕ(aa∗) = ϕ(a)ϕ(a)∗}the multiplicative domain of ϕ. If a lies in the multiplicative domain then

we have ϕ(ba) = ϕ(b)ϕ(a) and ϕ(ab) = ϕ(a)ϕ(b) for every b ∈ A. Notably

a u.c.p. map with domain a C*-algebra is a ∗-homomorphism if and only if

equality always holds in the Schwarz inequality.

2.3 Dilations

Let A be a unital C*-algebra, S ≤ A a unital operator system, H and K

Hilbert spaces, and ϕ : S → B(H) a u.c.p. map. A u.c.p. map ψ : S → B(K)

dilates ϕ if there is an isometry V : H → K with ϕ(a) = V ∗ψ(a)V for

all a ∈ S. By identifying H with the Hilbert space V H ≤ K we hereafter

6

2. Background

require that H ⊆ K and that ϕ(a) = PHψ(a)|H for all a ∈ S where PH is the

projection to H. That is, ϕ is a compression of ψ. If ψ dilates ϕ we write

ϕ � ψ.

Stinespring’s dilation theorem asserts that if ϕ : A → B(H) is a u.c.p. map

whose domain is a C*-algebra A, then ϕ � π for some unital ∗-homomorphism

π : A → B(K). Further there is a minimal Stinespring dilation π with

the property that K = π(A)H which is unique up to a unitary equivalence

which is the identity on H. Arveson’s extension theorem asserts that if S ≤A is an operator system and ϕ : S → B(H) is a c.p. map, then there is

a c.p. map ψ : A → B(H) which extends ϕ. Taken together, any u.c.p.

map ϕ : S → B(H) is dilated by a map of the form π|S where π : A →B(K) is a unital ∗-homomorphism. However a given operator system may be

represented completely isometrically inside of various C*-algebras and so the

minimal Stinespring dilation of a map S → B(H) need not be unique.

2.4 The Bounded Weak Topology

Completely positive representations of an operator system S on a Hilbert space

are bounded and so live as a subspace of the bounded operators B(S,B(H)).

Equipped with the usual norm topology,

UCP(S,B(H)) := {ϕ : S → B(H) | ϕ is u.c.p.}

is a closed and completely bounded subset but generally fails to be compact.

To furnish compactness arguments it will be essential to build a locally convex

topology on B(S,B(H)) in which UCP(S,B(H)) is compact. This will allow

us to–for instance, extract a convergent subnet from a collection of u.c.p.

dilations of a u.c.p. map.

Definition 2.3. Let X and Y be normed vector spaces. The bounded weak

(BW) or pointwise weak-∗ topology on B(X, Y ∗) is the locally convex topol-

ogy induced by the bounded linear functionals

fx,y : T 7→ T (x)(y),

7

2. Background

for x ∈ X and y ∈ Y .

If Z = span {fx,y | x ∈ X, y ∈ Y } then the dual pairing B(X, Y ∗)×Z → Cdefined on generators of Z by (T, fx,y) 7→ T (x)(y) = fx,y(T ) extends to an

isometric isomorphism B(X, Y ∗) ∼= Z∗. Furthe,r through this isomorphism

the BW topology clearly coincides with the weak-∗ topology on Z∗. Hence by

the Banach-Alaoglu theorem closed balls in B(X, Y ∗) are compact in the BW

topology.

If H is a Hilbert space then H∗ ∼= H and so B(H) ∼= B(H,H∗) is a

dual space by the above argument. This weak-∗ topology on B(H) is unique

and is the same as that induced by viewing B(H) as the dual of the trace

class operators on H. Thus we can equip B(S,B(H)) with the BW topology

whenever S is a normed linear space.

Proposition 2.4. Let S be an operator space and H a Hilbert space. Given

ϕ ∈ B(S,B(H)) and a norm-bounded net (ϕλ)λ∈I ⊆ B(S,B(H)), the following

are equivalent.

1) ϕλ → ϕ in the BW topology.

2) For each a ∈ S and x, y ∈ H we have 〈ϕλ(a)x, y〉 → 〈ϕ(a)x, y〉.

Proof. For x, y ∈ H consider the trace class operators Tx,y ∈ B(H) given by

Tx,y(z) = 〈z, y〉x. The span of such operators is norm dense in the trace class

operators. Hence if ϕλ is a bounded net and 〈ϕλ(a)x, y〉 = tr (ϕλ(a)Tx,y) →〈ϕ(a)x, y〉 = tr (ϕ(a)Tx,y), it follows that tr (ϕλ(a)A) → tr (ϕ(a)A) for all

trace class A, so ϕλ → ϕ BW. Conversely if we have BW convergence ϕλ → ϕ,

then 〈ϕλ(a)x, y〉 = tr (ϕλ(a)Tx,y)→ tr (ϕ(a)Tx,y) = 〈ϕ(a)x, y〉 for every x, y ∈H.

The BW topology has nice compactness properties and interacts nicely

with the matrix norm and order structures discussed this far.

Proposition 2.5. Let S be an operator system, H a Hilbert space, and fix

8

2. Background

r > 0. The following subsets of B(S,B(H)) are BW-compact.

CBr(S,B(H)) := {ϕ : S → B(H) | ‖ϕ‖cb ≤ r},CC(S,B(H)) := CB1(S,B(H)),

CPr(S,B(H)) := {ϕ : S → B(H) | ϕ c.p. and ‖ϕ‖ ≤ r},UCP(S,B(H)) = {ϕ : S → B(H) u.c.p.}.

Proof. These are all subsets of the BW-compact set

Br(B(S,B(H))) = {ϕ : S → B(H) | ‖ϕ‖ ≤ r}

for either some r > 0 or r = 1, so it suffices to prove they’re BW-closed. Let

ϕλ, ϕ ∈ B(S,B(H)) and suppose ϕλ → ϕ BW. Given any n ∈ N, we can iden-

tify Mn(B(H)) with B(H⊕n) and it follows that (ϕλ)n ∈ B(Mn(S), B(H⊕n))

converges BW to ϕn by Proposition 2.4 since ‖(ϕλ)n‖ ≤ r and for A = (aij) ∈Mn(S) and x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ H⊕n we have

〈ϕn(A)x, y〉 =∑i,j

〈ϕn(aij)xj, yi〉 = limλ

∑i,j

〈ϕλ(aij)xj, yi〉 = limλ〈(ϕλ)n(A)x, y〉 .

So for x, y ∈ H and a ∈ S,

| 〈ϕ(a)x, y〉 | = lim | 〈ϕλ(a)x, y〉 | ≤ r‖a‖‖x‖‖y‖

so ‖ϕ(a)‖ ≤ r‖a‖ and ‖ϕ‖ ≤ r. Since (ϕλ)n → ϕn BW the same argument

shows ‖ϕn‖ ≤ r for each n and this proves CBr(S,B(H)) is compact. Now

suppose ϕλ are c.p. maps. Given x ∈ H and a ∈ S+ we have 〈ϕ(a)x, x〉 =

lim 〈ϕλ(a)x, x〉 ≥ 0 since [0,∞) ⊂ C is closed. Again (ϕλ)n → ϕn so the same

argument in H⊕n shows ϕ is c.p., so CPr(S,B(H)) is compact. Lastly suppose

each ϕλ is unital. Then for x, y ∈ H we have 〈ϕ(1)x, y〉 = lim 〈ϕλ(I)x, y〉 =

lim 〈x, y〉 = 〈x, y〉 which proves ϕ(1) = I, so UCP(S,B(H)) is compact.

9

3. The C*-envelope and Boundary Representations

3 The C*-envelope and Boundary Represen-

tations

3.1 The C*-envelope

Given a subset U of a unital C*-algebra A, C∗(U) denotes the smallest unital

C*-subalgebra of A containing U .

Definition 3.1. [1, 11] Let S ≤ A be an operator system in a unital C*-

algebra A. Given a u.c.p. map ϕ : S → B, we say B is a C*-envelope of S

and write B = C∗e (S) if

1) ϕ is completely isometric,

2) B = C∗(ϕ(S)),

3) Whenever ψ : S → C is a unital complete isometry into a unital C*-

algebra C with C = C∗(ψ(S)), there is a ∗-homomorphism π : C → Bsuch that the diagram

S C

B

ψ

ϕπ

commutes.

Remark 3.2. The ∗-homomorphism π : C → B is necessarily surjective and

unique: Its range is a C*-algebra and contains ϕ(S) = π(ψ(S)) so π(C) = B.

If π′ : C → B is another ∗-homomorphism with π′ ◦ ψ = π ◦ ψ = ϕ then

π′ and π are equal on the ∗-subalgebra generated by ψ(S) which is dense in

C∗(ψ(S)) = C, so π′ = π by continuity.

Remark 3.3. One can define the C*-envelope of an operator algebra by re-

quiring that the complete isometries involved are also unital algebra homo-

morphisms [11]. Except for selected examples and remarks to follow, this

paper will focus entirely on the situation for operator systems. All of the

analogous results for operator algebras follow upon making only the obvious

modifications.

10


The C*-envelope is unique up to isomorphism.

Proposition 3.4. Let S be a unital operator system and let ϕ : S → B =

C∗(ϕ(S)) and ψ : S → C = C∗(ψ(S)) give C*-envelopes of S. There is a

∗-isomorphism π : B → C with π ◦ ϕ = ψ.

Proof. There are unique onto ∗-homomorphisms π : B → C and σ : C → Bwith π ◦ ϕ = ψ and σ ◦ ψ = ϕ. Then σ ◦ π is a ∗-homomorphism B → B with

σ ◦π ◦ϕ = σ ◦ψ = ϕ, and by the uniqueness in Remark 3.2 σ ◦π is the identity

map idB. An identical argument gives π ◦ σ = idC so π and σ are mutually

inverse ∗-homomorphisms.

