CONVEXITY THEORY AND ITS APPLICATIONS IN FUNCTIONAL ANALYSIS

L.M.S. MONOGRAPHS Editors: D. EDWARDS and H. HALBERSTAM

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16. Convexity Theory and its Applications in Functional Analysis by L. Asimow and A. J. Ellis

Published for the London Mathematical Society by Academic Press Inc. (London) Ltd.

CONVEXITY THEORY AND ITS APPLICATIONS IN FUNCTIONAL ANALYSIS L. AsIMow University of Wyoming, Laramie, Wyoming, USA

A. J. ELLIS University College of Swansea,

Swansea, Wales, UK

1980

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British Library Cataloguing in Publication Data Asimow, L Convexity theory and its applications in functional analysis. —(London Mathematical Society. Monographs; 16 ISSN 0076—0560).

1. Convex sets 2. Functional analysis I Title II. Ellis, A I III. Series 517'.7 80—40648 QA640 ISBN 0—12—065340—0

PRINTED IN GREAT BRITAIN BY J. W. ARROWSMITI4 LTD. W1NTERSTOKE ROAD, BRISTOL

To Marilyn Asimow and Jennifer Ellis

Preface

With the appearance in 1966 of Lectures on Choquet Theory, Phelps [172], the representation theory of compact convex sets became accessible to a large, and as it develops, enthusiastic audience. The geometric appeal of the Choquet simplex is irresistible; it has led quite naturally to an exploration of related geometric structure in more general state spaces. This enterprise, while somewhat interesting in its own right, would be of little note but for the remarkable insights it has provided in appreciating various analytical aspects of the underlying function spaces. This feature was already apparent in the comprehensive treatment in 1971 of Choquet Theory in the book by Alfsen [5]. Since that time the geometric study of compact convex sets has rewarded

its devotees not only with generalizations of individual theorems in functional analysis, but with the prospect of a unified geometric theory that yields a clearer understanding of a reasonable variety of classical results.

Our object here is to promote this geometric perspective. We hasten to make the standard disclaimer concerning the lack of totality in our selection of topics. Granting the necessarily idiosyncratic nature of our subject matter

we have attempted to illustrate the means by which a fairly elementary geometric theory, based on partially ordered Banach spaces and duality, can

be applied in a systematic fashion to concrete function spaces (real and complex) and unital Banach algebras.

In Chapter 1 we have gathered together the functional analytic minaries that constitute our basic tools. The reader with only a basic knowledge of abstract real analysis will find there statements of the main results as well as short but self treatments of the Krein—Sinulyan Theorem, the basic Choquet Theory and the Bishop—Phelps Theorem. Chapter 2 gives the basic duality results, lattice theory and concrete representation theorems for order unit spaces and Banach lattices of type M and L. Since much of the book treats subspaces of continuous functions on

viii

PREFACE

compact Hausdorif spaces we felt the classical representation theorems of abstract ordered Banach spaces were especially pertinent. We continue the treatment of real affine function spaces in Chapter 3 by examining in detail the case where the state space is a Choquet simplex. In Chapter 4 we show how the study of real A(K) spaces can be employed even for complexvalued function spaces by means of a complex state space. Chapter 5 gives a survey of the application of the theory to the study of (non-commutative) Banach algebras.

Our thanks are due to Caroline Johnson and Paula Melcher for their expert typing of the manuscript. We are very grateful to Dr K. F. Ng for reading the manuscript and making many helpful suggestions. We have also

benefited from useful discussions of parts of the book with Dr T. B. Andersen. The second author wishes to thank the University College of Swansea for sabbatical leave during the Autumn Term of 1976, enabling him to visit the University of Wyoming.

July 1980

L. Asimow A. J. Ellis

Pref ace

V

1. PrelimInaries 1. Separation and Polar Calculus 2. Krein—Smulyan Theorem and Corollaries 3. Gauge Lemma and Completeness 4. Subspaces of C(X) and Aøine Functions 5. Representing Measures 6. Maximal Measures and the Choquet—Bishop-de Leeuw Theorem 7. Bishop—Phelps Theorems

I

2. Duality in Ordered Banach Spaces 1. Positive Generation and Normality 2. Order Unit and Base Norm Spaces 3. Directedness and Additivity 4. Homogeneous Functionals on Cones and the Decomposition Property 5. Banach Lattices and the Riesz Interpolation Property 6. Abstract L and M Spaces 7. Choquet Simplexes 8. Decomposition of Dual Cones 9. Stability of Split Sets and the Characterization of Complemented

1

4 7 11 15

18

26 30 31 35 39

44 48 57 69 74 83

Subcones

10. Application to A(K) 11. The Riesz Norm 3. Simplex Spaces 1. The Facial Topology for K and the Centre of A(K) 2. Facial Characterizations of Simplexes

91

99 102 102 108

x

CONTENTS

3. Ideals in Simplex Spaces 4. Stone—Weierstrass Theorems for Simplex Spaces 5. Prime Simplexes and Anti-Lattices

117 122 124

6. Topological Characterization of the Extreme Boundary of a 126 Compact Simplex 7. Poulsen's Simplex—its Uniqueness and Universality 8. Non-Compact Simplexes and Convex Sets

130 138

4. Complex Function Spaces 0. The Complex State Space 1. Complex Representing Measures 2. Interpolation Sets 3. Gauge Dominated Extensions and Complex State Space 4. Peak Sets and Al-sets 5. Dominated Interpolation 6. Decomposability and Exact Interpolation 7. M-Hulls and Function Algebras 8. Facial Topologies and Decompositions 9. Lindenstrauss Spaces

208 219

5. Convexity Theory for 1. The Structure Topology and Primitive Ideas in A(K)

224 224

2. The Ideal Structure of a Unital

and the Facial 226

Structure of its State Space 3. Commutativity and Order in

4. The Centre of a

146 146 148 154 159 164 169 179 193

232

Weakly Central Algebras, Prime 234

Algebras 5. Unit Traces for

6. The State Spaces of Jordan Operator Algebras and of

239 242

Algebras

References

252

Index of Definitions

261

Subject Index

263

CHAPTER 1

Preliminaries

We bring together in this chapter most of the general material in functional analysis that will be relied on in subsequent chapters. We have simply stated some of the standard results and have provided complete proofs of others. The distinction is based mainly on which theorems are pretty much standard in a graduate level analysis course (say, for example Royden [185] or Rudin [187]) and which are left for more advanced courses.

Thus we omit proofs of the Separation Theorem and the bipolar calculations (which are used repeatedly throughout the remainder of the text)

and the Krein—Milman Theorem. We do prove the Krein—Smulyan Theorem (also an invaluable tool in our later study) in Section 2. In Section 3

we present a technical lemma which incorporates most of the iteration procedures in Banach spaces (including the Open Mapping Theorem) that arise later on. In Sections 4—6 we introduce state spaces and prove the de Leeuw Theorem on representing measures. We conclude with a proof of the Bishop—Phelps Theorem.

1. SEPARATION AND POLAR CALCULUS Let (E, F) be a pair of (real or complex) linear spaces in duality, with each total over the other; i.e. if (a, x) =0 for all x E F then a 0 and if (a, x) =0 for all a E E then x =0. Let (E, F) be endowed with the weakest topologies for which the linear functionals and (a, .) are continuous on E and F for each a and each x. Then E and F are locally convex linear topological spaces with each being the (dual) space of continuous linear functionals over the other. 1.1. THEOREM (SEPARATION THEOREM). If A and B are disjoint convex subsets of E with A compact, and B closed then there exists an x E Fsuch that

max{re(a, x): a eA}
2

PRELIMINARIES

Let X be a subset of E or F. We denote the convex hull of X by Co X and the closed convex hull by X. We write bal X for {ax: x X and al = 1} and cob X for the convex balanced hull of X. We write Px, or p(X), for the support function of X, which is defined on the dual space by

px(a) = sup {re (a, x): x X}

if Xc F (and dually for X c E). Note that Px may be an extended function. If 0€ A and A is convex we define the Minkowski functional of A by pA(or

p(A))(a) = inf {r 0: a

rA}.

If (B', F') is another dual pair and T : B B' is a continuous map then the adjoint : F' F is given by (a, T*y) = (Ta, y). (The continuity of T guarantees T*y c F.)

ForAcE we define the polar

A°={x €F: re (a, x) 1

for alla €A}.

If A c F then A° is the subset of B defined similarly. 1.2. THEOREM (POLAR CALCULUS).

(i) A° is a closed convex set containing 0 and A°° =

(A u {0}).

(ii) A c B implies B° c A°. (iii) (AA)° = (l/A)A° for 0
(T(A))O=(T*ii(AO). (vi) If 0€ A then p(A°) =p(A).

Part (1) follows from the Separation Theorem and (ii)—(v) are straightforward. For (vi) we have first that p(A)(x) =00 if and only if for all 0< A
p(A)(x)
(vii) If A, B are closed convex sets containing 0 then (A B)° = (A° u B°). If, in addition, A is a cone (AA cA for all A >0) then so is A°

SEPARATION AND POLAR CALCULUS

and (A

We say x is an extreme point of X if whenever x = Ày + (1— A)z (y, z X and 0
Similarly, the convex subset F or X is called a face if whenever x F is a convex combination of y and z in X then y and z belong to F.

If Xis a compact convex set in E (orF) Yand Yis compact then OXc Y

1.3. THEOREM (KREIN—MILMAN).

In almost all cases we will be taking E to be a normed linear space with dual space F = E*. We write

A,={a EA:IIaIIr}. Thus the (closed) unit ball is denoted E1 and the unit ball of E* is

B?.

The dual topologies of our previous discussion are called the weak and weak* (w*) topologies respectively. Since convex sets in B have the same closure in the weak and norm topologies we generally will not distinguish these cases. 1.4. THEOREM. (ALA0GI.u). A subsetXofE* is

Xis

if and only if

and

1.5. THEOREM. Let A and B be closed convex sets containing 0 with A compact. (i) The set A + B is closed.

(ii) The set co ((A +PB)u B)

(Au B), where

PB ={x €B: p(B)(x)=0}. Proof.

We show (ii): Let (x,) be a net in co (A u B) with Xa Xc, =Ac,yc,

+(1 —Ac,)zc,;

1,

y., €A,

x. Let

€B.

Since A is compact, by passing to a subnet, we can assume and A

<1 then eventually Ac, < 1 so that Zcv = (xc,—Ac,yc,)/(1—Ac,)-'(x—Ay)/(l

A)EB.

Thusx €co (AuB). IfA = 1 and O
4

PRELIMINARIES

Hence x — y E rB for all r> 0 so that p(B)(x — y) =0. in this case

THEOREM AND COROLLARIES

2.

We briefly discuss various topologies on the dual, E*, of a normed linear space E for which the elements of E remain continuous as linear functionals on E*.

Definition. A non-empty collection of sets in E is called a topologizing family for E* jf (i) A €4 implies A is norm-bounded, (ii) A, B €4 implies there exists C €4 such that ALl B c C, (iii) A E 4 implies aA €4 for scalars (real or complex) a, (iv) A finite implies A €4. 2.1. THEOREM.

If 4 is a topologizing family in E then

is a neighbourhood base at 0 for a locally convex topology on E* for which the elements of E are continuous linear functionals on E*.

Proof. We just note here that the standard von Neumann axioms are satisfied. For example, to check that given A° there is a B° such that AB° c A° for all A 1 take, using (ii) and (iii), B

Then B° c so that for A

(iA°) (—iA°)]. Thus x E B° implies ±x, ±ix c A°. 1, Ax = (re A )x + i(im A )x

Three important examples of topologizing families are on (1) 4= (A c E: A finite} which yields the (2) 4= (A c E: A norm-compact}, (3) 4= (A c E: A is norm-bounded} which yields the strong, or norm, topology on E*. (bw*) topology We call the topology of Example (2) the derives from its close connecon E*. The importance of the tion with the 2.2. THEOREM. relatively

The set V c E* is

if and only if V

in E,* for each r 0.

We give a proof based on the following two lemmas.

is

KREIN.-SMULYAN THEOREM AND COROLLARIES

5

s. Then the 2.3. LEMMA. Let V E* and x E* be given. Let following are equivalent: -neighbourhood of x in E? for all r s, is a relative (i) V of 0 in E? for all r 0. is a relative (ii) (V — x)

Proof.

Note that

and x+

Cx+

(x+

c(x+ of 0, such that

If (1) holds and r 0, choose W, a C V. Then x + W E? c V (x + W) ri argument for (ii) implies (1) is quite similar.

so that W

C V — x.

The

2.4. LEMMA.

of 0 then V n E? is a relative

(i) If V is a

for all r 0.

neighbourhood of 0 in is a relative (ii) If V then V is a

for all r 0

neighbourhood of 0 in of 0.

For (i) choose A norm-compact in E with A° c V and let r >0 be given. Choose A0 finite such that Proof.

A cA0+ E(1/2)r. Then A C co [2A0 u E11,] so that

construct a sequence is finite, (a) each

(2A0)0 A° c V. For (ii) we shall

such that

for n1,

(b) (c)

Take A0 finite and Ag Er

V. Assume A0,... , A,,_1 (n 1) are chosen

satisfying (a), (b), (c). Let

(A0u'. .

K

c

K

Vc= 0.

=

Using a standard compactness argument, we choose a convex neighbourhood Uof 0 such that W = neighbourhood + U is a of disjoint from K. Thus W° is a finite-dimensional set in the interior of c and we can choose a finite set such that W0 C co Thus (a), (b) hold and

0= W n K

(A0

u

Vc.

Thus

ri (A0 u.

tm

= (A0 u.

•

u

c

PRELIMINARIES

6

so (c) is satisfied. Now take A

=norm-ci(ci

A is compact and

c

V.