Remark 3.5. Given an operator system S one may form a category of C*-

extensions of S whose objects are C*-algebras A equipped with complete

isometries ϕ : S → A, where A = C∗(ϕ(S)). The right notion of morphism

between C*-extensions ϕ : S → A and ψ : S → B is a ∗-homomorphism

π : A → B with π ◦ ϕ = ψ. The C*-envelope of S is a simply a final object in

this category.

3.2 Boundary Representations

Let A be a unital C*-algebra, S ≤ A an operator system with A = C∗(S),

and H a Hilbert space. Arveson’s extension theorem asserts that there is a

u.c.p. map ϕ : A → B(H) that extends S. A u.c.p. map ϕ : S → B(H) is

said to have the unique extension property if ϕ has a unique u.c.p. exten-

sion, and this extension is a ∗-homomorphism. Further, call ϕ a boundary

representation if the unique extension ϕ is an irreducible representation of

A [1, 6].

One might object that the same operator system S can be completely

isometrically represented inside many C*-algebras A = C∗(S), and so the

definition of the unique extension property requires a particular choice of A.

The next result shows that the unique extension property is invariant under

complete isometries, so is really determined by the order isomorphism class of

S alone.

11


Proposition 3.6. [1] Let S ≤ A be an operator system with A = C∗(S).

Suppose ψ : S → B is a unital complete isometry, where B = C∗(ψ(S)).

If ϕ : S → B(H) is a u.c.p. map with the unique extension property, then

ϕ ◦ ψ−1 : ψ(S) → B(H) also has the unique extension property. Further if ϕ

is a boundary representation, so is ψ.

Proof. By concreteness we may assume B ≤ B(J) for some Hilbert space J .

By Arveson’s extension theorem we can extend ψ : S → B to a u.c.p. map

ψ : A → B(J). Let π : A → B(H) be the unique ∗-homomorphism extending

ϕ. By Arveson’s extension theorem, there is an extension ρ : B → B(H)

of ϕ ◦ ψ−1. We must show ρ is unique and multiplicative. If we show mul-

tiplicativity, uniqueness follows because B = C∗(ψ(S)) and so any two ∗-homomorphisms which agree on ψ(S) are equal. Further, if the unique exten-

sion π is irreducible, then π(S) = ρ(ψ(S)) has no proper invariant subspaces

and since ψ(S) generates B, ρ is an irreducible representation, too.

Let C = C∗(ψ(A), so B ≤ C ≤ B(J). Let ρ be a u.c.p. extension C → B(H)

of ϕ◦ψ−1 : ψ(S)→ B(H). Take a minimal Stinespring dilation θ : C → B(K)

of ρ, i.e. a ∗-homomorphism with θ(C)H dense in K. Given a ∈ S,

PHθ(ψ(a))|H = ρ(ψ(a)) = ϕ(a).

Since ϕ has the unique extension property, we must have

π(a) = PHθ(ψ(a))|H

for a ∈ A.

Let u ∈ A be unitary and let x ∈ H. Compute

‖θ(ψ(u))x− π(u)x‖2 = ‖θ(ψ(u))x‖2 + ‖π(u)x‖2 − 2 Re⟨θ(ψ(u))x, π(u)x

⟩= ‖θ(ψ(u))x‖2 − ‖π(u)x‖2 (π(u)x = PHπ(u)x)

= ‖θ( ˜ψ(u))‖2 − ‖x‖2 (π(u) unitary)

≤ 0

by complete contractivity. Thus θ(ψ(u))|H = π(u) = PHθ(ψ(u))|H so H is

12


invariant for θ◦ψ(u). SinceA is spanned by its unitary elements, H is invariant

for

θ(C) = θ(C∗(ψ(A)) ≤ C∗(θ(ψ(A)).

Since θ(C)H = H is dense in K, K = H and so ρ = θ is its own minimal

Stinespring dilation. This shows ρ = ρ|B is a ∗-homomorphism.

Theorem 3.7. Let S be an operator system. The boundary representations

of S completely norm S. That is, given A ∈ Mn(S) there is a boundary

representation ϕ : S → B(H) with ‖ϕn(A)‖ = ‖A‖.

In [1], Arveson called operator systems S which satisfy the conclusion of

Theorem 3.7 “admissible”, and proved that admissible operator systems have

a C*-envelope. However, the correctness of Theorem 3.7 was left open. The

proof of Theorem 3.7 was finally given by Davidson and Kennedy in 2015 in

[6]. We postpone the proof until Section 6. Meanwhile, we show how Theorem

3.7 can give the C*-envelope. The next result dictates our strategy.

Theorem 3.8. Let S be an operator system. If ϕ : S → B(H) is a unital

complete isometry with the unique extension property, then ϕ : S → C∗(ϕ(S))

is a C*-envelope for S.

Proof. Let ψ : S → A be any unital complete isometry into a unital C*-algebra

A = C∗(ψ(S)). Because ϕ : S → B(H) has the unique extension property,

so does ϕ ◦ ψ−1 : ψ(S) → B(H), by Proposition 3.6. So there is a unique

∗-homomorphism π : A → B(H) extending ϕ ◦ψ−1. Thus π ◦ψ = ϕ. Because

π is a ∗-homomorphism,

π(A) = π(C∗(ψ(S))) ⊆ C∗(ϕ(S))

so the image of π lies within C∗(ψ(S)).

In fact, the converse of Theorem 3.7 holds in the following sense. If ϕ : S →B = C∗(ϕ(S)) is a C*-envelope, and we assume B ≤ B(H) is represented as a

concrete C*-algebra, then ϕ is a complete isometry with the unique extension

property. This is because ϕ is multiplicative on S and so any extension to

13


C∗(S) is a uniquely determined ∗-homomorphism, whose image lies within

C∗(ϕ(S)).

In the next proof, we assume Theorem 3.7. The proof given in Section 6

avoids any circularity.

Theorem 3.9. Let S be an operator system. Then S has a C*-envelope.

Proof. By Theorem 3.7, S is completely normed by its boundary representa-

tions. Enumerate⋃n≥1Mn(S) = {xα | α ∈ I}, and with choice let {ϕα :

S → B(Hα) | α ∈ I} be a collection of boundary representations with

‖(ϕα)nα(xα)‖ = ‖xα‖ whenever xα ∈ Mnα(S). Because boundary represen-

tations extend to irreducible representations of A = C∗(S), we have dimHα ≤ℵ0 dimS so the direct sum

⊕α∈I Hα is a set. Consider the map

ϕ :=⊕α∈I

ϕα : S → B

(⊕α∈I

Hα

).

Identifying (⊕Hα

)⊕n ∼= ⊕H⊕nα ,

it is clear that ϕn =⊕

α(ϕα)n for any n ≥ 1. Thus each ϕn is isometric and

ϕ : S → B(H) is a unital complete isometry into H :=⊕

αHα.

We will show ϕ has the unique extension property. Let π : A → B(H) be

any u.c.p. extension toA = C∗(S). Since each Hα is reducing for ϕ(S) = π(S),

each Hα is reducing for π(A) ⊆ C∗(π(S)). Thus π =⊕

α∈I πα for some u.c.p.

maps πα = PHαπ(·)|Hα . But clearly πα|S = ϕα, and because ϕα has the unique

extension property, πα is a ∗-homomorphism. Thus π is a direct sum of ∗-homomorphisms, so is itself a ∗-homomorphism. Thus any u.c.p. extension is

multiplicative. Because S generates A as a C*-algebra, all ∗-homomorphisms

which agree on S are equal, so π is unique.

For now, we discuss another perspective on the C*-envelope. Instead of

considering ∗-homomorphisms, we instead think about closed ideals (their ker-

nels). This will reveal the situation in the commutative case. We take the

convention that all ideals in a C*-algebra must be closed.

14


Definition 3.10. Let S ≤ A be an operator system in a unital C*-algebra

A = C∗(S). An ideal I ≤ A is a boundary ideal for S if the natural

projection A → A/I is completely isometric when restricted to S. Call a

maximum (with respect to inclusion) boundary ideal the Silov ideal for S in

A [1, 2].

The Silov ideal is the kernel of the projection to the C*-envelope, if it

exists.

Proposition 3.11. Let S ≤ A = C∗(S) be an operator system in a unital C*-

algebra. If ϕ : S → B = C∗(ϕ(S)) is a C*-envelope for S, then J := kerϕ ≤ Ais the Silov ideal for S. Conversely, if J ≤ A is a Silov ideal for S, then the

quotient map S → A/J = C∗(S/J) is a C*-envelope for S.

Proof. Suppose the C*-envelope ϕ : S → B = C∗e (S) exists. Let I ≤ A be a

boundary ideal. The map

ι : S → A/I

with ι(a) = a+I is a unital complete isometry. So there is a ∗-homomorphism

π : A/I → B with π ◦ ι = ϕ. If a ∈ I, then ι(a) = 0 + I so ϕ(a) = π(ι(a)) = 0.

This shows I ≤ kerπ = J , so J is the Silov ideal.

Conversely suppose a Silov ideal J ≤ A exists, and ψ : S → A/J be the

projection ψ(a) = a+J . Since J is a boundary ideal, ϕ is completely isometric.

Assuming Theorem 3.9, let ϕ : S → B = C∗(ϕ(S)) be a C*-envelope for

S. There is an onto ∗-homomorphism π : A/J → B with π ◦ ψ = ϕ. Let

ρ : A → B be the ∗-homomorphism ρ(a) = π(a + J). Then J ⊆ ker ρ. Since

ρ|S = ϕ is completely isometric, ker ρ is a boundary ideal so ker ρ ⊆ J . Thus

kerπ = (ker ρ)/J = {0 + J} is trivial and π is a ∗-isomorphism.

Proposition 3.12. If S ≤ A is an operator system in a simple (lacks non-

trivial ideals) C*-algebra A = C∗(S), then A = C∗e (S) is the C*-envelope of S

when equipped with the inclusion map S → A.

Proof. There is an onto ∗-homomorphism A → C∗e (S) which commutes with

inclusion. Since A is simple, the kernel must be trivial and this map is a

∗-isomorphism.

15


Example 3.13. Proposition 3.12 gives some easy finite dimensional examples.

For instance, the operator system

S = {A ∈Mn | A is tridiagonal} ≤Mn

and the unital operator algebra

T = {A ∈Mn | A is upper triangular} ≤Mn

both generate Mn = C∗(S) = C∗(T ). Since Mn is simple,

C∗e (S) = C∗e (T ) = Mn.

Example 3.14. The Cuntz algebraOn is the universal C*-algebra generated

by n isometries {S1, . . . , Sn} with pairwise orthogonal ranges that satisfy [5,

4]n∑k=1

SkS∗k = I.

Any family of isometries satisfying this relation we will call a Cuntz family.

Define the Cuntz system

Sn := span {I, S1, . . . , Sn, S∗1 , . . . , S

∗n} ≤ On,

the smallest operator system containing the generators of Sn. The Cuntz

algebra On = C∗(Sn) is simple [4]. Hence by Proposition 3.12, On = C∗e (Sn).

3.3 The Commutative Case

Every commutative unital C*-algebra is of the form C(X) for some compact

Hausdorff space X [5]. The closed ideals of C(X) are in bijective correspon-

dence with closed subsets Y ⊆ X, where the corresponding ideal is the space

of functions in C(X) which vanish identically on Y , which is the kernel of the

16


restriction map

C(X)→ C(Y )

f 7→ f |Y .

Consider an operator system S ≤ C(X). If S separates points, then C(X) =

C∗(S) by the Stone-Weierstrass theorem. Conversely if C∗(S) = C(X) then

S must separate points, so C∗(S) = C(X) if and only if S is point-separating.

Given a subspace S ≤ C(X), call a closed subset Y ⊆ X a boundary for S

if the restriction map C(X)→ C(Y ) is isometric (hence completely isometric)

on S. Call a boundary which is minimum with respect to inclusion amongst

all boundaries the Silov boundary for S, and denote it ∂SX ⊆ X. The Silov

boundary for any point-separating subspace that contains the constants exists

[1]. In the correspondence between ideals and closed subsets, boundary ideals

correspond exactly to boundaries. Further, the maximum boundary ideal–the

Silov ideal, is associated to the minimum boundary–the Silov boundary.

In summary, when S ≤ C(X) is an operator system with C∗(S) = C(X)–

i.e. S separates points, then the C*-envelope of S exists and is the C*-algebra

C(∂SX), equipped with the completely isometric restriction map

S → C(∂SX)

f 7→ f |∂SX .

Boundary representations of S are restrictions of certain irreducible represen-

tations of C∗e (S) = C(∂SX). Every irreducible representation is of the form

f 7→ f(x), for x ∈ ∂SX. By the Riesz-Markov representation theorem such

irreducible representations are also represented by Radon measures on ∂SX.

The boundary representations only correspond to those points of the Silov

boundary of S which have unique representing measures, necessarily the point

masses. This is the Choquet boundary of S in X. Though the Choquet

boundary is dense in the Silov boundary, the Choquet boundary need not be

closed and so may be a strictly smaller set.

Example 3.15. An important commutative operator algebra is the disk al-

17

4. The Injective Envelope

gebra

A(D) := {f ∈ C(D) | f |T is holomorphic} ≤ C(D).

Here D = {z ∈ C | |z| < 1} is the open disk and T = ∂D = {z | |z| = 1}is the circle. By the maximum modulus principle, any f ∈ A(D) attains its

norm only on ∂D = T, so the restriction map A(D) → C(T) is (completely)

isometric. Thus T is a boundary, so contains the Silov boundary for A(D).

For real θ,

fθ(z) := 1 + e−iθz

lies in the disk algebra and attains its maximum norm only at eiθ ∈ T. Hence

no proper closed subset of T is a boundary, so T is the Silov boundary of A(D)

and the restriction map A(D) → C(T) : f 7→ f |T is the C*-envelope. This

same argument shows that the finite dimensional operator system of harmonic

functions

S = span {1, z, z} ≤ C(D)

(here z means the identity function D→ D) also has C(T) as its C*-envelope.

4 The Injective Envelope

The notion of injectivity for operator systems might look familiar to some

readers from module theory, but it’s really a concept that makes sense in any

concrete category.

Definition 4.1. Let S be an operator system. Then S is injective if whenever

T and R are operator systems with T ≤ R and ϕ : T → S is a u.c.p. map,

there is a u.c.p. map ϕ : R→ S extending ϕ.

Proposition 4.2. Let S ≤ B(H) be a concrete operator system. The following

are equivalent:

• S is injective.

• There is a c.p. projection ϕ : B(H)→ B(H) with ϕ(B(H)) = S. (Note

that since 1 ∈ S any such map is unital.)

18


Proof. Suppose S is injective. The identity map S → S dilates to a u.c.p.

map ϕ : B(H)→ S and by composing with the inclusion map S → B(H) we

view ϕ as a map B(H)→ B(H). Since ϕ|S = idS we have S ⊆ ϕ(B(H)). But

ϕ(B(H)) ⊆ S so ϕ has S as its image. For any a ∈ B(H) we have ϕ(a) ∈ Sso ϕ(ϕ(a)) = ϕ(a) and ϕ is idempotent.

Conversely suppose a c.p. projection ϕ : B(H) → B(H) onto S exists.

Let ψ : T → S be any u.c.p. map where T is an operator system contained

in a larger operator system R. If ι : S → B(H) is the inclusion map then

ι ◦ ψ : T → B(H) extends by Arveson’s extension theorem to a u.c.p. map

σ : R → B(H). Then ϕ ◦ σ : R → S is u.c.p. and extends ψ since if t ∈ Tthen ϕ(σ(t)) = ϕ(ψ(t)) = ψ(t) because ψ(t) ∈ S.

Remark 4.3. Proposition 4.2 also shows that if we replace “u.c.p.” in Defini-

tion 4.1 with simply “c.p.” or “u.c.c.” the resultant injective operator systems

are unchanged.

Remark 4.4. Arveson’s extension theorem asserts that B(H) is an injective

operator system for any Hilbert space H.

The search for further injective operator systems is greatly simplified once

one identifies that they are all C*-algebras.

Theorem 4.5. [11] Every injective operator system is completely order iso-

morphic to a C*-algebra.

Proof (Sketch). Let S ≤ B(H) be a (concrete) operator system with projection

ϕ : B(H) → S as in Proposition 4.2. One defines a new multiplication on S

by a ◦ b := ϕ(ab), where ab is multiplication in B(H). One then checks this

yields an associative algebra product. Given a ∈ S we have

a∗a ≤ ‖a∗a‖I = ‖a‖2I

and since ϕ is u.c.p.

ϕ(a∗a) ≤ ‖a‖2I

19


so ‖ϕ(a∗a)‖ ≤ ‖a‖2. Yet the Schwarz inequality gives a∗a = ϕ(a)∗ϕ(a) ≤ϕ(a∗a) so ‖a‖2 = ‖a∗a‖ ≤ ‖ϕ(a∗a)‖. So

‖a∗ ◦ a‖ = ‖ϕ(a∗a)‖ = ‖a‖2

and the C*-identity holds. Thus (S, ∗) is a C*-algebra with the same norm.

Definition 4.6. Let S be an operator system. An injective envelope of S

is an injective operator system E equipped with a unital complete isometry

ι : S → E such that whenever F is an injective operator system with ι(S) ≤F ≤ E we must have F = E.

Theorem 4.7 (Hamana). [9, 11] Every operator system has an injective en-

velope. Further, injective envelopes are rigid: If S is an operator system with

ι : S → E an injective envelope then the only u.c.p. map ϕ : E → E with

ϕ ◦ ι = ι is the identity..

Before presenting a proof of Theorem 4.7 we’ll show how the injective

envelope yields the C*-envelope.

Proposition 4.8. Let S be an operator system and ι : S → E an injective

envelope for S, then ι : S → C∗(S) ⊆ E is a C*-envelope for S.

Proof. Let ϕ : S → A be a complete isometry, where A is a C*-algebra

satisfying A = C∗(ϕ(S)). We may assume A ≤ B(H) for some Hilbert space

H. The map ϕ(S)→ E given by ϕ(a) 7→ ι(a) is completely isometric and by

injectivity extends to a u.c.p. map π : B(H)→ E with π ◦ϕ = ι. The inverse

map ι(a) 7→ ϕ(a) also extends to a u.c.p. map ψ : E → B(H) by injectivity

of B(H). Then π(ψ(ι(a))) = ι(a) for any a ∈ S so by rigidity π ◦ ψ = idE.

We claim that π|A is a unital ∗-homomorphism. Since ψ is u.c.p. it is linear,

unital, and self-adjoint so we only need to show it’s multiplicative on A. Given

a ∈ S the Schwarz inequality gives ϕ(a)∗ϕ(a) = ψ(ι(a))∗ψ(ι(a)) ≤ ψ(ι(a)∗ι(a))

and upon applying π and the Schwarz inequality again

π(ϕ(a))∗π(ϕ(a)) ≤ π(ϕ(a)∗ϕ(a)) ≤ ι(a)∗ι(a).

20


Hence ι(a)∗ι(a) = π(ϕ(a))∗π(ϕ(a)) = π(ϕ(a)∗ϕ(a)). This shows that ϕ(S) lies

in the multiplicative domain of π. Since A = C∗(ϕ(S)), π is multiplicative on

all of A.

So to prove existence of the C*-envelope we may first prove that every

operator system has an injective envelope. In light of Proposition 4.2 we may

assume we are working with a concrete operator system S ≤ B(H) and search

for a c.p. projection ϕ ∈ B(H) whose range contains S. Towards this end we

first widen our scope to only require maps that fix S.

Definition 4.9. Let S ≤ B(H) be an operator system. Call a bounded linear

map ϕ : B(H)→ B(H) an S-map if ϕ|S = idS. Call a seminorm p on B(H)

an S-seminorm if

p(a) = pϕ(a) := ‖ϕ(a)‖

for an S-map ϕ.