Hence

A°= U

V.

and V with lxi! = s. Then Proof of Theorem 2.2. Let V be OE V—x is is a relative so that by Lemma 2.4(i), for all r 0 and hence, by Lemma 2.3, V of 0 in for all r s. is a relative of x in For the converse choose x E V. We show V — x is a bw *neighbourhood of for all t 0 then Lemma 2.4(u) in 0. is relatively is a applies. But y E (V — x) implies x + y V so that V neighbourhood of x+y for all r Lix + yfl and hence (Lemma 2.3, (i) implies (ii)) [V — (x + y)] n E? is a w *..neighbourhood of 0 in for all r 0.

Finally, this says (Lemma 2.3 (ii) implies (i)) (V—x) neighbourhood of y in

for all r liyIi. But y

is a

so that t fyff.

We can now state some of the very USCI ul characterizations of convex sets. The first corollary is continuous functionals and a direct consequence of Theorem 2.2. 2.5. COROLLARY. is relatively

The function f is

on E* if and only if

for all r 0.

E** be the canonical embedding. Then f is an isometry and, since E** is a Banach space, the completion of E can be identified as norm-cl 4(E) in E**. Let qS: E

COROLLARY (BANACW-GROTHENDIEK THEOREM).

If a E** then a is

on E* if and only if a norm-cl b(E). Proof.

then, by Corollary 2.5, for each r>0 there is

If a is

c{x: re (a, x) l}. Then taking polars an AcE with A finite and A° in E** gives (since A°° = co b(A) c çb(E)) x

Hence x e norm-cl j(E).

co

GAUGE LEMMA AND COMPLETENESS

7

E with 1kb (a,,)-- all 0. For r >0 let X For the converse let (a,,) and let A C(X), taking b Then 0 is an isomorphism onto a subspace of C(X) and lObil=rllbil. Hence SUPx b(a,,)—al=rlkb(a,,)-—alI— of a. 0 so that a E C(X). Thus Corollary 2.5 yields the

If E isa Banach space then the following are equivalent: linear functional on

2.6.

(i) a is a

(ii) a€bE, (iii) a is THEOREM).

Let F be a Banach

space and let K be a convex subset of E*. Then K is r 0. K

if and only if

2.7. COROLLARY

Let V = KC so that the condition implies V

is relatively Thus K is and hence is an intersection of hw*_closed half-spaces. But the previous corollary guarantees these half-spaces are in fact and therefore, K is Proo/

w*..open for all r0 and hence V is

3. GAUGE LEMMA AND COMPLETENESS It is convenient to formulate here a general iteration scheme in Banach spaces, which will allow us to conclude a variety of duality results later on. This is expressed in terms of gauges, i.e. Minkowski functionals of convex sets containing 0 and possessing certain convergence properties.

If p is a positively homogeneous, sub-additive functional on a normed linear space E such that p(x,,) < implies x = exists and p(x) then p is called a gauge. It is the Minkowski functional of the set

Definition.

B ={x€E: p(x) 1}. If p has the weaker property that p(x)

p(xn)

whenever x

exists

then p is called a pre-gauge. We say the convex set B (containing 0)is a gauge set, or a pre-gauge set if PB has the corresponding property. It may happen that B is a proper subset of {x:

8

3.1. PRoposiTioN. Let A and B be closed convex sets containing 0 in the normed linear space E and let T : E space F.

Fbe a hounded linear map to the normed

sets. (i) A and B are (ii) If B is complete and bounded then B is a gauge set. (iii) If B is a gauge set then A + B and Co (A u B) are (iv) If B is a gauge set then T(B) is a pre-gauge set.

Proof.

sets.

Note first that the Minkowski functional PA is lower-semi-

continuous if and only if A is closed, since

{xEE:pA(x)_
E: PA(X) = 0} = fl aA. a >0

Thus, if x =lim s,,, where Sn = then pA(x)lim inf PA(Sn.) shows (i). If B is complete and This hm inf E1=1 PA(Xi) =

bounded then, for some M> 0, pB(x,,) <00 implies MPB; thus x = x,, exists. For (iii), let C = co (A u B) and let x = pc(x,,) <00. Let a,, be such that 0 pc(Xn) x,, with
+/.L))(y/A) + (/h/(A

(A

A+

Since A,

(A

co

(Au B).

are arbitrary, (Ifl

The argument for A + B is similar. Finally, if C = T(B), y = E

and

Pc(Yn) <00 then choose a,, such that

Pc(yn)< a,,
a,,. Hence x = y,,

x,, exists and

T(x,,)= T(x)andpc(y)=

It will frequently happen that the gauge, or pre-gauge sets that arise in practice are not closed, as in Proposition 3. l(iii) and (iv). Nevertheless, with some additional hypotheses it turns out the Minkowski functionals of the sets and their closures are the same. To formulate this we require the notion

GAUGE LEMMA AND COMPLETENESS

9

of the distance from a point x to a set A via a gauge set B, defined by

d(A, B)(x) = inf {r 0: x A + rB}. We will abbreviate this by d8, or just d, if the sets A and B are understood. We also require the closure of d, given by

c7(A, B)=inf{rO: x e(A+rBY}. The Gauge Lemma gives sufficient conditions for d(A, B) = J(A, B) on a domain set D. Let A, B, D be convex sets containing 0 in the normed linear space E and satisfying the following hypotheses: (i) D +(B) c D, where (B) is the linear subspace spanned by B, 3.2. LEMMA (GAUGE LEMMA).

(ii) ii(A, B) 0, (iii) any one of (a) A is closed and B is a gauge set, set, or (b) A is compact and B is a (c) A is complete and B is a bounded pre-gauge set.

Then d(A,B)=d(A,B) onD. (Note that d(A, E1) in (ii) is the usual distance from a point to A.) Denote d(A, E1) by ô. We have d d by definition. Let x0 ED with d(xo) < ro < r < 00. Choose a sequence of positive numbers with Proof.

r1>r0 and Since d(xo) <

we have

a1 +b1+c1; Then, by(i),

a1€A,

pB(bl)< r1

and

b1 ED and x0—b1 = a1 +c1implies

x0—b1 = a2+b2 -I-c2;

a2€A,

IIciII
pB(b2)
11c211<

r3/a.

Continuing by induction we obtain sequences (a,,), (ba) and (xe) such that

A,


r,,+i/a

and

=

+(b1 +- . - +

c,,.

Thus, if (iii) (a) holds then there exists b=>

with PB(b) PB(bn) < r.

Hence converges to a A and so converges in A and so subsequence case (c) we have

E A + rB. If (b) holds then a b,,

= b exists with pB(b) < r. In

PRELIMINARIES

10

where M is a bound for B. Thus (a,1) is Cauchy so that (a,,) converges to < r. a E A and consequently b = b,, exists with In making applications of the Gauge Lemma we shall adopt the convention of writing A 4* ',B .+ ',etc. where we fill in the blanks with the names of the sets currently playing the roles of A, B, D. The meaning of d, d and 8 will follow accordingly. We say B is D-regular if p(B) =p(B) on D. 3.3. COROLLARY. LetD be a closed convex cone and letB be a pre-gauge set with ±B CD. If D1 c aBfor some a >0 then B is D-regular and D1 c a 'B

for alla'>a. Proof.

and

1.

Apply the Gauge Lemma with A Thus D1 c aB says

{0}. Then d, d, 8 are p(B), p(B)

d a6 so that the conclusion follows. 3.4. CoROLLARY If D isa complete subspace and B a pre-gauge in D such that D = nB then D1 c a 'B for all a' greater than some a. Also, B is, D-regular.

Since D is a subspace, ±B CD. Since 0 E cor (B) in D, the Baire Category Theorem shows D1 c aB for some a. Thus Corollary 3.3 applies. Proof.

Let E, F be Banach spaces with T a bounded linear map from E onto

a dense subspace of F. Let T*:F* E'1' be the adjoint and denote balanced and convex. Let N be the Then X is X linear span (X) = U nX. We now give some of the standard completeness results: for example, the equivalence of (1) and (ii) is the Open Mapping Theorem and the equivalence

of (I) and (v) is the Closed Range Theorem. 3.5. THEOREM.

The following are equivalent:

(i) T:E-*Fis onto; (ii) F1 c

(iii) (T(E))1c

for all s greater than some r;

for alis>r;

(iv) T* has w*_closed range N; (v) T* has norm-closed range N; (vi) N1 C rX, some r >0.

SUBSPACES OF CX) AND

FUNCTIONS

11

Let B = T(E1). Then Corollary 3.4 applies (with D 4-*F) and hence (i) implies (ii). That (ii) implies (iii) is trivial. (iii) implies, for a
proof.

Pa = [cl T(E)]1

ci [T(E)13]c ci [T(E,35)J.

Thus F1 C r13 so Corollary 3.3 yields (ii) and hence, (i).

Now (iv) implies (v) and (v) yields (vi) by applying the equivalence of (i) is and (ii) in the dual spaces. If (vi) holds then N1 = (rX) so that (iv), (v) Therefore, by the Krein—Smulyan Theorem, N is and (vi) are equivalent. Finally, (vi) says precisely that

which, by the polar calculus, is equivalent to However, as we have seen, this is equivalent to (ii), using Corollary 3.3.

4. SUBSPACES OF C(X) AND AFFINE FUNCTIONS

In this section we introduce affine function spaces and discuss their representations as real subspaces of C(X). To begin, we take X as a compact

Hausdorif space and let C(X) denote the (real or complex-valued) continuous functions on X. Then C(X) is a Banach space with the usual sup-norm: lit II = sup {ff(x)l: x E X}. We first discuss state spaces in the more general setting of subspaces of complex-valued functions. However, the results are all valid in the real space in the last part as well and we shall restrict ourselves to subspaces of of this section. Thus, let M be a (complex) subspace of C(X) such that M separates the points of X and contains the constant functions. Let M* be the dual Banach

space. If L E M* we say L 0 providing L(f) 0 whenever f is a nonnegative real-valued function in M. 4.1. PROPOSITION.

If L E M* satisfies L(1)

1 = liLli then, for eachf EM,

L(f) E

Proof. We observe that

in C is the intersection of the closed discs D(a, r) (centre a, radius r) which contain f(X). For each such disc we have

llf—alIlsup{lf(x)—aI: x EX}r so that

L(f)—aI IL(f)—aL(1)l = IL(f—ai)liLiI If Hence L(f) E D(a, r).

PRELIMINARIES

12

We say M is seif-adjoint if M implies the complex-conjugate M. In this case ref and im f belong to M and L E M* with L 0 implies re L(f) = L(re I).

4.2. PRoposiTioN. If L E M* then L(1) = IILII implies L 0. If M is selfadjoint then L 0 implies L(1) = IIL1I. Proof.

If L(l) = ILII >0 and f0 then, by Proposition 4.1,

cR

L(f)/jLII E Hence

L(f)0.

Let L 0 and flfII 1, with M seif-adjoint. Choose a E C such that Ia =1 and IL(f)I = aL(f). Then

L(af) = re L(af) = L(re

af)L(l).

Hence ILl

L(1)

Definition.

The state space of M, denoted SM, equals

ILl.

{L E.M*: L(1)

1= LII}.

hyperplane with the unit ball Since SM is the intersection of a Mr we see that SM is and convex. We define (the evaluation map) by 4)(x) = with =f(x). Then it is easy to see that 4) is continuous and 4)(X) is a compact subset of SM. 4.3. THEOREM.

The state space SM =

Let 50 hence, for L0 SM, Proof.

4)

(X). If L0(f)

then f(X) c {L(f);

L E So)

and

f(X) ct(S0).

But then re L0(f) max {re L(f): L e S0}

so

that, by the Separation

Theorem, L0€ So. The Krein-.Milman Theorem yields the following. 4.4. COROLLARY.

ext SM c

The subset 4)'(ext SM), denoted 8MX, is called the Choquet boundary of M. In general it need not be closed or even a Bore! set. As we shall see, however,

it does have the property of being a boundary, meaning that each f€M attains its norm on I3MX.

SUBSPACES OF C(X) AND AFFINE FUNCTIONS

4.5. THEOREM.

13

1ff EM then there is an xo E OMX such that 11th.

Proof.

Choose x X such that lifli = af(x), with IciI = 1. Let T={LESM: L(af)=hh/11}.

E T and hence T is a non-empty compact convex set. Moreover, since T is the subset of SM where the continuous linear functional crf attains its maximum, it is easy to see that Then

ext T C ext SM. Thus, the Krein—Milman Theorem guarantees an x0 aMX with

=

ext T. 4.6. COROLLARY.

Proof.

The unit ballMr equals cob q5(aMX).

Given f €M there is an

E

aMX with

if(xu)i = lit II. Thus

for L E

re L(f) 11111= af(xo)

c (Mr )° and hence

(iai =1) = re

Thus

Mr K is a compact convex subset of a locally convex space F. It is convenient to assume the duality relationship of Section 1 between = E and F. The set K remains compact in the weak topology on F induced by E. consisting of We define the space A(K) as the (Banach) subspace of Suppose

the continuous real-valued affine functions on K. By afline, we mean f(Ax +(1 —A)y) =Af(x)+(l —A)f(y), where x, y eK and OA 1. We again have the evaluation map K-3'SA(K)

= SCA(K)*.

We let

O:E-+A(K);

(Oa)(x)=(a,x).

We will assume first that the function 1 =

Oa

for some a E E. This just

amounts to having K contained in a closed hyperplane of F disjoint from 0. Let N denote the linear subspace of F spanned by K. Thus

N=IJ n co(Ku-K).

PRELIMTNARLES

14

4.7.

(i) The evaluation map is an affine homeomorphism of K onto S; (ii) The map extends (uniquely) to a linear isoinorphism (also denoted co(Ku—K) with of N onto A(K)*, co(S¼J—S);

(iii)

0*th

1;

(iv) tIE is dense in A(K).

It is easy to see that cfr is affine, one-to-one and continuous. Thus Theorem 4.3 shows that S = 4(K) = qS(K). Since the spaces are real, (ii) follows from Corollary 4.6, where Proof.