Order the seminorms on B(H) by the pointwise ordering where p ≤ q if

and only if p(a) ≤ q(a) for every a. This gives a partial ordering.

Proposition 4.10. [9, 11] Let S ≤ B(H) be an operator system. There exist

minimal S-seminorms with respect to the ordering ≤. If pϕ is a minimal S-

seminorm, then ϕ is a projection.

Proof. We’ll use Zorn’s lemma on the set of S-seminorms on B(H). This set

is nonempty since the identity map is an S-map so the usual norm is an S-

seminorm. Consider a decreasing chain of seminorms C = {pψ | ψ ∈ F} where

F is some set of S-maps. Since any u.c.p. map is norm 1 the set F is bounded

and so has a BW-limit point ϕ of a cofinal subnet (ϕλ). For x, y ∈ H and

a ∈ S we have

〈ϕ(a)x, y〉 = limλ〈ϕλ(a)x, y〉 = lim

λ〈ax, y〉 = 〈ax, y〉 .

Thus ϕ|S is the identity, whence ϕ is an S-map.

I claim pϕ is a lower bound. Let ψ ∈ F and a ∈ B(H) and x, y ∈ H

with ‖x‖ = ‖y‖ = 1. Given ε > 0 we can–using cofinality, find a ρ ∈ F with

21


pρ ≤ pψ and

| 〈ϕ(a)x, y〉 | < | 〈ρ(a)x, y〉 |+ ε ≤ ‖ρ(a)‖+ ε ≤ ‖ψ(a)‖+ ε.

Since ‖ϕ(a)‖ = sup{| 〈ϕ(a)x, y〉 | | ‖x‖ = ‖y‖ = 1} and ε was arbitrary we

have ‖ϕ(a)‖ ≤ ‖ψ(a)‖. As a was arbitrary this shows pϕ ≤ pψ. Thus each

decreasing chain has a lower bound and so minimal S-seminorms exist.

Clearly ϕ2 is an S-map and since ‖ϕ‖ = ‖ϕ(I)‖ = 1 we have ‖ϕ2(a)‖ ≤‖ϕ(a)‖ for any a. Thus by minimality pϕ2 = pϕ. Inductively it is clear that

ϕk is another S-map with ‖ϕk(a)‖ = ‖ϕ(a)‖ for any a ∈ B(H). Clearly

the set of S-maps is a convex subset of UCP(B(H), B(H)) and so Φk :=

(ϕ+ ϕ2 + · · ·+ ϕk)/k are also S-maps with

‖Φk(a)‖ ≤ ‖ϕ(a)‖+ · · ·+ ‖ϕk(a)‖k

= ‖ϕ(a)‖.

Again using minimality, pΦk = pϕ. Now for any a ∈ B(H)

‖ϕ2(a)− ϕ(a)‖ = ‖ϕ(ϕ(a)− a)‖= ‖Φk(ϕ(a)− a)‖

=‖ϕk+1(a)− ϕ(a)‖

k≤ 2‖a‖

k.

Taking k →∞ shows ϕ2 = ϕ. Hence ϕ is a u.c.p. projection.

Call an S-map ϕ that’s a projection an S-projection. (Note that the

range of ϕ may be larger than S.) The S-projection found in the proof above

turns out to provide an injective envelope.

Proof of Theorem 4.7. Let S be an operator system. Up to a complete or-

der isomorphism we can assume S ≤ B(H) for some Hilbert space H. Let

ϕ : B(H) → B(H) be an S-map with pϕ a minimal S-seminorm found

via Proposition 4.10. Thus ϕ is an S-projection. By Proposition 4.2 the

range E := ϕ(B(H)) is an injective operator system containing S. Suppose

S ≤ F ≤ E with F an injective operator system. Then by the other direction

of Proposition 4.2 there is a u.c.p. projection ψ : B(H)→ B(H) with F as its

22


range.

Since F ≤ E we have ψ = ϕ ◦ ψ. Thus ψ ◦ ϕ is a u.c.p. projection, too,

and because

‖ψ(ϕ(a))‖ ≤ ‖ϕ(a)‖

for any a, ‖ψ(ϕ(a))‖ = ‖ϕ(a)‖ by minimality of pϕ. Thus ψ acts isometrically

on E = ϕ(B(H)). Given a ∈ B(H),

ψ(ϕ(a)− ψ(ϕ(a))) = 0,

and since ϕ(a) − ψ ◦ ϕ(a) ∈ ϕ(B(H)), ϕ(a) − ψ(ϕ(a)) = 0 because ψ|E is

injective. Therefore ψ ◦ ϕ = ϕ, whence E ⊆ F . This shows E = F and so F

is an injective envelope.

Finally we claim E is rigid. Let ρ : E → E be any u.c.p. map with

ρ|S = idS. We must show ρ is the identity map. Since ρ fixes S, ρ ◦ ϕ is an

S-map with ‖ρ(ϕ(a))‖ ≤ ‖ϕ(a)‖ for a ∈ B(H). Hence pρ◦ϕ = pϕ and so by

the proof of Proposition 4.10 ρ ◦ ϕ is a projection. Given a ∈ B(H) we have

ρ(ϕ(a))− ϕ(a) ∈ E and so

‖ρ(ϕ(a))− ϕ(a)‖ = ‖ϕ(ρ(ϕ(a))− ϕ(a))‖= ‖ρ ◦ ϕ(ρ(ϕ(a))− ϕ(a))‖= ‖(ρ ◦ ϕ)2(a)− ρ ◦ ϕ(a)‖ = 0

because ϕ ◦ ρ = ρ (ran ρ ≤ E) and ρ ◦ ϕ is a projection. Hence ρ acts by the

identity on all of E = ϕ(B(H)).

Remark 4.11. One may order the u.c.p. projections on B(H) by ψ � ϕ

whenever ψ ◦ ϕ = ψ = ϕ ◦ ψ. In the proof above we showed that if ϕ is an

S-map with pϕ a minimal S-seminorm then ϕ is a minimal S-projection with

respect to the ordering �.

Though we didn’t need it to obtain the C*-envelope, it would be remiss to

not mention another important property of the injective envelope: essentiality.

Definition 4.12. Let S be an operator system. Call an extension ϕ : S → T

(a unital complete isometry into an operator system T ) an essential extension

23


of S if for all u.c.p. maps ψ : T →M–where M is an operator system, if ψ ◦ϕis a complete isometry this implies ψ is itself a complete isometry.

Proposition 4.13. Let ϕ : S → E be an injective envelope of an operator

system S. Then E is an essential extension of S.

Proof. Suppose ψ : E → M is a u.c.p. map with ψ acting completely iso-

metrically on ϕ(S). Then ψ−1 : ψ(ϕ(S)) → ϕ(S) ≤ E extends to a u.c.p.

map ρ : M → E. Now note ρ ◦ ψ acts by the identity on ϕ(S), so by rigidity

ρ ◦ ψ = idE. Hence for a ∈Mn(S),

‖a‖ = ‖ρn(ψn(a))‖ ≤ ‖ψn(a)‖ ≤ ‖a‖

by complete contractivity, so ψ is a complete isometry.

Remark 4.14. Suppose further that in the previous proof ψ ◦ ϕ : S → M is

itself an essential extension. Because ψ is isometric, the map ρ acts completely

isometrically on ψ◦ϕ(S) and so is itself completely isometric. Because ρ◦ψ◦ρ =

ρ, it follows that ψ ◦ ρ = idM and so ψ = ρ−1 and ψ is onto.

Remark 4.14 shows that the injective envelope is a maximal essential

extension: Any further extension which is essential (with respect to S, always)

is necessarily an isomorphism. This is a clue that the injective envelope is

actually characterized by maximal essentiality–or rigidity.

Theorem 4.15. [11] Let S be an operator system and ϕ : S → E a unital

complete isometry. The following are equivalent.

1) E is an injective envelope for S, i.e. a minimal injective extension.

2) ϕ : S → E is a maximal essential extension of S.

3) ϕ : S → E is a maximal rigid extension of S.

4) E is injective and ϕ : S → E is an essential extension of S.

5) E is injective and ϕ : S → E is a rigid extension of S.

24

5. Maximal Dilations

Proof. We omit almost all of the proof. Remark 4.14 establishes that (1)

implies (2). So, let’s prove (2) implies (1). By Theorem 4.7, there is an injective

envelope ψ : S → F for S. The complete isometry ψ ◦ϕ−1 : ϕ(S)→ F lifts to

a u.c.p. map ρ : E → F . By essentiality of E, ρ is a complete isometry. We

have a tower of extensions

S E F.ϕ ρ

Because ρ ◦ ϕ = ψ : S → F is an essential extension by Proposition 4.13, ρ

must be an isomorphism as ϕ : S → E is maximal among essential extensions.

It easily follows that E ∼= F is an injective envelope.

5 Maximal Dilations

Let S ≤ A be an operator system in a unital C*-algebra. Recall that a u.c.p.

map ϕ : S → B(H) is dilated by a u.c.p. map ψ : S → B(K) when K ≥ H

and ϕ(a) = PHϕ(a)|H for a ∈ S; and in this case we write ϕ � ψ. A dilation

ψ � ϕ is trivial if ψ = ϕ⊕ ρ for some u.c.p. map ρ : S → B(H⊥). The map

ϕ is maximal if all dilations of ϕ are trivial.

Proposition 5.1. Let S be an operator system and ϕ : S → B(H) a u.c.p.

map. The following are equivalent.

1) ϕ is maximal.

2) Whenever ψ � ϕ, H is a reducing subspace for ψ.

3) Whenever ψ � ϕ, H is an invariant subspace for ψ.

4) Whenever ψ � ϕ, we have ψ(a)x = ϕ(a)x for a ∈ S and x ∈ H.

Proof. If ϕ is maximal and ϕ � ψ then ψ = ϕ ⊕ ρ for some u.c.p. map

ρ, so H is reducing. Conversly suppose each dilation ψ of ϕ has H as a

reducing subspace. Then if ψ : S → B(K) dilates ϕ both H and H⊥ are

ψ-invariant and so ψ = ϕ ⊕ ρ for some linear map ρ : S → B(H⊥). Here

ρ(a) = PH⊥ψ(a)|H⊥ = V ∗ψ(a)V where V : H⊥ → K is the inclusion map.