= co (S U —5).

cob

For (iii), (a, 0* o ç6z) = (0a, q5z)= (a, z). Hence 0* o

(oE)° = This proves (OE)°° = (OE)

'(0) = q5(O)

= id :N -+N. Finally,

{0}.

A (K).

If K is not a closed hyperplane of F to begin with, we can let K0 = K

K x {1} cF x R. Clearly

x

x

is a

defined by

((a, r), (x, s)) = (a, x)+ rs.

If K is a compact convex subset ofF (locally convex) and E = F* then tIE + R' I is dense in A(K). 4.8. COROLLARY.

Let of the form Proof.

Ex

A(K x {l}). Then Theorem 4.7(iv) shows functions

f(x) = (a, x) +r

((a, r), (x, 1))

are dense in A(K).

In essence, Theorem 4.7 shows that K can be realized as the state space of a subspace (A(K)) of continuous functions. This is quite convenient and we

shall frequently assume, without further explicit mention, that K is so situated. Conversely, if K is a state space to begin with, then the next result shows nothing changes. 4.9. THEOREM. If M is a closed subspace of C'R(X), containing 1 and separating points, then M is isometrically isomorphic to Proof.

Since S = SM is in a closed hyperplane in the duality (M, M*) we

have 8M dense in A(S). But in this case Corollary 4.6 shows 0 is an isometry, and hence onto, since M is closed.

REPRESENTING MEASURES

5. REPRESENTING MEASURES Let K be a compact Hausdorif space and let 'e0 denote the collection of all compact Ga's (countable intersections of open sets) in X. Then is a set lattice (closed under finite intersections and unions) and it follows easily consisting of sets that are finite disjoint from this that the collection unions of differences, A\B (A, B E is an algebra of sets. If is the smallest tT-algebra containing (and hence the oS-algebra spanned by arc referred to as Baire sets. It can also be described as then elements of

the smallest o-algehra for which the continuous functions on X are measurable. A finite (non-negative) measure j.t on is called a Baire it is easy to see that for any B measure. By building up from A c B} = inf A V) : X\V and B c V}. p,(B) = sup Hence we say a Baire measure is regular. The Baire sets are a sub-class of the Borel sets, comprising the smallest o-algebra containing all open sets. A (finite non-negative) Borel measure is said to be regular if, for any E sup {j.t(K)

K compact and K cE)

and

= inf {p(U): U open and E U).

In contrast to the Baire measures, Borel measures are not automatically regular. We say a regular Borel measure of total mass one is a probability measure on X and denote the set of such measures by Mt (X). If e (X) then corresponds to a state L,4, E S, the state space of C(X), given by LM(f) =

That the converse is also true is the content of our next theorem. For a proof of this fundamental result, see for example, Rudin [184].

If L€ S, the state space of C(X), then there exists a unique probability measure E Mt (X) such that f E C(X) implies L(f) = 5.1. THEOREM (RIEsz REPRESENTATION THEOREM).

We note that the uniqueness follows from the fact that A

is an f€ C(X) with =lim =J XA fact, for U open,

on A, 0 1 on X\A. Hence Thus

1.4(U) = sup

fU

implies there

xA so that

is determined by its values on

: A c U and A E

and in

PRELIMINARIES

16

We also have that for any E E

there is a B

and a set N with

NcDEP4 and The essence of the Riesz Representation Theorem is that the cor(X) and the state space, S, of C(X) is affine and respondence between bi-unique. Henceforth we shall tacitly identify the elements of S as probability measures on X. Theorem 5.1 leads to an identification of the space of regular signed Borel measures with the real linear span N of S in C(X)*. Then Theorem 4.9 shows that, with N1 = co (S u —S), N and are isometric to A(S)* and A(S) respectively. If /L 0 in MR(X) then Proposition 4.2 shows = p(1) = In general IlL Q =

= inf {a + j3: a, /3 0 and

=

—

13lL2;

lL2

S}.

This in turn gives

where = — is the Jordan decomposition of Thus we will always identify MR(X) with In Chapter 4 we will extend this identification to the space M(X) of regular complex-valued Borel measures as Cc(X)*. If we have a subspace M (containing constants and separating points) of

C(X), with the state space SM, then the elements of SM can also be associated with probability measures, although the correspondence is no longer one-to-one. To see this, let (= M°) be the annihilator subspace of M in C(X)*. We let q: C(X)* -*

be the quotient map and recall that C(X)*/Ml is isometric to M* with continuous, the unit ball C(X)r is (qL)(f) = L(IL Because q is to mapped onto In particular, from Proposition 4.2 q maps S onto SM. Thus we have the following result. 5.2. THEOREM. Let M be a separating subspace, containing the constants, in C(X). Then L 6 SM if and only if there is a probability measure lL on X with

L(f)=ffdlL. In this case, we say lL represents L, meaning = L. If 4 X SM is the evaluation map, then 4(X) is a homeomorphic copy of X as a compact subset of SM containing ext SM (Theorem 4.3 and Corollary 4.4). Hence we can carry a MT (X) to SM by defining 11(4'(B)) = lL(B)

REPRESENTING MEASURES

17

and I.i(SM\cb(X)) = 0. Since f(x) = 4)(x)(f) =

we can write

ISPA

where, in the second integral, we evaluate f on elements of SM via the duality

As we have seen, if X = K, a compact convex set, and M A(K) then 4) identifies K with SM and, in this case, each L E SM can be represented by the point mass measure on K where 4)(x) = L.

On the other hand, if we start with K, embedded as the state space of A(K), then we can take X = (ext K is a compact convex subset of a locally convex space and X = (ext K) then for each z e K there is a probability measure with Proof.

Let 6: A(K) -*Mc C(X) be the restriction map. Since X is a

boundary for A(K), 0 is an isometry onto M. Let : M* -* A(K)* be the adjoint. Clearly, 6* maps SM one-to-one onto K as the state space of A(K). Mt (X) with 0*qp = z. Hence, we can choose

Then for f€A(K), f(z) = (f, z)= (f, 0*qp) = (Of, qp)=J f d1.t =f

I dp,

where in the last integral we consider X as a subset of K with p(K\X) =0. 5.4. COROLLARY. Let Mbe a separating subspace of C(X) with constants. Then L E SM if and only if there is a Mt (X) representingL and supp c (= (ext SMY).

Proof.

By definition äM(X) = ext SM and Theorem 5.3 applies with

K=SM. Of course Theorem 5.3 and Corollary 5.4 are two sides of the same coin. This raises the question of whether we can further restrict the representing measure to live precisely on ext K. For example if K is a finite-dimensional polytope then each z E K is a convex combination of extreme points and the corresponding representing measure is the same convex combination of point-mass measures. The problem in the general setting is complicated by the fact that ext K need not be a Borel set, let alone compact. Thus we have gone as far as we can with the Riesz Representation Theorem. If, however, K is metrizable then ext K is in fact a G8 set, and we can find a representing measure K) =1. This is the Choquet Theorem for K on K with

PRELIMINARIES

18

metrizable and the corresponding result in the more general setting is called the Choquet—Bishop--de Leeuw Theorem.

6. MAXIMAL MEASURES AND THE CHOQUET-BISHOP-DE LEEUW THEOREM

To prove these facts requires some machinery involving the convex functions on K and their graphs in K x To begin with, if E is a locally convex space and f is an extended real-valued function on E we call

{(a, r)eExR:f(a)=r} the graph of/, Since f is extended, the domain of the graph may be proper. If F€ (E x R)* then F has the form

F(a, r) = (a, a *) + rs

for some a* E* and s R. We say F is vertical if F(O, 1) =0 (equivalently, s = 0). If H is the hyperplane F c, where F is not vertical then H is the graph of a uniquef = (•,

+ t (translate of a linear functional) on F. In fact

(a, r) H if and only if (a, _a*/s) + c/s = r so that f = (. , + c/s. Now consider K as the state space of A(K). A real function f on K is convex if f(Ax + (1 — A )y) Af(x) + (1 — A)f(y) whenever x, y K and 0

1. We think of f as identically +X off K so that the graph of f is a subset of K x R in A(K)* + R. Define the epigraph off by A

epi f = {(x, r) E K x

r f(x)}.

Thenf is convex if and only if epi f is convex. Let Q(K) denote the subcone of C(K) consisting of the continuous convex functions. An important subset of Q(K) consists of functions of the form a1 v • v a, E A(K), I = 1,.. . , n. We say f is lower-semicontinuous (l.s.c.) if

{x€K:f(x)r} is closed for all real r. Thenf is l.s.c. if and only if epi I is closed. We let O(K)

denote the l.s.c. convex functions on K. Since K is compact, elements of are bounded below. 1ff is any function on K, bounded below, let

Aj={a €A(K):af}. Let I denote the collection of finite subsets of Af. Then I is a net directed by inclusion and for each a €1 we let = v{a: a e a}. Since

a c $ implies ba

we have

ba = sup {a: a A1}, which

19

MAXIMAL MEASURES

defines the lower-envelope, I, of f. The following theorem is a compilation of some of the important properties of f. 6.1. THEOREM.

(i) 1ff E Q(K) then there is a sequence (ba) c Q(K) such that each bn iS the supremum of a finite family in Af and 1' f uniformly;

(ii) if f is bounded below on K then 1€ inKxR; (iii)

(iv) if g E Q(K) and g f then g (v) (fi +f2) /i +/2 (fi, fz bounded below);

(vi) a 0 implies af= (af); (vii) if f is continuous,

= sup UK ba dM: a (viii) if (K) then E below); (ix) f E O(K) is affine if and only if

I} (f bounded

{aEA(K): a
is a probability

measure on K representing x; (x) 1ff is convex and u.s.c. then

is a net, directed down by , with

f=lim{g: gEA'}=inf{g: (i) Given e >0, for each xo E K and r
F(x0,r) r. Thus if r = f(xo) —

a(x)>f(x)—e. Then a simple compactness argument shows there are

a1,.. Since the supremum of a finite number of such functions is of the same type,

(i) follows.

PRELIMINARIES

20

(ii) Since / is an increasing limit of the net (ba)aEI in Q(K), / is l.s.c. convex, and hence in Q(K). (iii) Since, for each a, epi is a closed convex set containing epi f we have

epi /= fl epi

(epi f).

(epi f) then the above graph-separation argument produces If (xo, a E A1 with a(xo)> r. Hence (Xo, r) epi land (iii) is shown.

Part (iv) follows directly from (ii) and (iii). Parts (v) and (vi) are consequences of Af, + A12 C A(f,+12)

and aA1 = Aaf.

(vii) Let 1(x) be given by the right-hand side. Then 1(x) = inf {r: (x, r) e Co (epi f)}

and hence f is the convex function defined by co (cpu). Thus epi epi/ so that J f. Now, given E >0, for each x E K choose a compact convex neighbourhood of x in K on which If —f(x)I < e/2. Let Ni,,.. . , be a finite subcover of K. Let = {r R: r f(x,) — e/2}. Then epi fc x I, so. x that epi fc x I,) by virtue of the fact that each co is compact convex. Thus, for x E K, r,);

y'

r,

Hence,

and

(viii) Choose an increasing subsequence

(,h(b) means j'

= SUp

Let fo = sup b,, = urn

such that b

Then, by the Monotone Convergence Theorem IL(fo)

> If then, since rational r such that

SUP

is regular, there is a compact set F and a and

1(x)> r fo(x)

for all x E F.

MAXIMAL MEASURES

21

Using the compactness of F and the lower-semicontinuity of/we can find b0

I) and >0 such that

(fJ

b8r+e onF. =

If

v

b,, + e on F and

then c,1 =

But then

sup

SUp p,(ba) sup

+

sup

Thus sup

=

SUP

(ix) 1ff is affine and a1, a2
a2)

is compact, convex and disjoint from epi f so that graph separation yields an a such that a1 v a2 < a
= sup

JK

= sup

= sup {a(x): a
: a E A(K) and a

=f(x) =f(x). (x) If h is continuous and h f+ 6 then, given x with

x=Y.

A1x,,

and x1 c K, we have A1f(x,) + e

e

A(K) with so that, by part (vii), h f+ s. Now for each x E K choose Then x belongs to a neighbourhood with h and a finite set a1,..., am in A(K) such that if g A

Q(K) and f+


f+s for some 6>0. Hence the above shows that A1 is directed

and, sincef is the infimum of a family of continuous functions A1 converges

tof.

PRELIMINARIES

22

1ff is bounded above then we can define the upper envelope of f by

f(x) = Then

1(x) = inf {a(x) : a f, a

A(K)}

and (rf)A

(xi)

= rf (r 0).

We note also that if a E A(K) then =a

(a

(xii)

and

rfs(rf)

(xiii)

(rO).

(To obtain (xiii), apply (xi) to 0= (rf—

Let q: MT (K) -+ K, where q is

the restriction to Mt (K) of the quotient map C(K)* C(K)*/A(K) = A(K)*. We let if and only if jL represents x; i.e. = Thus e a = a(x) for all a E A(K). We use the notation x. Note that the, point mass

E

6.2. PRoPosITIoN. 1ff is continuous then, 1(x) = sup {p,(f)

Proof. We have a sf implies a(x) = x= implies = 6.1(vii) shows equality. 6.3. THEOREM.

E

S

and

and

so that /(x) But A1f(xj). Hence Theorem

The following are equivalent:

(i) xEextK; (ii)

(iii) 1(x) =f(x) for allf€ C(K); (iv) /(x) f(x) = f(x) for all f€ C(K). If (i) holds then = q'(x) is a face of MT (K). Thus ii e ext implies v ext Mt (K). But then, by Corollary 4.4 v = e,,, y K. Then a(y) = a(x) for all a €A(K), so that y = x. Hence Since 1(x) = inf : we have (ii) implies (iii). If (iii) holds for all f€ C(K) then Proof.

1(x) = —(—f)"(x) = —(—f)(x) =f(x).

If (iv) holds at x and x = Ày +(1 —A)z with z}, choose fE C(K) such thatf(x) = 1 and f(y) =0= f(z). Then 1(x) sO and f(x) 1. Hence x = y = z and so x e ext K.