25


Since the maps ψ and T 7→ V ∗TV are u.c.p., ρ is u.c.p.. Hence (1) and (2)

are equivalent.

Since S is self-adjoint and ϕ is a self-adjoint map, ϕ(S) is self-adjoint and so

invariant subspaces are automatically reducing. Thus (1)-(3) are equivalent.

Now suppose ϕ is maximal and ψ � ϕ. Then ψ = ϕ ⊕ ρ and so certainly

ψ(a)x = ϕ(a)x for a ∈ S and x ∈ H. Conversely, if ψ satisfies (4) then H is

clearly an invariant subspace for any dilation.

We are interested in maximal u.c.p. maps because they have the unique

extension property.

Proposition 5.2. [2] Let S be an operator system in a unital C*-algebra

A = C∗(S) and let ϕ : S → B(H) be a u.c.p. map. Then ϕ has the unique

extension property if and only if it’s maximal.

Proof. First suppose ϕ is maximal. By Arveson’s extension theorem, there

is a u.c.p. map π : A → B(H) that extends ϕ. By Stinespring’s dilation

theorem there is a unital ∗-homomorphism ρ : A → B(K) which dilates π

with K = ρ(A)H. But then ρ|S � ϕ so H ≤ K is reducing for ρ|S. Since

A = C∗(S), this implies H is reducing for all of ρ(A). Hence ρ(A)H = H, so

K = H, and so ρ = π is a unital ∗-homomorphism extending ϕ.

If ψ : A → B(H) is any other u.c.p. map that extends ϕ, the same

argument shows that ψ is a unital ∗-homomorphism. Since ψ and π agree on

S and A = C∗(S), ψ = π, showing π is unique. This proves that ϕ has the

unique extension property.

Now instead suppose ϕ has the unique extension property. Let ψ : S →B(K) be a dilation of ϕ. By Arveson’s extension theorem find a u.c.p. map

θ : A → B(K) which extends ψ. Consider the compression π : A → B(H)

of θ where π(a) = PHθ(a)|H . Since θ extends ψ and ψ compresses to ϕ, π

restricts to ϕ on S. By the unique extension property we conclude π is a

∗-homomorphism.

Let a ∈ A. By the Schwarz inequality ψ(a)∗ψ(a) ≤ ψ(a∗a). Upon com-

26


pressing to H,

PHψ(a)∗ψ(a)|H ≤ PHψ(a∗a)|H= π(a∗a)

= π(a)∗π(a) = PHψ(a)∗PHψ(a)|H .

By precomposing with PH ,

PHψ(a)∗ψ(a)PH ≤ PHψ(a)∗PHψ(a)∗PH

or rather

PHψ(a)∗(I − PH)ψ(a)PH = ((I − PH)ψ(a)PH)∗((I − PH)ψ(a)PH) ≤ 0.

Hence (I − PH)ψ(a)PH = 0. This implies that H is invariant for ψ. By

Proposition 5.1, ϕ is maximal.

Remark 5.3. Let A ≤ A be a unital operator algebra in a C*-algebra. The

C*-envelope of A is defined in an analogous way: We form a category

of C*-extensions, which consist of unital completely isometric algebra homo-

morphisms ϕ : A → B, where B is a C*-algebra with B = C∗(ϕ(A)). The

C*-envelope B = C∗e (A) is final in this category, i.e. whenever ψ : A →C = C∗(ψ(A)) is a completely isometric homomorphism, there is a unique

∗-homomorphism π : C → B with π ◦ ψ = ϕ.

Proposition 5.2 shows that maximal u.c.p. maps have the unique extension

property and so are multiplicative on their domain. Thus all techniques that

follow yield analogous results for operator algebras. By dilating maps on the

smallest operator system A+ A∗ which contains an operator algebra A, we

obtain u.c.c. representations which are also algebra homomorphisms when

restricted to A. Importantly, multiplicativity is ensured throughout.

Dritschel and McCullough [8] proved that, fortunately, maximal dilations

of u.c.p. maps always exist.

Theorem 5.4. [8, 2] Let S be an operator system and ϕ : S → B(H) a u.c.p.

map. Then ϕ has a maximal dilation.

27


We will give a proof of Theorem 5.4 below, but for now we show it yields

the C*-envelope.

Theorem 5.5. [8] Every operator system has a C*-envelope.

Proof. Let S ≤ A be an operator system. By faithfully representing A we

may find a complete isometry ι : S → B(H) for some Hilbert space H. By

Theorem 5.4 take a maximal dilation ϕ : S → B(K) of ι. Given n ∈ N we

identify Mn(B(K)) = B(K⊕n). The projection to H⊕n ≤ K⊕n is

PH⊕n =

PH · · · 0. . .

0 · · · PH

so we have PH⊕nϕn(A)|H⊕n = ιn(A) for A ∈ Mn(S). Hence ιn � ϕn for all n.

Since dilations increase norm, for A ∈ Mn(S) we have ‖ϕn(A)‖ ≥ ‖ιn(A)‖ =

‖A‖. Since ϕ is c.c., ‖ϕn(A)‖ = ‖A‖ and ϕ is a maximal complete isometry.

By Proposition 5.2, ϕ is a complete isometry with the unique extension

property. By Theorem 3.8, C∗e (S) = C∗(ϕ(S)).

In light of condition (4) in Proposition 5.1, we say a u.c.p. map ϕ : S →B(H) is maximal on a subset U ⊂ A×H if whenever ψ � ϕ, ψ(a)x = ϕ(a)x

for (a, x) ∈ U . We say ϕ is maximal at (a, x) ∈ S×H if it’s maximal on the

singleton subset {(a, x)}. Evidently ϕ is maximal if and only if it’s maximal

on S × H. The proof of Theorem 5.4 will proceed in three stages: Given a

pair (a, x) ∈ S ×H, we show there is a dilation of ϕ that’s maximal at (a, x).

Then, by induction, we show that there is a dilation ψ : S → B(K) which

is maximal on S ×H. Finally, by gluing together an inductively constructed

chain of such dilations we arrive at a maximal one.

Lemma 5.6. [6, 8] Let S be an operator system, H a Hilbert space, and

ϕ : S → B(H) a u.c.p. map. Given (a0, x0) ∈ S × H there is a dilation

ψ : S → B(K) of ϕ which is maximal at (a0, x0). The Hilbert space K can be

chosen such that K ∼= H ⊕ C.

28


Proof. Any dilation ψ : S → B(K) satisfies ‖ψ(a0)x0‖ ≤ ‖a0‖‖x0‖ and so

α := sup{‖ψ(a0)x0‖ | ψ � ϕ}

is finite. Further if L = span {H,ψ(a0)x0}, then ρ : S → B(L) with ρ(a) =

PLψ(a)|L dilates ϕ and ‖ρ(a0)x0‖ = ‖ψ(a0)x0‖. By isometrically embedding

L into H ⊕ C, we can replace ρ with a dilation ρ′ : S → B(H ⊕ C). This

dilation satisfies ‖ρ′(a0)x0‖ = ‖ψ(a0)x0‖. This shows that

α = sup{‖ψ(a0)x0‖ | ψ ∈ UCP(S,B(H ⊕ C)), ψ � ϕ}.

We seek a dilation ψ : S → B(H ⊕ C) which satisfies ‖ψ(a0)x0‖ = α. For

each n ∈ N, choose a u.c.p. dilation ϕn : S → B(H ⊕ C) with

‖ϕn(a0)x0‖ > α− 1

n.

Since UCP(S,B(H)) is BW-compact, we may find a u.c.p. BW-limit point

ψ of a cofinal subnet {ϕnλ}λ∈I of {ϕn}n∈N. Note ψ also dilates ϕn, since for

x, y ∈ H and a ∈ S, we have

〈ψ(a)x, y〉 = limλ〈ϕnλ(a)x, y〉 = 〈ϕ(a)x, y〉 .

Now

ψ(a0)x0 = ϕ(a0)x0 ⊕ β,

where

β = limλ〈ϕnλ(a0)x0, 0⊕ 1〉

satisfies

|β|2 = limλ‖ϕnλ(a0)x0‖2 − ‖ϕ(a0)x0‖2

= α2 − ‖ϕ(a0)x0‖2.

So

‖ψ(a0)x0‖ =√‖ϕ(a0)x0‖2 + |β|2 = α

29


is the maximum norm amongst all dilations of ϕ. Given a dilation ρ : S →B(J) of ψ, ρ dilates ϕ by transitivity. Thus ‖ρ(a0)x0‖ ≤ α. By the Pythagorean

identity,

‖ρ(a0)x0 − ψ(a0)x0‖2 = ‖(I − PH⊕C)ρ(a0)x0‖2

= ‖ρ(a0)x0‖2 − ‖PH⊕Cρ(a0)x0‖2

= ‖ρ(a0)x0‖2 − ‖ψ(a0)x0‖2 ≤ α2 − α2 = 0.

So ρ(a0)x0 = ψ(a0)x0 whenever ρ � ψ.

Lemma 5.7. [8, 2] Let S be an operator system, H a Hilbert space, and

ϕ : S → B(H) a u.c.p. map. If U ⊆ S ×H, there is a dilation of ϕ which is

maximal on U .

Proof. We use transfinite induction. Note first that if V ⊆ S ⊗H and ψ � ϕ

is maximal on V , then any any dilation ρ � ψ also dilates ϕ and is clearly also

maximal on V . Choose an ordinal γ large enough that we may enumerate

U = {(aλ, xλ) | λ < γ}.

For ordinals λ ≤ γ we will define a dilation ϕλ : S → B(Hλ) of ϕ which is

maximal on {(aµ, xµ) | µ < λ}. Further we ensure that Hλ ≤ Hµ when λ ≤ µ.

Start by setting ϕ0 = ϕ.

Suppose λ + 1 < γ is a successor ordinal and that ϕλ has been defined

and is maximal on Uλ. Using Lemma 5.6, find a dilation ϕλ+1 of ϕλ that’s

maximal on (aλ+1, xλ+1). Then ϕλ+1 is a dilation of ϕ that’s maximal on

Vλ ∪ {(aλ+1, xλ+1)} = Vλ+1.

Now consider the case where β ≤ γ is a limit ordinal. Suppose dilations

ϕλ : S → B(Hλ) maximal on Uλ have been defined. Further assume the

Hilbert spaces Hλ, λ < β, form a chain. Define Hβ to be the completion of

the inner product space⋃λ<βHλ, the inner product defined by extending that

on each Hλ. Then Hβ is a Hilbert space. For a ∈ S and x ∈ Hλ, λ < β, the

assignment

ϕβ(a) : x 7→ ϕλ(a)x

30


is well-defined, linear, and bounded (by ‖a‖), so extends to a bounded linear

operator ϕβ(a) ∈ B(Hβ). The map ϕλ is positive because for a ∈ S+ and

x, y ∈⋃λ<βHλ there is some common ordinal λ < β with x, y ∈ Hλ, whence

〈ϕβ(a)x, y〉 = 〈ϕλ(a)x, y〉 ≥ 0.

This shows ϕλ(a) is positive on a dense subset, which is enough. Similarly

ϕβ(1) acts by the identity on the dense subspace⋃λ<βHλ, so ϕβ(1) = I.

Hence ϕβ is a u.c.p. dilation of ϕ with Hβ ≥ Hλ for λ < β. For (aλ, xλ) ∈ Uβ,

λ + 1 < β because β is not a successor. Since ϕβ � ϕλ+1, ϕβ is maximal on

(aλ, xλ) and so maximal on all of Uβ.

This completes the induction. Setting ψ = ϕγ gives a dilation of ϕ that’s

maximal on Uγ = U .

Remark 5.8. The construction in the proof of Lemma 5.7 is general: Consider

a chain (in dilation order) of u.c.p. maps ϕλ : S → B(Hλ) indexed by ordinals

λ < γ, where γ is a limit ordinal and λ ≤ µ implies ϕλ � ϕµ. Define Hγ

as the completion of⋃λ<γ Hλ, i.e. the colimit of the Hilbert spaces {ϕλ}λ<µ.

There is a unique map ϕγ : S → B(Hγ) which satisfies ϕγ(a)|Hλ = ϕλ(a)

such that ϕλ � ϕγ for every λ < γ. Following [8], we call ϕλ the spanning

representation or spanning dilation of {ϕλ}λ<µ.

Lemma 5.7 is all we need to show that maximal dilations of any u.c.p. map

exist.

Proof of Theorem 5.4. Let ϕ : S → B(H) be a u.c.p. map on an operator

system S. Set ϕ0 = ϕ and H0 = H. By Lemma 5.7, find a dilation ϕ1 :

S → B(H1) of ϕ0 which is maximal on S×H0. Proceeding inductively, define

ϕ2 : S → B(H2) as a dilation of ϕ1 that’s maximal on S × H1 ⊆ S × H2.

Continuing in this fashion, we arrive at a chain of dilations

ϕ = ϕ0 � ϕ1 � ϕ2 � · · · .

For each n > 0, ϕn : S → B(Hn) is maximal on S × Hn−1. As in Remark

5.8, let ϕω : S → B(Hω) be the spanning representation of {ϕn}n≥0. Here

Hω =⋃n≥0Hn.

31

6. Purity and Boundary Representations

Let ψ : S → B(K) be a u.c.p. dilation of ϕω. Let a ∈ S and x ∈ Hn, for

some n ≥ 0. Since ψ � ϕn+1 and ϕn+1 is maximal at (a, x), we have

ψ(a)x = ϕn+1(a)x = ϕω(a)x.

Hence ψ(a)x = ϕω(a)x for x ∈⋃n≥0Hn. Since

⋃n≥0Hn is a dense subspace,

ψ(a)|Hω = ϕω(a)|Hω . As a was arbitrary, ϕω is maximal by Proposition 5.1.

Remark 5.9. [6] In the special case where S is separable and H is finite

dimensional, we can refine the proof of Theorem 5.4 given above as follows:

At the kth step of induction, we perform the dilation process in Lemma 5.6

for only a finite set of pairs (ai, x(k)i ), 1 ≤ i ≤ Nk, hence keeping Hk+1 finite

dimensional. Here {x(k)1 , . . . , x

(k)Nk} forms a 1/k-net for the unit ball of Hk,

limkNk = ∞, and {a1, a2, . . .} is a dense subset of S. Thus each Hk is finite

dimensional and the colimit Hω =⋃nHn is separable. Since the set {(ai, x(k)

i ) |k ≥ 1, 1 ≤ i ≤ Nk} is dense in S × Hω, it follows by continuity that ϕω is

maximal.

6 Purity and Boundary Representations

Given operator systems S and T , recall that we partially order the c.p. maps

S → T by declaring that ϕ ≤ ψ if and only if ψ−ϕ is completely positive [1].

Call a c.p. map ϕ : S → T pure if for ψ ∈ CP(S, T ), ψ ≤ ϕ implies that ψ is

a scalar multiple of ϕ.

Remark 6.1. ϕ ∈ CP(S, T ) is pure if and only if the only decomposition of

ϕ as a sum ϕ = ϕ1 + ϕ2 of c.p. maps ϕ1, ϕ2, is when ϕ1 and ϕ2 are scalar

multiples of ϕ: If ϕ = ϕ1 + ϕ2 is pure, then ϕ1, ϕ2 ≤ ϕ so they are scalar

multiples. Conversely if ψ ≤ ϕ, then ϕ = ψ + (ϕ− ψ) as a sum of c.p. maps.

If the only decomposition of ϕ as a sum of c.p. maps via scalar multiples, this

implies ψ is a scalar multiple of ϕ.

We are most interested in pure maximal maps. Maximal maps have the

unique extension property, and purity guarantees the unique extension is irre-

ducible.

32


Proposition 6.2. [6] Let S ≤ A be an operator system with A = C∗(S). If

ϕ : S → B(H) is a pure maximal u.c.p. map, ϕ is a boundary representation.

Proof. By Proposition 5.2 ϕ has the unique extension property. Let π : A →B(H) be the unique ∗-homomorphism extending ϕ. Suppose π was not an

irreducible representation. Then there is a proper projection P ∈ B(H) that

commutes with π(A) ⊇ ϕ(S). Thus ψ : a 7→ Pϕ(a) = Pϕ(a)P is a c.p. map

S → B(H) with 0 ≤ ψ ≤ ϕ. Since ψ(1) = P is a proper projection and not

a scalar multiple of I, ψ is not a scalar multiple of ϕ. Thus ϕ is not pure,

contrary to hypothesis.

With this in mind, we have a path towards finally proving Theorem 3.7.

Firstly, we show that the inductive dilation process in Section 5 can be done

while maintaining purity at each stage. It follows that every pure u.c.p. map

extends to a pure maximal one, which is a boundary representation. Secondly,

with this in mind it suffices to show there are sufficiently many pure maps to

completely norm an operator system.

The following is a refinement of Lemma 5.6 for pure maps.

Lemma 6.3. [6] Let S be an operator system, H a Hilbert space, and ϕ :

S → B(H) a pure u.c.p. map. Given (a0, x0) ∈ S ×H, there is a pure u.c.p.

dilation ψ : S → B(K) of S which is maximal at (a0, x0). Further we can

choose K such that K ∼= H ⊕ C.

Proof. As in Lemma 5.6, let

α := sup{‖ψ(a0)x0‖ | ψ � ϕ}

and also define

η =√α2 − ‖ϕ(a0)x0‖2.

Consider two sets

Y := {ψ ∈ UCP(S,B(H ⊕ C)) | ψ � ϕ} ,

and

X := {ψ ∈ Y | ψ(a0)x0 = ϕ(a0)x0 ⊕ η} .

33


In the proof of Lemma 5.6 we showed that a BW-limit of dilations of ϕ dilates

ϕ, so Y is closed. It is easy to check that X and Y are convex with X a

BW-closed face of Y . Since Y is a BW-closed subset of UCP(S,B(H ⊕ C)),

Y is compact by Proposition 2.5. Lemma 5.6 shows that Y is nonempty

and contains a maximal map ρ with ‖ρ(a0)x0‖ = α. By conjugating with an

appropriate unitary map of the form I⊕eiθ, we can ensure 〈ρ(a0)x0, 0⊕ 1〉 = η

so X is nonempty. Note X is a nonempty convex BW-compact set and so we

may take an extreme point ψ via the Krein-Milman theorem. Since X is a

face of Y , ψ is also extreme in Y .

Now ψ is a dilation of ϕ which is maximal at (a0, x0). The remainder of

the proof will consist of checking that ψ is pure. Suppose that ψ = ψ1 +ψ2 as

a sum of c.p. maps. By Remark 6.1 it suffices to show ψi are scalar multiples

of ψ for i = 1, 2. Even though ψi are not assumed unital, we may assume

ψi(1) is at least invertible by replacing ψi with

ψ′i := (1− 2ε)ψi + εψ

for some small ε > 0. These still satisfy ψ = ψ′1 + ψ′2, but ψ′i(1) ≥ εI does not

have zero in its spectrum. By compressing to H,

ϕ = PHψ(·)|H = PHψ1(·)|H + PHψ2(·)|H

is a sum of u.c.p. maps. Because ϕ is pure, PHψi(·)|H = λiϕ for positive

scalars λ1, λ2. We have

λiϕ(1) ≥ εI,

λ1ϕ(1) + λ2ϕ(1) = I,

which because ϕ is unital imply λi ≥ ε > 0 and λ1 + λ2 = 1.