MAXIMAL MEASURES

23

Our object now is to find a mechanism for choosing measures in where mass is tending to concentrate on ext K. Toward this end we define an ordering on the elements of Mt (K) by > ii if and only if p,(f) v(f) for all f E 0(K). It follows from Theorem 6. 1(viii) that > ii implies p.(f) p(f)

for fe Q(K). 6.4. THEOREM. (i)

(ii) For each ii there is a

which is maximal with respect to > and

(iii) 1ff C(K) and p Mt (K) then there isa > vsuch

>v. v(f).

If a A(K) then ±a 0(K) so that p.(a) = v(a). The second part of (i) follows from Proposition 6.2. To show (ii). we show every chain with respect to> is bounded above. If fl is such a chain is a net in Mt (K) and hence there is a subnet, which to 1ff E Q(K)and i.' we denote (va) which converges then for each 8>0 we can choose > ii and (vo— i-'a)(f)I <6. Then Proof.

vo(f)

v(f)—s. = p(gA)

define a sub-additive, positive ii. For (iii), we let homogeneous function on C(K) such that Hence

çb(a+g)v(a)+qS(g) and (using (xi), (xii), and (xiii) above). Furthermore, if A = {g C(K) : 4j(g) 1} then

C(K)1—PcA (P={g:gOonK}) convex subset of non-negative measures in so that A° is a C(K)r. If feA(K) then f =7 so that we can take =v. If we can find e A(K) and = çb(f) — v(f). Thus if i = p + we have

+r/)=

v(a)+çb(rf)— q5(a +rf).

This shows, for M = A(K) + Rf, (A

Hence

M)° = A° + M1 (since A° is

-s-i with

€A° and

onA(K)

It follows that and

p.,(/)=i(f)=v(j).

Finally, if g E —0(K) and v(g)>0 then g = and g/v(g) E A. Since

A°,

p(g) v(g). But then g —0(K) implies v(g + c)>0 for c a sufficiently on 0(K) so large constant and = v(c) = c, since c A(K). Thus that

> v.

PRELIMINARIES

24

We can now characterize the maximal measures by their preservation of integral values for envelopes of functions. (K), then the following are equivalent: 6.5. THEOREM. Let E (i) is maximal; (ii) for allfE C(K); = = for alifE Q(K). (iii) p;(J) = That (i) implies the first equality in (ii) is immediate from Theorem 6.4(iii). The second equality follows from f= —(--f). If (iii) holds for and Proof.

then fE Q(K) implies fL(f) v(f). But —f€ 0(K) and Theorem 6.1(viii) shows

v(—f), or

Thus v(f) t'(f)

so that

> v. Hence ,u. is maximal.

If fE C(K) and B1 = {x : 1(x) = f(x)} =

fl {x : f(x) —f(x) < 1/nj

set containing ext K. On the other hand, if

then B1 is a

is maximal then

Theorem 6.5(u) shows p(K\B1) = 0 for all fE C(K). Moreover, ext K = {B1: fE C(K)} by Theorem 6.3. It would appear then that maximal measures live mainly on ext K. We next make some precise statements of this observation. First, if K is metrizable then K is separable and we can find a sequence

fl

of non-negative functions in A(K)1 which separates the points of K. Hence the function

is continuous and strictly convex, meaning h(Ax +(1

—A)h(y)

whenever 0
Bh=extK (aG8setinK). Thus, we have the Choquet Theorem: 6.6 THEOREM (CHOQUET'S THEOREM). If K is a metrizable compact (K) such that convex set then for each x e K there is a E x and 1.

25

MAXIMAL MEASURES

Let X (ext K). If G is compact in K and disjoint from X then there is an fE C(K) with

fIG=O and is maximal then

1 on K. Thus, if This shows supp C X.

implies

= 0.

=0 for every In the general case we show that maximal implies Baire set B disjoint from ext K. To do this, we need a lemma which amounts to a "maximum principle" for point-wise limits of convex functions. 6.7. LEMMA. Let O(K) and let h,1 Proof.

h0

be a decreasingsequence of bounded functions in 0 on K.

0 on K. If h0 =0 on ext K then h0

Let Xo E K be given and choose a sequence

in A(K) such

that and

—1/n
Let (f: K K0 c fl,, R; 4)(x) = convex (and hence metrizable) set in 00

00

Then K0 = çb(K) is a compact E A(K0) be the projection

R. Let

onto the nth coordinate so that °

=

If y Eext K0 then y = 4'(x) for some x Eext K. Hence urn sup

= h0(x) =0.

= urn sup a0(x) urn

Now, using Theorem 6.6, we choose E Mt (K0) such that represents 1. and p,o(ext K0) = ho(xo). But = = 4'(xo) Thus /.6o(Pn) = Pn(Yo) bounded above in A(K0) implies we can apply Fatou's Lemma to to conclude 0 Since

fext K0 urn sup was

arbitrary, h0

=

f

lim sup

urn

Pn(Yo) = ho(xo).

K0

0 on K.

If K is a compact convex set then for each x E Kthere isa (maximal) Mt (K) which represents x and has measure zero on any Baire set disjoint from ext K. 6.8. THEOREM (CHOQUET—BISHOP—DE LEEUW).

then and AcB}. Hence let A If B be a compact G5 disjoint from ext K. Let 0f 1, on A and f<1 on = 0 on K\A, f€ C(K). Thenf' XA and, since (f?) =f on ext K, (ffl) extK. and Proof.

PRELIMINARIES

26

Let X = (ext K). In some applications we begin with functions defined only on X. Many of the same results apply—we summarize the situation below. If f is bounded on X then the envelopes for f can be defined as before:

/(y) = sup {a(y): a E A(K) and a f on X} f(y)= inf {a(y): a €A(K) and a f on X}. Thus, for example, 1€ O(K) and (if =1's where the second is in the original sense of Theorem 6.1. The results of Proposition 6.2 and Theorem 6.5 still hold. 6.9. THEOREM. Let X = (ext K) (i) (ii)

Proof.

is maximal on K then

and f C(X). =

Assume, by translating f, that 1 f on X. Let A denote the

restriction

and let y E K be given. Then, using Theorem 5.3, choose Let

/L Mt (X) representing y with respect to so that

Af={aEA: Then where p(A1) is the support functional of Af. But p(Af) =

=

+A1)

where the latter set is w *closed since C(X)1 — P c Cf, implying that C1 is a set in M(X). Thus and Thus, ii

0, v(1) =

1, p

y and

To show (ii), let F be a continuous extension of I to K. Then P j But if maximal then supp c X and Theorem 6.5 shows

is

7. BISHOP-PHELPS THEOREMS

Let A be a closed convex set in the Banach space E and assume 0€ A. Let A* denote the domain of the support function PA in E*. Then A* is a convex

27

BISHOP-PHELI'S THEOREMS

cone consisting of the linear functionals in E* that are bounded above on A.

We say (a, x) forms a support pair for A if a A and (a, x)

pA(x).

We show here that the set of support pairs in A x A* is plentiful. More

precisely, the (product space) projection of support pairs to A is dense in the boundary of A and the projection to A* is (norm) dense in A*. The proof rests heavily on the notion of support cones. Let x A*(llxlI =1) and 0< y < 1. The support cone P(x, y) is the cone in E spanned by

Since Uxil = 1

and y <1 this set has non-empty interior and P(x, y) = {a: (a, x) yllafl}.

If a A we denote A(a, x, 'y) = A n (a +P(x, -y)).

If A(a, x, y) is the singleton {a} then the Separation Theorem yields y A* with (a, y) a support pair and y 0 on P(x, y). 7.1. THEOREM (BISHOP—PHELPS).

Let a A and x A* with (a>0).

Then for each 0< <1 there is a support pafr (b, y) llb — all

a/'y and

= 1) with

— xli

Let P = P(x, y) and let A(') denote A(', x, y) on A. We note that if b E A(a) then A(b) c A(a). Furthermore, if c E A(b) then Proof.

'ylic

—bfl (c—b, x)

{(c, x) : c €A(a)}—(b, x)

so that diam (A(b))

We construct a sequence

(2/y) sup (A(a)

b,

x).

in A with a0 = a and

a nested sequence of sets with diameters decreasing to zero. Hence the completeness of E gives {b}= Now c A(b) implies c

fl

for all n so that c =

b.

ThusA(b) ={b}. Since

PRELIMINARIES

28

mt P 4) this means the Separation Theorem can be applied to find y E E* = 1) with

yO onP

and

(b,y)=pA(y).

Thus (b, y) is a support pair and b A(a) shows

(A—a,

ylIb—all(b—a, The fact that y 0 on P shows

(c, y)0}c{c€E: (c,x)_
[0,

1/y]x c Er +

Hence

(1/'y)x=Ay+w; so

A0 and

that x—yAy—yw.

Thus lix

—yAyJiy

and 11—

1= I lix Ii — ll'yAyli I lix

yAylI y.

Consequently, IIx—yii=Hx--yAy —(1

7.2. COROLLARY. Let e >0 be given. (i) For each x A* there is a support pair (a, y) with IIy — xli < e. (ii) For each a in the boundary of A there is a support pafr (b, x) with iIb—all<e. Take x 0 and apply Theorem 7.1 with any a A to x/iixII with any For (ii) choose A and llc — aIl €/2. Then there is an x A* y (lixil = 1) with

Proof.

(c,

Thus 7=

(a, x) + (c — a,

x)>pA(x). x) + €/2 and Theorem 7.1 applies with

BISHOP PHELPS THEOREMS

29

Notes

Some general references for the background material discussed in this chapter are Day [75], Holmes [131] and Semadeni [191]. In particular Holmes [131; Section 17] has a discussion of completeness that incorporates some of the material in Section 3 concerning the gauge lemma and regularity of domain. The embedding of a compact convex set K as the state space of A(K) (Theorem 4.7) is due to D. A. Edwards [87]. The standard text sources for the material in Sections 4—6 are Phelps [169] and Alfsen [5]. The latter has an excellent bibliography that traces the historical development of the

Choquet theory. We merely note some of the highlights. The metrizable version, Theorem 6.6, was proved by Choquet [60] in 1956. The general case, Theorem 6.8, is in Bishop and de Leeuw [40]. The ordering> or close variations of it are found in Bishop and de Leeuw [40] and Choquet [61]. Theorem 6.4(iii), the main development in showing the existence of maximal representing measure, occurs in Bonsall [45] and Choquet—Meyer [63]. The characterization of maximal measures in Theorem 6.5 is due to Mokobodski [161]. The Bishop—Phelps Theorem of Section 7 was proved by them in 1961 [41]. See also Bishop and Phelps [42]. An updated treatment with applications may be found in Phelps [170].

CHAPTER 2

Duality in Ordered Banach Spaces

We now take up the study of real Banach spaces that are partially ordered by means of a closed convex cone. We say the set P is a convex cone if (i) AP C P

for all A 0 and

(ii) P+PcP.

If P is a closed convex cone in the normed linear space E then (E, F) is partially ordered by a

•

b

if and only b — a

P.

We note that properties (1) and (ii) of P imply that (i)' a b and c E E implies a + c b + c, (ii)' a b and A 0 implies Aa Ab, (iii)'

Conversely, if (iY, (ii)', (iii)' hold for a partial ordering "s" then P = {a: a 0} is the convex cone yielding "and P is referred to as the positive

cone of "s ". Our principal source of examples for ordered Banach spaces consists of

the A(K) spaces introduced in Chapter 1 Section 4 (or equivalently, containing constant functions). The ordering is the subspaces of natural one, induced by taking F as the cone of non-negative functions. The cone P in A(K) has the additional feature of having non-empty interior (indeed, 1 mt F). This will not be the case in our general study. Finally, observe that (E, F) induces a natural ordering on E* with dual positive cone F" = —P° = {x E*: (a, x) 0 for all a P}. Thus, for example, A(K)* is ordered by F", where, using Proposition 1.4.2 and the fact that K is the state space of A(K),

A 0}. 30

pOSrFIVE GENERATION AND NORMALITY

31

Notes Some general reference books on ordered vector space theory are Jameson [134], Peressini [168], Vulikh [213] and Wong and Ng [217].

1. POSITIVE GENERATION AND NORMALITY Every a e A(K) can be written as

a0 (i=1,2).

a=a1—a2; Indeed, if a1 is any element

a =a1—(a1—a).

The fact that a is bounded above on K guarantees the presence of many such

ais. The ordered space (E, P) is positively generated if each a E E can be written as Definition.

(i=1,2).

a=al—a2; Equivalently, E = P — P.

If E is a Banach space then we shall see that if E = P — P then in fact one can find a bound (depending only on flail) for the norms of the a (i = 1, 2). Accordingly, we say E is aE

a,()

a=a1—a2;

(i=l,2)

and ha ill + 11a211

hail.

1.1. PRoPosITIoN.

The following are equivalent: (i) (E, P) is a..generated; (ii) E1 c a co (P1 u —P1).

Proof.

a = a1 —a2EE1 with a1, a2O and

Let

0< r = IIaj II + hla2hl a hail a. Then (with 0/0 taken to be 0)

a=

r[(flalfI/r)al/hlalhh+(hla2hh/r)(—a2/Iha2Ih)]E a co (P1 Li

Conversely, given b

0 let a = b/hibhl. Then (ii) implies

a = a(Aai +(1 —A)(—a2)) (a, E P1) so that, with b1 b=

hlbhlaAai, b2 =hlbhla(1 —A)a2, b1 —b2;

b

0

and

hhbiIl+

ailbhl.

DUALITY IN ORDERED BANACH SPACES

32

If E is a normed space then we say a new norm p is equivalent to 1' if there are positive constants a and b such that

onE. Of course equivalent norms induce the same topology on E. 1.2. THEOREM. Let (E, P) be an ordered Banach space which is positively generated. Then there is an a >0 such that Eisa-generated. Moreover there is a new norm p, equivalent to such that

(i) p = II. II on P; (ii) (E, p, P) is A-generated for all A > 1.