Define Qi := ψi(1), i = 1, 2. Being positive, we can factor each Qi into a

34


Cholesky decomposition

Qi =

(√λiI 0

x∗i βi

)︸︷︷︸

T ∗i

(√λiI xi

0 βi

)︸︷︷︸

Ti

=

(λiI

√λixi√

λix∗i αi

)

as operator matrices on H⊕C. Here xi ∈ H ∼= B(C, H), and αi, βi > 0 satisfy

β2i + ‖xi‖2 = α2

i . (1)

Because Q is invertible, βi > 0. Because Q1 +Q2 = T ∗1 T1 +T ∗2 T2 = I, we have

the relations

λ1 + λ2 = α1 + α2 = 1, (2)

and √λ1x1 +

√λ2x2 = 0. (3)

For i = 1, 2, consider the c.p. maps τi : S → B(H ⊕ C) given by τi(a) =

T−1∗i ψi(a)T−1

i . Explicitly

T−1i =

(λ−1/2i −λ−1/2

i β−1i xi

0 β−1i

),

and so for a ∈ S,

τi(a) =

(λ−1/2i 0

∗ ∗

)(λiϕ(a) ∗∗ ∗

)(λ−1/2i 0

0 ∗

)

=

(ϕ(a) θi(a

∗)∗

θi(a) fi(a)

).

Here θi ∈ B(S,H∗) and fi : S → C is a state. Since the top left entry is ϕ(a),

35


both τ1 and τ2 are c.p. dilations of ψ. Note for later that

ψi(a) = T ∗i τi(a)Ti =

(λiϕ(a) ∗√

λi(x∗iϕ(a) + βiθi(s)) ∗

). (4)

We will show that x1 = x2 = 0. Because τi dilate ϕ, we have

|θi(a0)x0| = |PCθi(a0)x0| ≤ η (5)

because η =√α2 − ‖ϕ(a0)x0‖2 is an upper bound for |PCρ(a0)x0| for any

ρ � ϕ. Further

η = PCψ(a0)x0

= PCψ′1(a0)x0 + PCψ

′2(a0)x0

= (√λ1x1 +

√λ2x2)∗ϕ(a0)x0 +

√λ1β1θ1(a0)x0 +

√λ2β2θ2(a0)x0 (by (4))

=√λ1β1θ1(a0)x0 +

√λ2β2θ2(a0)x0. (by (3))

By (1), β2i ≤ αi for i = 1, 2 and so

η ≤√λ1β1|θ1(a0)x0|+

√λ2β2|θ2(a0)x0|

≤ (√λ1β1 +

√λ2β2)η (by (5))

≤ (λ1 + λ2)1/2(β21 + β2

2)1/2η (Cauchy-Schwarz)

≤ (λ1 + λ2)1/2(α1 + α2)1/2η

= η,

having used (2) in the last step. Thus each of the inequalities in the chain

above are equalities. So we conclude β2i = αi and so by (1) we must have

x1 = x2 = 0. Since equality holds in the Cauchy-Schwarz inequality, the

vectors (√λ1,√λ2) and (β1, β2) are collinear and so equal, since both have

norm 1. Thus αi = λi so

Qi =

(λiI 0

0 λi

)= λiI and Ti =

√λiI

36


are scalar.

What do we conclude? Well,

ψ = ψ1 + ψ2

= T ∗1 τ1(·)T1 + T ∗2 τ2(·)T2

= λ1τ1 + λ2τ2,

and this is a convex combination! Since τ1, τ2 ∈ Y and ψ is an extreme point,

τ1 = τ2 = ψ. This shows ψi = λiτi = λiψ are scalar multiples, so ψ is pure.

Theorem 6.4. [6] Let ϕ : S → B(H) be a pure u.c.p. map on an operator

system S. Then ϕ dilates to a boundary representation of S.

Proof. The proof uses transfinite induction and is essentially identical to the

proofs given for Lemma 5.7 and Theorem 5.4. Accordingly we skip many of

the details.

Firstly as in Lemma 5.7 we use transfinite induction to show that ϕ has a

dilation which is maximal on all of S×H. Enumerate S×H = {(aλ, xλ) | λ ≤γ} for some ordinal γ. Lemma 6.3 provides the inductive step for successor

ordinals. Otherwise suppose β ≤ γ is a limit ordinal and that we have defined

a chain of dilations ϕλ : S → B(Hλ) which are all pure and with ϕλ maximal

on {(aµ, xµ) | µ < λ}. Let Hβ =⋃λ≤βHλ and let ϕβ : S → B(Hβ) be the

spanning representation, cf. Remark 5.8. As in the proof of Lemma 5.7 it

follows that ϕβ is maximal on all of {(aλ, xλ) | λ < β}.We need only check that ϕβ is pure. Suppose 0 ≤ ψ ≤ ϕβ. Given λ < β,

if ψλ : a 7→ PHλψ(a)|Hλ is the compression to Hλ, then we have

0 ≤ ψλ ≤ PHλϕβ(·)|Hλ = ϕλ.

Since ϕλ is pure, ψλ = cλϕ for some constant 0 ≤ cλ ≤ 1. Further, the

constants cλ all agree, since if λ ≤ µ < β and I ∈ B(Hλ) is the identity, then

cλI = PHλψ(1)|Hλ = PHλ(PHµψ(1)|Hµ)|Hλ = cµI

and so cµ = cλ. Hence ψλ = cϕλ for some fixed 0 < c < 1 for all λ < β. Since

37


⋃λ<βHλ is dense in Hβ, it follows that ψ = cϕβ by continuity. Hence ϕβ is

pure.

Thus ϕ =: Φ0 has a dilation Φ1 := ϕβ which is pure and maximal on

S × H0, for H0 := H. Repeating this process inductively, as in the proof of

Theorem 5.4, we obtain a countable chain

ϕ = Φ0 � Φ1 � Φ2 � · · ·

with Φn+1 : S → B(Hn+1) pure and maximal on S × Hn. Let Φ : S →B(Hω) be the spanning dilation of this chain. By the argument in the proof

of Theorem 5.4, Φ is maximal. By the same argument as in the paragraph

above, a spanning dilation of a chain of pure dilations is pure. Altogether, Φ

is pure and maximal, so by Proposition 6.2 is a boundary representation.

Remark 6.5. In the separable case, the discussion in Remark 5.9 applies

verbatim. That is, if S is separable and H is finite dimensional, we can in

countably many inductive steps dilate ϕ to a boundary representation on a

separable Hilbert space.

Knowing that pure maps dilate to boundary representations, we can fi-

nally prove that sufficiently many boundary representations exist to completely

norm any operator system.

Proof of Theorem 3.7. [6] Let S ≤ A be an operator system in a unital C*-

algebra A = C∗(S). First we note that it is enough to show the pure maps

completely norm all positive elements: Given A ∈Mn(S), if there is a bound-

ary representation π : S → B(K) with ‖πn(A∗A)‖ = ‖A∗A‖ then because πn

is multiplicative

‖πn(A)‖2 = ‖πn(A)∗πn(A)‖ = ‖πn(A∗A)‖ = ‖A∗A‖ = ‖A‖2

by the C*-identity.

Hence suppose A ∈ Mn(S)+ ≤ Mn(A)+ is a positive element. It is a

standard result from basic C*-algebra theory that there is a pure state f :

Mn(A) → C with ‖f(A)‖ = ‖A‖ [5, Lemma I.9.10]. Being a pure state, it

38

7. Example: The Toeplitz-Cuntz System

follows that f is a pure u.c.p. map. By Theorem 6.4 there is a boundary

representation σ : Mn(A)→ B(K) which dilates f . Since dilations are norm-

increasing, ‖σ(A)‖ = ‖A‖. Let π : A → B(H) be the u.c.p. map found

by compressing σ to H := σ(E11)K, where E11 ∈ Mn(A) is the first matrix

unit. Since σ is an irreducible ∗-representation, it is standard that π is also an

irreducible ∗-representation and πn and σ are unitarily equivalent [11]. Hence

‖πn(A)‖ = ‖A‖.We claim ψ := π|S itself is a boundary representation of S. Suppose

θ : S → B(J) was any u.c.p. dilation of ψ. Then θn dilates ψn ' σ|Mn(S).

Thus ψn has the unique extension property and C∗(Mn(S)) = Mn(A), so

θn = ψn. Thus for a ∈ S, θ(a)⊗ I = θn(a⊗ I) = ψn(a⊗ I) = ψ(a)⊗ I which

implies θ = ψ. This shows ψ has the unique extension property. Since its

unique extension π is irreducible, ψ is a boundary representation.

7 Example: The Toeplitz-Cuntz System

7.1 Representations of the Toeplitz-Cuntz Algebra

We present a meatier example that builds upon Example 3.14. Consider a

family {T1, . . . , Tn} of n isometries with pairwise orthogonal ranges, whose

range projections satisfyn∑i=1

TiT∗i ≤ I.

The universal C*-algebra generated by such isometries is called the Toeplitz-

Cuntz algebra, but for now we assume concretely that T1, . . . , Tn ∈ B(H) for

a Hilbert space H. Our goal is to classify all possible actions of the operators

T1, . . . , Tn. That is, we seek to understand possible representations of the

Toeplitz-Cuntz algebra. Following Popescu [12], we’ll prove that T1, . . . , Tn

have a simultaneous Wold decomposition into a direct sum of a Cuntz system

with copies of creation operators arising from the left regular representation

of the free semigroup on n letters.

Let’s establish convenient notation. Let F+n be the free (unital) semigroup

in the n letters 1, . . . , n. The elements of F+n are words i1 · · · ik where ij ∈

39


{1, . . . , n}, k ≥ 0, and the semigroup law is concatenation. If w = i1 · · · ik ∈F+n , set

Tw := Ti1 · · ·Tik ∈ B(H).

The assignment w 7→ Tw is evidently a semigroup homomorphism. The iden-

tity of F+n is the empty word ∅ and accordingly we set

T∅ = I.

If w and v are words in F+n of the same length, we have the relation

T ∗wTv = δvwI (6)

where δvw = 1 if and only if v = w.