Let B = Co (P1 u —P1). Then, by Proposition 1.3.1(iii), B is a pregauge set. But Proof.

E=P-P=IJ nB=IJ ni so Corollary 1.3.4 yields an a such that C a'B

for all a' > a.

Thus E is a'-generating for all a'> a. Since B C E1 we have

BcEica'B foralla'>a. Thus if p = p(B) (the Minkowski functional of B) then p is a norm (since B is symmetric and absorbing) equivalent to 11. If a E P then a B if and only if a E E1 sop = on P. That (E, p, P) is a'-generating for any a'> I now

follows from Proposition 1.1 together with the fact that B is a regular pre-gauge set (p(B) = p(B), 1.3.3).

Example 1.6 below shows that Theorem 1.2(u) is not the same as being 1-generated. Our next goal is to determine when an element x E (E, can be decomposed as the difference of two non-negative functionals, that is, when is (E*, p*) positively generated? If we take the polar of the statement in Proposition 1.1(11), using some polar calculus we get the property

+p*)CE* whiCh is dual to a-generation in E. This leads us to the following definition. Definition.

The ordered space (E, P) is a-normal if

(El-P)rl(El÷P)cEa.

POSITIVE GENERATION AND NORMALITY

33

The next proposition gives the usual characterization of normality. The proof is straightforward and is omitted. 1.3. PRoPOsITION.

The following are equivalent:

(i) (E, P) is a-normal; (ii) then lixilsa max{IIcII, flbII}. 1.4. THEOREM. LetE be an ordered normed linear space. The following are equivalent: (1) E* is positively generated;

(ii) E* is a-generated for some a; (iii) E is a-normal. Proof. Since E* is a Banach space, (i) implies (ii) by Proposition 1.2, and (iii) follows as the polar of (ii). If (iii) holds then, since E1 — P and E1 + P are

regular pre-gauge sets, for any A >1 and hence

caEi=Ea which

by taking polars, gives (ii).

Theorem 1.4 establishes the duality between a-normality and a-generation in one direction. To complete the picture we investigate the situation when E* is a-normal. Theorem 1.4 yields that (E**, is a-generated but (E, P) does not quite inherit the full strength of this conclusion.

The space (E, P) is said to be approximately a-generated if (E, F) is a'-generated for all a'> a. Thus, the statement in Theorem 1 .2(u) could be rephrased as saying (E, p, F) is approximately 1-generated. Definition.

Let (E, P) be an ordered Banach space. The following are

1.5.

equivalent: (i)

E is approximately a-generated. Proof.

The dual E* is a-normal if and only if

(Er

+p*)CE*

are already we can take the polar of the Since the sets Er ± intersection and get the equivalent statement (iii) (P1 u —P1) E1 C a

DUALITY IN ORDERED BANACH SPACES

34

But then B = Co (P1 u —F1) is a regular pre-gauge set and hence (iii) Is equivalent to

Eica'co(P1u—P1) foralla'>a, which is precisely (ii).

The following example of an approximately 1 -generated space which is not 1-generated shows that Theorem 1.5 is the best conclusion. 1.6, EXAMPLE.

denote the Banach space of sequences x =

Let

=0 with

such that

lix ii = max

n = 1,

2,. . .}.

Let

E={xEco: Xl+X2 Then E is (1, 1, —1/2,

where x

E

a closed subspace (the zero set of the linear functional . Moreover, E is positively generated, 1' . .) 0. Let

0 means each

B = co (P1 u —F1) in E. We show x = (1/2, '-1/2,0,0,. . .)e AB\B for all A

y, z e P1 and a + (3

>1. If x = ay —(3z;

1, then

1/2 =

—f)zi

—1/2 = ay2—(3Z2. Thus, 1

But Yi

1, z2—z1

1, and y', z, 0(i =

1,

2). It follows that Yi =

1,

=0 and a = 1/2 = j3. But then

Y2 =

E

implies

=

and

1

= 1 = Zn+2 for all ii. Since y, z E

If 0
1

this is impossible. Hence

1/2k>r for some n then

rx = (1/2)(y —z)

where

y=fr,0,r/r',...,r/r',O,...,) and

z=(0,r,r/r',. . ,r/r',O,...,) .

belong to P1. Hence x eAB for all A = 1/r> 1.

ORDER UNIT AND BASE NORM SPACES

35

It can be shown in similar fashion that Pi — P1 in the above example fails to be closed. In fact

(1, —1, 0,0,. . . ,)eA(P1 —P1)\(P1 —P1) for all A >1.

2. ORDER UNIT AND BASE NORM SPACES We observed in the introduction that if E = A(K) then the function 1 E mt P. Indeed, we have E1

which clearly implies 1-normality. In general we say u is an order unit for the ordered norm space (E, P) if

E1 =(u—P)r'i(u+P).

(1)

In this case we call (E, F) an order unit space. We shall see shortly that the presence of an order unit essentially characterizes A(K) spaces. We say the ordering in a linear space is proper if P —P = {0}. This just

means that ±a 0 if and only if a =0. Let (E, F) be an order unit space. Then (i) E is 1-normal; (ii) u +E1 c P (hence u E mt P);

2.1. PROPOSITION.

(iii) the ordering in (E, F) is proper; (iv) if f is a non-negative (on F) linear functional on E then fE E* and Il/il

=f(u).

Proof. Parts (i) and (ii) are immediate from equation (1) above. If ±a 0 for all A and hence, a =0. For then —u Aa < u for all A >Oso that Aa

(iv) we have for any a E E1,

Hence f 0 on P implies If(a)I 11111

f(u)

Therefore IIfIIIIuII = IIfII.

With the aid of an order unit we can define the state space of E (in close analogy to Chapter 1, Section 4) by

S ={x EE*: x 0 on P and (u, x)= 1). From 2.1(iv) we have that x E S implies lxii = 1.

By an order isometry we mean an isometry between ordered normed spaces taking the positive cone of one onto that of the other.

DUALITY IN ORDERED BANACH SPACES

36

2.2. THEOREM. Let (E, P) be a normed order unit space. Then the state space S is -compact, convex and E is order isometric to a dense subspace of A(S). If E is a Banach space than E is order isometric to A(S). Proof.

From the definition of S we have that S is the intersection of a

hyperplane with the dual cone and is contained in Thus S is E implies and convex. Moreover ES. Since E is 1-normal, E* is 1-generating and hence

=co(Pr u—Pfl=co(Su—S). If 0: E

A(S) is the restriction map then IlOaII=sup{J(a,x)I: x ES}=sup{I(a,

Since Bu =

1

x

we have from Theorem 1.4.7 that OE is dense in, or if complete,

equal to A(S). Definition.

If P is a closed convex cone then a subset B of P is called a base

for P if (i) B is closed, convex and bounded, and (ii) for each 0 a E P there is a unique A >0 such that a/A E B. If (E, F) is an ordered normed space such that P has a base B and then (E, F) is called a base norm space. By Theorem 1.5 this implies E is approximately 1-generated so that in fact

E1= fl aco(Bu—B). We say (E*, is a dual base norm space if the base B is gives hence compact). In this case the

(and

=co(Bu-B). 2.3. THEOREM.

Let £ be a Banach space. Then the following are

equivalent: (i) (E, P) is an order unit space, (ii) (E*, is a dual base norm space. is the state space of E, E is order If these conditions hold then the base of isometric to A(S), and E* is order isometric to A(S)*. If (1) holds then the state space S of E is since fact a base for Proof.

S={xEP*: (u,x)=1}.

- compact, convex

and in

ORDER UNIT AND BASE NORM SPACES

37

Thus, the fact that

Er =co(Su—S) shows (ii). If (ii) holds and B is the base then the restriction map 9: E A(B) is an order isometry (into) since

Er =co(Bu—B). Moreover, if u is defined to be identically 1 on B and extended (uniquely) to

be linear on E* then Corollary 1.2.5 shows a E. Thus 0 is an order isometry onto A(B).

We now show that a linear space with an appropriately defined order unit can be made into a (normed) order unit space. This will provide a strictly order-theoretic characterization of (dense subspaces of) A(K) spaces. Let P be a proper convex cone in the linear space E. We say the ordering on £ induced by P is Arc hi:: if for a, b E,

Definition.

ra

b

for all r 0 implies a

0.

We say u is an Archiinedean order unit if for a e E (a) a E E implies there is an r 0 such that —ru a

ru, and

(b) ra a for all r 0 implies a <0. 2.4. PROPOSITION.

Let u be an Archimedean order unit for (E, F), P a

proper convex cone in E. Then (1) F is Archimedean;

(ii) u€coreP; (iii) a ePif and only if u +ra €Pforallr0; (iv) P is linearly closed.

Part (i) is immediate. For (ii) let a E. Then for 0< r, r sufficiently small, we have from (a) above —(1/r)u a. Proof.

Thus u + ra P. This shows u core P. For (iii), a + ra 0 for all r 0 if and only if r(—a) u if and only if —a 0 if and only if a 0. Now a E core F, so (iv) holds if

U + a P whenever a + Aa E P for 0
I=(u

+P)={a

—u

and let p = p(I), the Minkowski functional of I.

DUALITY IN ORDERED BANACH SPACES

38

2.5. THEOREM. Let u be an Arc himedean order unit for (E, P) and letland p be as above. Then p is a norm on E with unit ball I in which E is a normed order unit space with order unit u and closed convex positive cone P. The set I is symmetric and convex. Also Proposition 2.4(u) and (iv)

Proof.

shows 0€ core I and I is linearly closed. Hence p(I) is a semi-norm. If p(a) =0 then Aa €1 for all A 0 and hence

forallA0. Thus ±a 0 and, since P is proper, a =0 and p is a norm. Since I is linearly closed, I = {a: p(a) 1} is the unit ball of (E, p). Moreover, u + Ic P so that u mt P. Hence the fact that P is linearly closed implies P is closed in (E, p). Thus the definition of I shows (E, p, P) is a normed order unit space.

We conclude this section with a characterization of cones with a compact base, S. This is useful since such cones can be identified with the dual cone in A(S)*.

2.6. THEOREM. Let P be a proper cone in a locally convex space E. The following are equivalent: (1) P has a compact base S, (ii) P is locally compact, (iii) P is locally compact at 0. Proof. If (1) holds, {a E: f(a) 1} then

choose f€

such that inf f(S) = 1.

If U =

U Pc co (S u {0}) and hence is compact. Thus, for any 0 a {a

we have

P: a

aS, (ii) is shown. If (iii) is a compact neighbourhood in P. Since P = holds let U be a closed convex neighbourhood of 0 in E such that U P is compact. Let

S0=(U\int If 0

a

ext (U F) then a e S0. Since P is proper, 0€ ext

Hence, by the Krein—Milman Theorem (1.1.3),

Thus there is an f€ E* with f 1 on So. If S = {a F: f(a) =

1}

then S is a

DIRBCTF.DNESS AND ADDIT!VITY

39

closed subset of U P and hence compact. Moreover, if 0

a P then for

and hence f(a) A >0 so that a/f(a) e S. Thus S is a some A >0, a/A E compact base for P. 2.7. COROLLARY. P is closed.

If P is a locally compact cone in E (locally convex) then

Letf E E* such that f 1 on the base of P. Let a P and let U be a neighbourhood of a disjoint from the (possibly empty) compact set

Proof.

A = {b E F: f(b) f(a)± 1}.

Then U

f(b)
3. DIRECTEDNESS AND ADDITIVITY

Another way of expressing the fact that an ordered normed space E is positively generated is to say E is directed, that is, for each a, b E there is a c E with c a, b. As with positive generation it is possible to express the

degree of directedness: we say E is (a, n)-directed (n a positive integer greater than one) if whenever a1,. . , a,, cE1 there is an a €E,, with .

aa1

(i=1,...,n).

We also require an approximate version of directedness—thus we say E is approximately (a, n)-directed if E is (a', n)-directed for all a'> a. We give an example below of an approximately (1, 2)-directed space which is not (1, 2)-directed. The dual order property has to do with the degree to which the triangle inequality can be reversed for positive elements. Specifically, we say E is (a, n)-additive if whenever a1, . . . , a,, P then Ilaill+'

.

.

.

A simple induction shows that a (1, 2)-additive (directed) space is in fact (1, n)-additive (directed) for all n. The same holds for approximately (1, 2)-additive spaces. In general (a, 2)-additivity (directedness) implies (a', n)-additivity (directedness) for any n and some a' (depending on n and a). Finally, (1, 2)-additivity just amounts to fl being additive on P and we just say E is additive or 1-additive in this case. in general, if E is (a, n)-additive (directed) for all n and fixed a we will say simply E is a-additive (directed). 3.1. PROPOSITION. The space E is 1-additive {a F: hail = 1} is a (convex) base for P.

if and only if B =

40

DUALII'Y IN ORDERED BANACH SPACES

If a, b

Proof.

and

r =llalI-I-ljbll then

(a + b)/r = is

+ (Ilbll/r)(b/llbll)

a convex combination. The conclusions follow easily from this.

3.2. PRoPosITIoN. The following are equivalent: (i) The positive cone P has a base B; (ii) the dual cone has non-empty norm interior; (iii) E is (a, n )-additive for all n (fixed a). If E is an ordered Banach space for which these conditions hold then every non-empty subset S which is directed by converges as a net to l.u.b. S. Proof. If B is a base for P then by separating B from 0 we can find an element x0 such that x0(b) 1 for all b E B. If B Ea and x (1

then, for any beB, + x) 1— lb II lx 0.

(b,

Hence x0+ (1/13)E? cP* so (ii) holds. If + (1 /13)Er c and a c P then for any x E

(a, —x)+f3(a,x0)0. Hence 13(a, xo) (a, x) so that (a, xo) (11$ )ila Now, given llxollllai

.

II.

+ a,,,

.

. .

Hence E is (a, n)-additive with a =/3IIxoII.

If E is (a, n)-additive for all n take So={a €P:llall= 1}. Then ifS = co S0we have S is disjoint from the interior of (1/a)E1. Then the

Separation Theorem provides an XØ E* such that for any a

1',

la 1 (a,

= i} is a base for P. Let E be a-additive. Suppose (5, is not a Cauchy net. Then for some >0 and each a S there exists b, c a in S with lb — cu e. Hence, by the triangle inequality either — all or llc — all €12. Therefore we can construct an increasing sequence (a,,) c S with lla,,÷i — a,,ll €/2. But, if S is bounded in norm by M,

It follows that B = {a F: (a,

2M

—

aill =

—

k=1

ak)j

a1

llakl-1

akIl ne/2a.

Hence S is Cauchy and so has a unique limit a0 which clearly is l.u.b. S.

AND ADDITIVITY

41

To establish the duality between directedness and additivity we need to

formulate these properties as set inclusions. To do this we consider the s-fold Cartesian product which we make into an ordered formed space by a 0 if and only if a 0 hail = max {IIa1hl: i = 1,

. . .

,

n}.

Then (E)* = IInE* where (a, 1)

x,).

E* is given by

Then the dual cone P*

I

and

(a1,

=

0

if and only if each x, 11111=

0

hlxahl.

The diagonal, D, in E is given by

15={ãEE:a1=' Note that, since i.3 is a subspace,

(15)0={IEE*:

for allaEE}={IEE*:>.x1 =0}. The ordered normed space E is (cx, n)-directed if and

3.3. only if

Eica(Di -F). The proof is a simple verification and is omitted. 3.4. PROPOSITION.

(A)°

If A = L1 —P then

={

I E P*:

1, 1 = (xi,. .

.,

Proof.

p*

(A)° =

and

for alldEEi} (

=jIEE*:

fl

i—I

1

DUALITY IN ORDERED BANACH SPACES

42

3.5. THEOREM.

If E is an ordered Banach space then the following are

equivalent: (1) B is approximately (a, n)-directed,

(ii) E* is (a, n)-additive. Proof.

If (i) holds then

(A)°c fl

(iii)

= aEr.

But this says precisely that E* is (a, n)-additive. If (ii) holds then taking the polar of (iii) gives

E1ca cl(151—P). But then 131—P is E-regular so that Corollary 1.3.3 shows ca'(151—15)

for all a'>a

and hence (1) holds.

The most interesting case results when a = 3.6. COROLLARY.

1.

If E is an ordered Banach space then the norm is

additive on the dual cone

if and oniy if the open unit ball in E is directed.

Theorem 2.3 characterizes the dual base norm spaces (those with a compact base). We can now characterize the dual spaces that happen to be base norm spaces (with base not necessarily

Let (E, P) be an ordered Banach space. The following are equivalent: (1) E is 1-normal and approximately I -directed; (ii) E* is a base norm space. 3.7. COROLLARY.

Proof.

If (i) holds, the dual properties give

Er =co(P? u—Pr) and

is additive on

But then Proposition 3.1 says

B = {x E

Dxli = 1}

is a base. Hence (ii) holds. Conversely, Proposition 3.1 in the other direction gives E* is 1-additive. This together with the 1-generation of E* gives (i) by Theorems 3.5 and 1.4.

pIRECTEDNESS AND ADDITIVITY

43

To obtain the "dual" duality theorems the whole procedure is simply reversed. We take E as before but give E the 1' product norm, Dali

and

then E*

has

the

=

haul

norm,

liii = max

This direction is easier in that no approximate properties arise. We simply state the results. 3.8. THEoREM. equivalent:

If E is an ordered norm space then the following are

(i) Lis (a,n)-additive; (ii) E* is (a, n )-directed. The following are equivalent: (i) E is a base norm space;

3.9. COROLLARY.

(ii) E* is an order unit space. If these conditions hold and B is the base forP and u the order unit for E* then

B={aeP:(a,u)=l}. In the case when E = A(K)* it is easy to check that the order unit space E* coincides with the space Ab(K) of all bounded real-valued afline functions on K with the supremum norm.

3.10. EXAMPLE. We can construct an approximately 1-directed space along the same lines as Example 1.6. Let

E={i ec0: Let

a,1 v forces Ci = cz = c3 =1 and hence each If c E E1 and c a, b then c,, = 1, which is impossible if c e c0. Thus E fails to be 1-directed. It is not

difficult to check, however, that E is a -directed for all a >1.

DUALITY IN ORDERED BANACH SPACES

44

Notes The pioneering work in abstract ordered vector spaces was conducted by a Russian school led by Kantorovich in the mid to late 1930s. The first duality result of the type considered herein is due to Grosberg and Krein [122] who proved Theorem 1.4 in 1939. The reverse duality (Theorem 1.5) was shown by Andô [18] (without regard to the constant a) and, in its present form, by

Ellis [100]. The representations of order unit spaces and the results of Theorems 2.2 and 2.5 representing them as function spaces are essentially due to Kadison [136]. The notion of base-norm space and the duality of Theorem 2.3 is due to Ellis [100] and Edwards [87]. Theorem 2.6 on locally compact cones is shown by KIee [142]. The duality between additivity and directedness is established by Asimow [23]. The case a = 1 (Corollary 3.6)is proved in Asimow [22] and Ng [165]. Corollary 3.9 is shown in Ellis [100]. Example 3.10 appears in [22].

4. HOMOGENEOUS FUNCTIONALS ON CONES AND THE DECOMPOSITION PROPERTY

In this section we develop some techniques involving homogeneous functionals on cones which will be used in characterizing Banach lattices and discussing their duality properties.

We take E to be an ordered norm space with closed convex proper positive cone P. We say P has the decomposition property if whenever

(b1,b2O) there exist a1, a2 0 such that

a=a1+a2 and

a1_
a2b2.

We say P is a lattice cone if for each a, b P the least upper bound, a v b, and the greatest lower bound, a A b, exist. Since P is proper, a v b, and a A b are unique. A lattice cone always has the decomposition property, as we can take

a1=anb1 and a2=a—a1. Then

and

as(a+b2)A(b1+b2)=a Ab1+b2 so that 4.1. LEMMA. 1fF has the decomposition property and

a

a, h

b1

with

=

(as, b1eP).

ON CONES

HOMOGENEOUS

45

The case n = 1 is obvious. Assume the conclusion holds for n and let

Proof.

a+b=

nt-i

i= 1

Then d a + b so that the decomposition property yields d = a' + b'; a' a, b' b. The induction hypothesis gives Let d =

a'=Ea,, Let

a — a'

and

withc,=a,+b1. b—

b'.

Let p be a non-negative, positive homogeneous functional on P. We say p is bounded if there is a constant a such that

p(a)aIIalI

for all a eP.

If p is a non-negative, positive homogeneous functional then we can define the lower envelope of p by

The homogeneity of a functional p guarantees that p is convex on P if and

only if p(a + b) p(a) +p(b); a, b E P. In other words p is sub-additive. Similarly p is called super-additive if p(a + b) p(a) + p(b); a, b E P. Thus p is super-additive if and only if p is concave. 4.2. THEOREM. Let p be a non-negative, positive homogeneous functional on the closed proper cone P. (i) is a sub-additive, non-negative, positive homogeneous functional on P.

p on P and if q is any other sub-additive, positive homogeneous then q functional on P with Oq (iii) lIP has the decomposition property and p is super-additive then j3 is (ii)

linear. Proof.

If a, b E P and

a

=

< r, j3(b) < s then let aj

and

b

=

with

a + b = a, +

b•

(a,, b,

0)

DUAIJTY IN ORDERED BANACH SPACES

46

so that j5(a

Hence p is sub-additive. Since p is non-negative and positive homogeneous,

it follows easily that p is as well. For (ii), it is clear that fl

q(a)

so that q c,

p(a1)

For (iii) let a, b E P and r > p(a ± h). Then choose a ± b =

p(c1)< r. Then Lemma 4.1 gives

with

a=

a,

b=

and

=

a, ± b,.

Hence r.

Of particular interest will he functionals p of the form p is bounded, non-negative, positive homogeneous and super-

additive. It is convenient to say p is the point-wise infimum of x1,. .. , x,, and write p = inf {x,: i = 1,. . . , n}. We will reserve the notation x1 A x,, in when it exists. for the greatest lower bound of Xj, . . , If p = inf then it is easy to see that j5 takes the form .

(a,,xj: a

a,;

c E* where (E, P) We now focus on functionals defined on a dual cone is an ordered normed space. Letp be a non-negative positive homogeneous functional on We define j5 by = sup {(a, x): a E E and a p (point-wise) on p*}, We observe that is defined in close analogy with the I of Chapter 1, Section 6. Indeed, if base K then p is determined by its values has a on I( (by positive homogeneity) and, if f =PIK, then =f and

as

in Theorem I .6.1(vii).

4.3. THEOREM. Let (E, P) be an ordered normed space and let p be a non-negative positive homogeneous functional on (i)

and j5

are

p p on

sub-additive positive homogeneous functionals with 0

HOMOGENEOUS FUNCTIONALS ON CONES

(ii) Let B(p) {x e p*: Then B(p)° = A(p),

47

1} and A(p) {a E E: a

B(p) c B(j3)c

=

=

(B(p)).

(iii) j3 is the Minkowskifunctional of B(p)°° and the support functional of

A(p). Theorem 4.2(i) shows is sub-additive homogeneous. The homogeneity of p and the definitions show B(p)° = A(p). By definition j5 is the support functional of A(p), a closed convex subset of E containing —P, so that jY is 1.s.c., sub-additive and positive homogeneous on its domain of with x = x (x, e definition. But a e A(p) and x e imply Proof.

(a, x)

(a, x) =

p(x,)

so that,

In particular B(p) c B(fi) BQ) and BQ3) is convex since p

is sub-

additive. That B(j5) = B(p)°° follows since B(p)° = A(p) and j5 is the support functional of A(p), hence the Minkowski functional of A(p)° = B(p)°°. This completes (i), (ii) and (iii).

We conclude with some criteria for Let

4.4. THEOREM.

F) be an ordered Banach space which is positively

generated.

i = 1,..., such that

(i) Ifp =inf{a1: existx1,. . , -

x= (ii) If fl-Il — mt

to equal jY.

x,

andforeachxeP* there

and p(x) =

(a,, x,).

0 in E** then p = whenever p is a non-negative positive homogeneous functional on

Proof. We use the fact that E is in fact a-generated for some a to conclude that E is pt)-directed for some g3 and hence E* is n)-additive. Let K= Then if and let x E

x=

A,y1;

A,

and

= 1;

yEK

DUALITY IN ORDERED BANACH SPACES

48

since if r = Thus

11x41

then r ,3iIxII

= inf

/3 and we can take y'

(r/11x111)x1, A1 = jJx,I1/r.

(a,, x,): x = {

A1a,(y,):

where we

an

identify a, with its restriction to K,

element of A(K).

We now

invoke Theorem 1.6. 1(vii) to conclude that

= sup Since each a e A(K)

p is shows Also,

{a(x): a e A(K)

is

and

a

on

on

positive homogeneous, the Krein—Smulyan Theorem (1.2.7)

is w-Ls.c.

But then j5 and

the fact that

hence equality holds.

whenever x =

11x111

x in

implies that if

with

I

fl (a,,


of then we can choose, by the Alaoglu Theorem, limit points x1, . . . , k k inP * (X1)k,. .. For (ii) we use the hypothesis to conclude via Proposition 3.2 that E* is /3-additive. Thus the same argument applies to x = x, for any positive integer n. •

•

—

5. BANACH LATTICES AND THE RIESZ INThRPOLATION PROPERTY Let (E, P) be an ordered normed space such that for each a, b E E, a A b and a v b exist and are unique. We then say (E, F) is a lattice. If (E, F) is a lattice then P must be proper, since ±a E P implies both a and —a equal a n (—a). The uniqueness of a A (—a) then forces a = —a and hence a =0. In addition E is positively generated since a = a v 0— (a v 0— a) P — P.

5.1. PRoPosrrioN. Let a, b (E, F) be such that a A b exists. (i) for any c, (a + c) A (b + c) exists and

(a+c)A(b+c)=a Ab+C; (ii) if A 0 then (Aa) A (Ab) = A (a A b); (iii) (—a) v (—b) exists and equals —(a A b).

5.2. PROPOSITION. Let (E, F) be an ordered normed space which is positively generated with P proper. The following are equivalent:

BANACH LATFICES AND THE RIESZ INTERPOLATION PROPERTY

49

(i) (E, P) is a lattice; (ii) a, b e P implies there is a unique a n b E P. Proof. Assume (ii) holds. Then given any c, d E there is an e GE such that both C + e, d + e e P. Then CA d

shows

=(c +e)A(d+e)—e

c A d exists for any pair c, d. Then

cvd=—[(—c)A(—d)].

Let (E, F) be a lattice. We define

a4=a vO a=—(aAO) al = a v—a. 5.3.

PRoiosrrioN. Let (E, P) be a lattice. Then

(i) a+b=avb+aAb; (ii)

a —(a vb)=a+(—a A—b)=OA(a—b)=OA(a —b)+b--b =b Aa —b. Then (ii) follows with b =0 in (1). (iii) is immediate from the definitions of a4

anda. Now lal+a =a v—a +a =(2a)vO=2a4 so

that

lal= 2a4 —(a4--ai=

Thus laIO

and so

laI=laIvO=(a v—a)vO=(a vO)v(—a vO)=a4v a. Finally (v) follows from (i) applied to a4 and a. We have reserved the term normed lattice or, if E is complete, Banach lattice for lattices (E, F) with further relations between the order properties and the

norm. Specifically we require that E be a-generated (or just positively generated if E is a Banach space) and j3-normal for some constants a and j3.