Consider the closed subspace

V :=

(n∑i=1

TiH

)⊥=

n⋂i=1

(TiH)⊥

It follows from (6) that V is a wandering subspace for the family {Tw}w∈F+n

,

in the sense that

TwV ⊥ TvV

whenever w 6= v. Set Vw = TwV for w ∈ F+n and consider the closed subspace

K :=⊕w∈F+

n

Vw ≤ H.

K is clearly invariant for T1, . . . , Tn. In fact, it is reducing because

T ∗i Vw ⊆

Vv, w = iv,

0, else.

Let {eλ | λ ∈ Λ} be an orthonormal basis for V . The unit vectors

eλw := Tweλ, λ ∈ Λ, w ∈ F+

n

40


form an orthonormal basis of K. If Wλ := span {ewλ | w ∈ F+n }, then

K =⊕λ∈Λ

Wλ

as a direct sum of separable reducing subspaces for Tn.

How does Tn act on each summand Wλ? Define the left regular repre-

sentation of F+n as the separable Hilbert space

`2(F+n ) := span {ew | w ∈ F+

n }

with isometries Lw ∈ B(`2(Fn+)) that act on this basis by Lwev = ewv,

w, v ∈ F+n . Then L1, . . . , Ln is a family of isometries with pairwise orthogonal

ranges, another Toeplitz-Cuntz family. The notation thus far should make the

following clear: For λ ∈ Λ, the unitary that identifies

eλw ∈ Vw 7→ ew ∈ `2(F+n )

gives a unitary equivalence Tw|Vw ' Lw for all w ∈ F+n simultaneously (i.e. a

unitary equivalence of semigroup representations). So

Tw|K ' (Lw)⊕α

acts via a direct sum of α := |Λ| copies of the left regular representation.

Now consider the action of Tn on the invariant complement J := K⊥. Since

V = V∅ ⊆ K,

J ⊆ V ⊥ =n∑i=1

TiH.

Since T ∗i Tj = δijI, it follows that∑n

i=1 TiT∗i acts by the identity on V ⊥ and

hence on J . Thus

Si := Ti|J

is a family of isometries with∑n

i=1 SiS∗i = I. This shows Sn = {S1, . . . , Sn} is

a Cuntz family in the sense of Example 3.14.

41


7.2 The C*-envelope of the Cuntz System

The Toeplitz-Cuntz algebra T On is the universal C*-algebra generated by n

isometries T1, . . . , Tn with pairwise orthogonal ranges. The previous discussion

shows that every representation of T On on a Hilbert spaceH has an orthogonal

decomposition of the form

H ∼= `2(F+n )⊕α ⊕ J,

Ti ∼= (Li)⊕α ⊕ Si,

where Li ∈ B(`2(F+n )) is the left regular representation, α is some cardinal,

and S1, . . . , Sn form a Cuntz family [7, 12].

What is the C*-envelope of the operator system

Tn := span {I, T1, . . . , Tn, T∗1 , . . . , T

∗n} ≤ T On

generated by T1, . . . , Tn? Call Tn the Toeplitz-Cuntz system. We assume a

faithful representation T On ⊆ B(H). Decompose H = K⊕J = `2(F+n )⊕α⊕J

with equality

Ti = L⊕αi ⊕ Si

up to complete isometry. By universality of the Toeplitz-Cuntz algebra (or, by

compression to J) there is a ∗-homomorphism π : T On → On with π(Ti) = Si,

where Sn = {S1, . . . , Sn} is a Cuntz family. We claim π is completely isometric

on Tn.

We will construct a u.c.p. map ρ : Sn → Tn ≤ B(H) with ρ ◦ π|Tn = idTn .

Since both ρ and π are u.c.p., it follows that π is completely isometric. Let Xn

be the set of bi-infinite sequences x = (in)n∈Z with letters ik ∈ {0, 1, . . . , n} of

the form

x = (· · · , 0, 0, iN , iN+1, iN+2, · · · ), ij ∈ {1, . . . , n}.

That is, there is some N ∈ Z with ik = 0 for all k < N , and ik > 0 for

k ≥ N . There is a natural semigroup action of F+n on Xn where if x =

42


(· · · , 0, iN , iN−1, · · · ) ∈ Xn and w = j1 · · · jk ∈ F+n ,

w · x := (· · · 0, 0, j1, . . . , jk, iN , iN−1, · · · ).

That is, w · x is the sequence found by prepending the letters of w to the

sequence of nonzero terms in x. We will identify F+n with the subset of Xn of

sequences of the form

i1 · · · ik ∈ F+n ↔ (· · · 0, 0, i1, . . . , ik, 1, 1, · · · ) ∈ Xn

where the first 1 after ik appears in position 0 in the sequence. With this

identification the semigroup multiplication in F+n and the semigroup action

of F+n on Xn agree. Form the Hilbert space `2(Xn) with orthonormal basis

{ex}x∈X . For w ∈ F+n , there is an isometry Qw ∈ B(`2(Xn)) which acts on the

basis by

Qwex = ew·x.

Having identified F+n ⊆ Xn, we have an inclusion `2(F+

n ) = span {ew}w∈F+n

.

Further, `2(F+n ) is invariant for all Qw and

Qw|`2(F+n ) = Lw, w ∈ F+

n .

This construction of `2(Xn) and the isometries Qw–which give a representation

of F+n , comes from Davidson and Pitts’ [7] classification of atomic representa-

tions of T On. This representation is the direct sum of all the “infinite tail”

representations described in [7, Section 3]

Why form a larger Hilbert space? Q1, . . . , Qn is another family of isometries

with pairwise orthogonal ranges, the range of Qi being spanned by those basis

vectors whose semi-infinite sequences start with i. Further, the sum of all the

ranges of Q1, . . . , Qn is all of `2(Xn), because for x ∈ Xn we have

x = i · y, hence ex = Qiey ∈ ran Qi

where i ∈ {1, . . . , n} is the first nonzero term in x and y ∈ Xn comprises

the remaining entries. Thus the range projections satisfy∑n

i=1QiQ∗i = I,

43


so {Q1, . . . , Qn} is a Cuntz family. By universality there is a unique ∗-homomorphism τ : On → C∗(Q1, . . . , Qn) which sends Si to Qi for each i.

Let θ : On → B(`2(F+n )) be the u.c.p. compression

θ = P`2(F+n )τ(·)|`2(F+

n ).

θ satisfies

θ(Si) = Qi|`2(F+n ) = Li

for i = 1, . . . , n.

Now define a u.c.p. map

ρ := θ⊕α ⊕ idSn : On → B(H) = B(`2(F+n )⊕α ⊕ J).

For i = 1, . . . , n, we have the relations

ρ(Si) = L⊕αi ⊕ Si = Ti, π(Ti) = Si.

Thus ρ and π satisfy ρ ◦ π|Tn = idTn and π ◦ ρ|Sn = idAn . So π|Tn is a u.c.p.

map with u.c.p. inverse, meaning π must be completely isometric on Tn. In

total, we have a unital complete isometry

π|Tn : Tn → On.

Since On = C∗(Sn) = C∗(π(Tn)) is simple [4], Proposition 3.12 tells us that

On = C∗e (Tn) is the C*-envelope of the Toeplitz-Cuntz system On. For pos-

terity we record this result as a theorem.

Theorem 7.1. Let Tn ≤ T On be the smallest operator system containing

the generating isometries {T1, . . . , Tn} of the Toeplitz-Cuntz algebra T On. If

π : T On → On = C∗(S1, . . . , Sn) is the unique ∗-homomorphism sending Ti to

Si for i = 1, . . . , n, then π|Tn : Tn → On is a C*-envelope for Tn.

Remark 7.2. In the special case n = 2, we can perform the dilation of the

left regular representation {L1, L2} to a Cuntz system {Q1, Q2} in separable

44

References

fashion. Instead let M = C⊕ `2(F+n ). Then define Q1, Q2 ∈ B(M) by

Qi|`2(F+n ) = Li, i = 1, 2,

Q1(1⊕ 0) = 1⊕ 0,

Q2(1⊕ 0) = 0⊕ e∅.

Then {Q1, Q2} is a Cuntz family on the separable Hilbert space M ⊇ `2(F+n )

that compresses to {L1, L2}.

Popescu [13] showed that all operator algebras generated by any Toeplitz-

Cuntz family {T1, . . . , Tn} of n isometries are completely isometrically isomor-

phic. Thus if An ≤ T On is the unital subalgebra generated by T1, . . . , Tn ∈T On, there is a completely isometric homomorphism An → On sending Ti to

Si for each i. The analogous version of Proposition 3.12 for operator algebras

shows that C∗e (An) = C∗e (Tn) = On.

References

[1] William B. Arveson. Subalgebras of C*-algebras. Acta Mathematica 123.1

(1969), pp. 141–224.

[2] William B. Arveson. Notes on the unique extension property. Unpub-

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[3] William B. Arveson. The noncommutative Choquet boundary. Journal

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[4] Joachim Cuntz. Simple C*-algebras generated by isometries. Comm.

Math. Phys. 57.2 (1977), pp. 173–185.

[5] Kenneth R. Davidson. C*-algebras by example. Vol. 6. American Math-

ematical Soc., 1996.

[6] Kenneth R. Davidson and Matthew Kennedy. The Choquet boundary of

an operator system. Duke Math. J. 164.15 (Dec. 2015), pp. 2989–3004.

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reflexivity for free semigroup algebras. Proceedings of the London Math-

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[8] Michael A. Dritschel and Scott A. McCullough. Boundary representa-

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Journal of Operator Theory 53.1 (2005), pp. 159–167.

[9] Masamichi Hamana. Injective envelopes of operator systems. Publica-

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by Hari Bercovici and Ciprian I. Foias. Basel: Birkhauser Basel, 1998,

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[11] Vern Paulsen. Completely bounded maps and operator algebras. Vol. 78.

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[12] Gelu Popescu. Isometric dilations for infinite sequences of noncommut-

ing operators. Transactions of the American Mathematical Society 316.2

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[13] Gelu Popescu. Non-commutative disc algebras and their representations.

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