DUALITY IN ORDERED BANACH SPACES

50

Our aim is to prove that (E*, p*) being a Banach lattice is a property that is essentially dual to (E, F) having the decomposition property (DP) of Section 4. It is convenient here to reformulate this as the Riesz Interpolation Property (RIP). To say (E, F) has the RIP means that whenever

c, d a, b there is an element e such that

5.4. PROPOSrIION.

Let (E, F) be an ordered normed space. The following

are equivalent: (I) P has the DP;

(ii) (E, P) has the RIP. Proof. Let (i) hold and apply Lemma 4.1 to the equation

(a—c)+(b—d)=(a—d)+(b--c) to obtain a1, a2, b1, b2

0 such that

a—d b—c

a—crai ja2 [&2 with the rows and columns of the 2 x 2 array summing as indicated. Now let e = a2+c and note that a—c = a1 +a2 implies

a2+c = e = a —a1. Now —c

so that

Ce Now

e—d=(a —a1) —d=(a —d)—a1 =b10 and

b—e =b—(a2+c)=(b—c)—a2b20 so that

AND THE RIESZ INTERPOLATION PROPERTY

BANACH

51

as well. If (ii) holds and (b1,b2_>0) then

Oa,b1

a—b2,

so that there is an a1 such that a—b2,

and hence 0 a — a, bz so that a2 =

a

—

a1

decomposes a as required.

We can easily establish sufficient conditions for (E*,

to be a Banach

lattice.

5.5. THEOREM. Let (E, F) be a normed ordered space which is a-generated and p-normal for some a and /3. 1fF has the (DP) then (E*, is a Banach P then lattice. Furthermore, if x, y

inf {(a1, x)+(a2, y) :a = a1+ a2; a1, a20}.

(a, x A

IfxEE* andaEPthen (a,x Proof.

The dual properties show that (E*, p*) is /3-generated and a-

P

normal. Let x, y E w

= [inf {x, y11

as in Section 4. Thus, for any a E P.

w(a)=inf{(a1,x)+(a2,y):a=a1+a2;a1, a20}. Then Theorem 4.2(i) and (iii) show w is linear on P. In addition w is bounded

on P since

0 w(a) (a, x)

A

(a, y)

flaIl(IIxfl A ilyll).

Since B is a-generated w has a unique extension to P—P = B such that if c = a — b;

a, b E P and IIaII+IIbII audi

then iw(c)i

w(a) + w(b) (iixii A

< a(flxfl A lIyfl)IIcIi

that w and the formula Also Theorem 4.2(u) shows w = x A y in for (a, w) holds. If x e E* and y any element such that y x, 0 then so

y —(xv 0) = y +(—x) A 0= (y —x)A y

DUALITY IN ORDERED BANAC}1 SPACES

52

and (a, y)—(a, x vO)=inf{(a1, y —x)+(a2, y):a =

a1

+a2; a1, a2O}

=inf{(a,

=(a, x v 0 is shown.

If (E*,

is a Banach lattice then there are a number of interesting

consequences (in fact, characterizations, as Theorem 5.7 below will show) that will be quite useful in later applications to A(K) spaces. To begin with, we say (E, F) is a complete Banach lattice if it is a Banach lattice such that every non-empty subset which is bounded above (in order) has a (unique) least upper bound. Let F be a convex subset of the cone P. Then, using the definition of face in Chapter 1 Section lit is easily checked that F is a face of P if and only if F is a subcone of P for which

a e F and 0 b

a implies b E F.

We say the closed subcone F of P is complemented if there is a map 0F, or just denoted u if F is understood, such that o: P -+ F and (1) o- is positive homogeneous and additive, (ii) o2a = cra a for all a E P. If F is complemented under cr then the complementary subcone

G = {a

P

:

= O}

and every a E P has a unique representation

a=bi-c;

b=uaEF and ceG.

In particular both F and G are faces of P. Conversely, if F and G are subcones of P such that every a E P can be represented uniquely as b + c, b E F, c E G, then F and G are complementary with map = b.

5.6. PRoPosrrloN. Let (E, F) be an ordered normed space which is agenerated and j3-normal for some a and f3.

(i) If F is a complemented subcone of P then 0F extends uniquely to a bounded projection of E onto the subspace F — F with null space G — G. In particular, F—F and G — G are i'll-closed subspaces of E. (ii) If F1 and F2 are complemented in P under cr1 and o'2 then F1 F2 is

complemented under

ocr2 = cr12 =

°cri and

F1 + F2 is complemented under

+ o'2 —

2.

RANACH LATTICES AND THE RIESZ INTERPOLATION PROPERTY

53

Each x F1 + F2 has a unique representation where G1, G2 are the faces complementary to F1, F2.

a projection Proof. (i) Since E is positively generated the extension of onto F — F with null space G — G is automatic from the properties (i) and (ii) If a E P then 0 oFa a implies, by normality, that lkrFall of 1311a11. If

a,bEP,with la 114— lb II

a IlcIl

then Ikrpcll

lluFall + l1(TFblI

apIIcfl.

(ii) For any x E P let y = cr2x and write y

oiy +(y —oiy).

Since F2 is a face and y E F2, we have

and

It follows easily that F1 F2 is complemented under O'i ° which, by symmetry, must equal 02° 01. That F1 + F2 is complemented under 01+ — O'12 is easily verified using the fact that 01IF20121F2

and

In particular, for x = x1 + X2 E F1 + F2, x =(cri+o'2—cr12)(X) =cri2(Xi +X2)+(XI —u2x1)+(x2—cr1x2)eF1

which is easily seen to be unique.

Definition. We say a face F of

P

F={xEP*: (a,x)=0}. 5.7. THEOREM. Let (E, P) be an ordered Banach space which is positively generated and normal. The following are equivalent: (i) (E*, is a Banach lattice; (ii) has the DP;

DUALITY TN ORDERED BANACH SPACES

54

(iii) (E*, is a complete Banach lattice; (iv) each exposed face of P* is complemented; (v) for any a1,. . . , a,, P the functional [inf{a1}] is linear on (vi) (E, P) has the RIP. Proof.

(i) implies (ii): as noted in Section 4, if

is a lattice then

has the

DP. The rest is by definition. (ii) implies (iii): let S be a non-empty set and bounded above by u. Let S0 = u — S. Then So is a non-empty subset of it suffices to show the greatest lower bound, w, of S0 exists in Then u — w is the least upper bound of S. Let T = {x e : x S0}. Since 0 E T, T is not empty, and we let D = {(a, j3) : a, /3 non-empty finite subsets of T, So resp.}. Since P* has the DP we

can use Proposition 5.4 (and a straightforward induction) to define a net

{w(a,fl):(a,/3)€D} such that w(a, /3) interpolates the families a and /3:

f3)y Now, for any fixed

forallxEa, y E/3.

S0. the collection of (a, /3) D such that

/3 is

co-final in D so that, by normality and the Alaoglu Theorem we can find a wE subnet of (w (a, /3)) which converges Since Pk is it follows that for any a eP, x E T and y E So,

(a,x)(a, w = g.l.b. S0. Since E* is positively generated this w shows in particular (via Proposition 5.2) that E* is a lattice, and hence, a complete Banach lattice. face of P*. For each xo E (iii) implies (iv): let A be any the set

A(xo) = {y E A: y xo}

is bounded above and non-empty so that we can define yo = o-xo = l.u.b. A(xo).

Now A(xo) is directed by (since x, y E A(xo) implies x v y E A(xo)) and there hence is a net in A. Since A(xo) is bounded above and A is is a subnet converging to z €A. Hence A(x0) z so that Yo z. But, given any a e F, (a, x) (a, yo) for all x E A(xo) so that (a, z) (a, yo) and hence z yo. Thus Yo = z e A and clearly Yo Xo.

If xi,x2EP* then A(x1)+A(x2)=A(x1 +x2) by virtue of the DP and the fact that A is a face. Also, for A 0, AA(x) = A(Ax)

BANACH LA1TICES AND THE RIESZ TNTERPOLATION PROPERTY

55

SO that Cr is linear. Clearly Cr = id on A SO that A is complemented in p* under (iv) implies (v): Given a, b E P let p = inf {a, b} so that fl(x) = inf {(a, Xi) + (b, X2) : x = x1 + x2; x1,

x2 e

From Theorems 4.3 and 4.4 we have j7 = j3, the support functional of

A(p)={cEE:ca,b}. We note first that if F1 and F2 are exposed faces of such that arc both linear then I F2 is also linear. This follows since F1, PIFI and F'2 are complemented (under cr2) and hence, by Proposition 5.6 F = F1 +F2 is complemented under u = cr1 + Cr2— Cr12. But then the functional q defined by q pocr1+pocr2—p0cr12 is

linear and equals

on F1 u F2. Moreover, if

then qx

follows from this that j3 q a, b on F. But if x F and x = Xt + x2; then Xi, x2 F so that X1, X2 It

qx (a, x1) + (b, x2).

Thus q p

F and therefore equality holds. We show that whenever

a support pair for A(p) then Xo belongs to a which j3 = is linear. Since a, b the sets (Ce,

face F0 on

is

p*: (c0, x) = (a, x)}

Fa = {x

are in fact

and F,, = {x

(Ce,

x) = (b, x)}

exposed faces. Now, by Theorem 4.4(i) we can choose

x0=x1+x2 with

= (a,

+ (b, x2).

But then (c0, x1) + (cc), x2) = (c0, x0) = j3(xo) = (a, Xi) + (b, x2)

and (c0, x2) = (b, x2). Thus x0e Fa +Fb. and hence (Ce, x1) = (a, = c0 on Fa u Fb the above remark shows jY = c0 on Fa + F,,. Finally, the Bishop—Phelps Theorem (1.7.1) shows that {x: (c, x) support pair for A(p)} is fl-dense in Thus

(F:

complemented face of

and PIF linear)

DUALITY IN ORDERED J3ANACJ-I SPACES

56

is a

11-dense sub-cone Q of P*. But if q

then q extends (uniquely) to a

bounded linear functional (also denoted q) on the dense subspace 0— Q c E**. Since q S a, b we have q Since and therefore an element of The general case of (v) is proved is l.s.c. on we have that = q similarly. (v) implies (vi): given a, b E P we wish to show that the set

A(p)={ceE:ca,b} is directed, where p = inf {a, b} on

It follows from (v) that if

q—inf{a,EP:i——1,..,n} then c7 is linear. To show A(p) is directed we can use the same technique as in

Section 3. Thus let E E xE with jJ(ai, a2)II = max {IIa ill, 11a211}

and P = P x P

let 15 denote the diagonal and let a = (a, a) and E (b, b). Let q = inf {a, b} p* so that by Theorems 4.3 and 4.4 7 is the support functional of

It is easy to see that A(p) is directed in E if and only if

mE. Now, the support functional c7 of A satisfies for (x, y)E y) = inf {((a, a), (xi, Yi)) + ((b, b), (X2, Y2)): x = Xi + X2, y = Yi

+ Y2}

= inf {((a, x1)+(b, x2))+((a, y1)+(b, y2)): x = x1 +x2, y = Yi +y2}

=fl(x +y). On the other hand so that if fl is the support functional of A 15 and (x, y) j5(x, y) = sup

{((c, c), (x, y)): (c, c) E A 15}

=sup{(c,x+y): ca,b}=j3(x+y). Thus 7=

so

that

We deduce that

A 15—P is closed.

ABSTRACT L AND M SPACES

57

E and

Given (c, d) A choose

(c,d)=(z, z)—(pj,p2)+(wl, w2); where z

a, b;

p1,p2O and

w1, w2€E1.

Since E is (M, 2)-directed (some M) there is a q P with q w1, w2 and M. Hence we have

or The remark atthe start shows that if p' inf {a, b, z + q} then the procedure can be applied to p' (or a suitable translate to make p' 0). Thus we can and construct sequences such that IIq,1÷1I1

and

a+

c, d, z,, z,, ÷ i + q,,

+i, b + qn + i, z,, +

+

In particular, the normality of E shows, since Zn÷j —

Zn

that (zr) is Cauchy and hence (zn) converges to z interpolating c, d and a, b.

(vi) implies (i): This is immediate from Proposition 5.4 and Theorem 5.5.

6. ABSTRACT L AND M SPACES A normed lattice (E, F) is regular if

IaIIbI

implies

This property implies that the norm is increasing on P and that fall = lI(lal)ll. Conversely, if the norm is increasing on P and hail = ll(lai)hl then E is regular.

We will see (Section 11) that in general normality and positive generation ensure that E can be re-normed to be regular. In this section we will present versions of the Kakutani Representation Theorems for regular Banach lattices of type L and type M. The object is to give order-theoretic characterizations of two of the most common types of function spaces with lattice structure. A regular Banach lattice is an abstract M-space if whenever 0 a, b then Ia vhff = laD v IfbIl.

In the absence of completeness we will use the term pre-M-space. Every M-space E can be realized as a closed Banach lattice-ordered subspace of Cft(X) (X compact Hausdorif). We content ourselves here with a proof of this fact in the event that E has an order unit, in which case E = CR(X).

DUALITY IN ORDERED BANACH SPACES

58

A regular Banach lattice for which the norm is additive on the positive cone is called an abstract L-space. The prototype for L-spaces is the space az). of functions integrable with respect to a measure space (X, We first give simple characterizations of L and M spaces in terms of their order properties and then demonstrate the duality between them. 6.1. THEOREM. An ordered normed space (E, F) is a pre-M space if and only if (i) a, b €Pimplies an b exists in F; (ii) E is 1 -directed; (iii) E is 1-normal. Let (E, F) be a pre-M space. Then clearly (i) holds and if a, b then c = al v lbl E E1 and c a, b. If c a b then Proof.

—(Ib:vcI)a and hence

laIlblvlcl and (iii) follows.

If (i), (ii) and (iii) hold then clearly (E, F) is a normed lattice. Then (iii) shows that is increasing on P and, since

jaj, ll(IaI)II. But (ii) shows that there is b a, —a with lbll lIafl. Thus v (—a)ll = ll(laI)Il

and it follows that E is regular. Now, given 0 a, b, choose c hell = hall v IlbIl by (ii). Then 0 a v b c and so, using (iii), ha v flail v llbll

a, b and bhl

IlcIl =

la v bhi.

6.2. THEOREM. An ordered Banach space (E, P) is an L-space if and only

if (i) a, b E P implies a n b exists in P, (ii) E is 1-generated, (iii) E is 1-additive. Proof.

If E is an L-space then (i) and (iii) are automatic. Also, hall

=

=

+ ahl = hla'll+hlall

shows E is 1-generated. Let (1), (ii) and (iii) hold. Then E is a Banach lattice

ABSTRACT 1. AND M SPACES

59

with norm additive on P. It remains only to check that E is regular. Property (iii) shows Ills increasing on P. Also, given a E, hail

But, by (ii), a = a1

=

+ hail = hIdai)hI.

—

a2 (a1 0) with Ia ii = ha ill + hIa2hI.

However,

0

a a2 so that ia II

iI(Ia 1)11.

We can now apply the duality theorems of the previous sections to L and M

spaces and immediately deduce the following theorem. 6.3. THEOREM.

(i) If E is an L-space then E* is an M-space. (ii) If E is a pre-M-space then E* is an L-space. We next characterize dual L and M spaces.

6.4. THEOREM. Let (E, P) be an ordered Banach space. Then E* is an M-space (if and) only if E is an L-space. If E* is an M-space then we can conclude that (i) P has the DP, (ii) E is i-additive, (iii) E is approximately 1-generated.

Proof.

In view of (i) and (ii), together with Proposition 3.2, we see that E is in fact a

complete Banach lattice. In particular £ is then -generated for, given a a = a1 — a2 (a4 0) we then have 0 Thus (iii) shows hail = IiaiI+Iiail. Hence, by and ha Theorem 6.2, E is an L-space. 6.5. THEOREM.

Let (E, P) be an ordered Banach space. Then E* is an

L-space if and only if (i) P has the DP, (ii) F is 1-directed,

(iii) Eis 1-normal. If (I), (ii) and (iii) hold then their dual properties show, via 6.2, that E* is an L-space. Conversely, if E* is an L-space then (i) and (iii) hold but we Proof.

conclude only that E is approximately 1-directed. However an iteration using (i) and (iii) will yield (ii). Thus let a, b E1 and let = (1 + (n 1). Choose c1 a, b, 0; IlcilI

A1, and suppose c1,. ..

,

are given with

DUALITY IN ORDERED RANACH SPACES

60

112k (k = 1,. and IICk+l — a, b, 0; such that E1 so that there exists a,

a, b, 0;

, n —1). Then

E

which interpolates. Thus

implies there exists

b,

..

and

Thus —C,, —(1— 1/Ajc,, and therefore

IIc,,+i—c,,I1(1-- 1/A,,)A,, =A,, —1

=2".

Hence (c,,) is Cauchy and clearly has a limit c E E with c

a, b.

We next note some facts about normed lattices which will be useful in discussing the representation theorems below.

Let

6.6. PRoPosITIoN.

(E, F) be an a-normal and 19-generated ordered

normed space. The following are equivalent: (1) E is a normed lattice; (ii)

for each

a, b E

P and

p = inf {a,

and extends uniquely to a A

Proof.

xEP*,

If E

is

a lattice then F"

has

b} on b P.

the functional

the DP so that

p(x)=sup{(c,x): ca, b}=(a Ab,

is

is additive on

additive and, given

x).

then by definition Conversely, if j3 is the restriction of an element c E B to is a normed lattice. = a n b. Hence Proposition 5.2 shows E c

If (B, F) is a normed Lattice the elements x F" which preserves the lattice operations are called lattice homomorphisms. In view of Propositions 5.2 and 5.3 it suffices that

(a nb,x)=(a,x)n(b,x)

for all a, b

€1'.

6.7. THEOREM.

Let

(B, P) be a normed lattice. Then x E F" is a lattice

homomorphism if and only if the ray

x isa homomorphism and 0 y so that y) +(a y) implies, since y 0, that

Proof.

If

(a,x)vO=O=(a,x) (a

x is a face of P".

,

x let (a, x) = 0. Then (a4, x) = Hence (a, y) =0. Thus y = Ax for some

ABSTRACT 1. AND M SPACES A

0. If

61

is a face of JJ* and a, b E P then, by Proposition 6.6 and

Theorem 4.4, (a A b, x) = inf ((a, xj) + (1.',

: Xi + X2 =

x}

: A + = 1}

= inf {(a, Ax) ÷ (b, = (a, x) n (b, x).

6.8. PRoposiTioN. Let (E, F) be a normed lattice and let S be a non-empty subset for which a = I.u.b. (S) exists. Then for any b GE, bA a

= l.u.b. {b A s: s E S}.

Clearly for alisES. Let cbns for ailsES. We use Proposition 5.3(i) to obtain c b + s — b v s. Thus c + b v a c + b v s b + s for all S so that Proof.

c +b V a b +a

andhencecbAa. Thus bAa =l.u.b.{bAs:5ES} We next prove the Kakutani representation of an M-space E with order unit by showing E = CR(X) where X is the set of extreme points of the state space S of E. This is tantamount to the Stone--Weierstrass Theorem (lattice version). 6.9. THEOREM (STONE—.WEIERSTRASS).

Let X be a compact Hausdorff

space and let the subspace E be a sub-lattice of CR(X) which contains 1 and separates the points of X. Then E is dense in CR(X). Proof.

and f(y)

For each x y in X and a, f.3 eR there is anfE E such that f(x)= a i9• In fact if r =fo(x)—fo(y) 0 for foE E then

f = (a/r)(fo fo(y)' 1)— (f3/r)(fu Leth E CR(X)ande >Obegiven. Letx E be such that each y x let = h(x) and

y X let be a neighbourhood of y on which such that if we can choose . .,

1)

= h(x) 1.For = h(y). For each > h — e. By compactness

•

then gx(x) = h(x) and g, > h — r. For each x let be a neighbourhood on which the (as above) satisfies > h + 6. Again, by compactness, we choose Xi,. . . , Xm such that if

g=gx1A then

h—e
•

DUALITY IN ORJ)ERED BANACII SPACES

62

Let (E, P) be a normed order unit space and recall from Section 2 that E is order-isometric to a dense subspace of A(S), where S is the state space of E. If in addition E is a lattice then we can make a much stronger assertion. By a

lattice-isometry we mean a linear isometry which preserves the lattice operations. 6.10. THEOREM. Let (E, P) be an order unit space and a lattice. Let X = ext 5, where S is the state space of E. Then Xis weak *..compact and E is

lattice-isometric to a dense sub-lattice of is lattice-isometric to Cn(X). Proof.

If E is a J3anach space then E

Let e denote the order unit of (E, F). Since

S={xEP*:(e,x)= 1} it is immediate from Theorem 6.7 that X = ext S is the set of lattice homomorphisms whose value at e is 1. Thus X is the intersection of the weak*_compact sets

S(a, b) ={x ES: (a A b, x)= (a, x) A (b, x)} (a, b E P)

Let 0: E C'R(X) be the restriction map. Since the restriction of E to S is isometric to a dense subspace of A(S) the and hence is

Krein—Milman Theorem assures that 0 is an isometry. Moreover each x E X being a lattice homomorphism shows

0(a Ab)=(Oa)A(Ob)

and

d(a vb)=(Oa)v(Ob).

Since OE is a separating sub-lattice containing 1 = Theorem 6.9.

Ge

the rest follows from

Observe that (E, F) being an order unit space ensures that properties (ii) and

(iii) of Theorem 6.1 hold. Thus the hypotheses of Theorem 6.10 are equivalent to (E, F) being a pre-M-space with order unit, although this formulation is somewhat redundant, in view of Theorem 6.1. Y), Y compact Let E be a linear sub-lattice of Hausdorff, which contains 1 and separates points. Then the evaluation map 6.11. COROLLARY.

0: is a homeomorphism onto X = ext S.

Proof. We have from Corollary 1.4.4 that X 4 Y. But each qSy homomorphism and hence belongs to X.

is a

lattice

6.12. COROLLARY. If X and Y are compact Hausdorff with CR(X) order isometric to Y) then X is homeomorphic to Y.

ABS I'RACI' L AND

Proof.

If iii

:

63

SPACES

Y)

CR(X) is an order isometry then the adjoint 111* yields

are the and and where a homeomorphism between respective evaluation maps onto the extreme points of the respective state spaces. It follows from Corollary 6.11 that X and Y are homeomorphic. An interesting and useful application of 6.10 can be made in connection with the Stone Representation Theorem for Boolean algebras. This will also be of

vaiue in the development leading to the characterization of L-spaces. A l3oolean algebra is a partially ordered set ( is transitive, reflexive and anti-symmetric) such that is a distributive lattice ((A A B) v C = (A v C) n (B v C)), (i) (ii) g.l.b. and l.u.b. 4 (denoted 0 and I) exist in 4, (iii) for each A c 4 there exists a unique element AC such that A A

=0

The axioms are, of course, modelled after set-theoretic operations

and

u. We shall see that each Boolean algebra 4 has a realization as the collection of clopen (both open and closed) subsets of a totally disconnected (TD) compact Hausdorif space. (We say X is totally disconnected if each point has a neighbourhood base of clopen sets.)

A subset A of X is clopen if and only if the characteristic function, denoted xA, is continuous. We observe that f is a characteristic function of a

subset of X if and only if f has values of 0 and 1 only. This in turn is equivalent to fA

The subspace in (not closed in general) spanned by the continuous characteristic functions, denoted 9'(X), consists precisely of the continuous simple functions, that is, functions taking only finitely many values. Such a

function f is associated with a partition of X into a disjoint union of non-empty clopen sets A1,. . . , A,, withf constant on each The subspace is a sub-lattice of CR(X) containing the constant functions.

6.13. PRoposiTioN. LetXbe a compact Hausdorff space. Then Xis TD if and only if .9'(X) is dense in Proof.

If X is TD then SP(X) separates the points of X and hence, by

Conversely, given x E U, U open, there is a Theorem 6.9, is dense in continuousfunctionf such that 1,f(x) =1 andf = Oon X\U. Let a be

an element of ,9'(X) such that Ill— all neighbourhood of x contained in U.

Then {x: a(x)

is a clopen

The collection of clopen subsets of X forms a Boolean algebra, under u and which is lattice isomorphic to the subset of characteristic functions

DUALITY IN ORDERED BANACH SPACES

64

The induced lattice operations on the characteristic functions are the natural ones inherited from CR(X). Now let be a Boolean algebra. We wish to construct a formal replica of of simple functions on a compact Hausdorif space and use the space in

this to identify the elements of

with the continuous characteristic

functions, denoted To do this we define a (formal) simple function a on to be a finite collection of pairs (A,, a,)71 with 0 A, E a R and A1 v•

A A, = (I

v

(We say such a collection of We say a = (A,, is

(i

forms a partition of sit.)

equivalent to 6 = (B1,

if for each real

number c V{A,: a=c}=V{B1: b,=c}.

Modulo this equivalence relation, the set of formal simple functions, denoted .5"(A), forms a normed order unit space which is a lattice, that is, a pre-M space with order unit. The operations are defined by Àã = (A,, Aa,)71

a +6 = (A, A B,, a, + hail = max {Ia,I:

A B,

0)

I = 1,. . . , n}

dAb=(A,AB1,a,Abj)

e=(I,1) where a, b are as above. The simple function a is called a (formal) characteristic function if each is zero or one: equivalently,

ãA(e—a)=0. Let be a Boolean algebra and let be the associated space of (formal) simple functions. LetX = ext S, S the state space Then X of is a TD compact Hausdorff space such that (i) is lattice-isometric to 11(X) and (ii) is (lattice) isomorphic to If Y is a TD compact Hausdorff space with isomorphic to then V is homeomorphic to X. 6.14. THEOREM (STONE).

From Theorem 6.10 we have is lattice-isometric to a dense corresponding to sub-lattice of CR(X). Let a denote the element of ci€ Proof.

ABSTRACT L AND M SPACES

Now

65

a is a characteristic function if and only if SA(ë—S)=O,

showing that a is then a characteristic in which case a n (1—a) =0 in function on X. Hence each S .9'(d) corresponds to a E with the same values a,. On the other hand, if f is a continuous characteristic function on X then, since {a: a &t'(s4)} is dense, there is an a 9'(.91) with la —fU< Then Let A=V{Aj: and let 5be the (formal) either characteristic function of A. Then lb—Ill

lIb — all + la —Ill = llb— all + ha —111< 1.

Since both b and f are characteristic functions, b = f. This shows .9'(X) =

In particular, from Proposition 6.13, X is TD, and (i) is {a: S shown. The identification A -+ a,

S = {(A, 1), (Ac, O)}

is a lattice isomorphism from onto the characteristic functions of X, and showing (ii). The uniqueness assertion follows from the lattice.. hence isometry of S°(.s4) and S°(Y) induced by .+ Y). For then and CR(Y) are order-isometric and Corollary 6.12 applies. We now consider the representation theory for abstract L-spaces. We call an

element u of the L-space (E, F) a weak order unit (also known as a Freudenthal unit) if 0

u E P and

a A u =0

implies a =0.

The first observation about L-spaces is that the conditions of Proposition 3.2 are met and this allows us to conclude a facial structure for P analogous to that of a dual lattice cone described in Section 5.

Let (E, F) be an abstract L-space. (i) Each closed face F of P is complemented with complementary face

6.15. THEOREM.

G={b: bAa =Oforalla eF} (ii) If a P and Fa is the smallest closed face of P containing a then the complementary face is Ga = {b : b A a = 0}.

(iii) The collection of closed faces ofPforms a Boolean algebra with

F1 A F2 = F1

F2, F1 v F2 = F1 + F2

and complementation being the complementary face. The associated collection of projections is an isomorphic Boolean algebra with

o'1vu2u1+cr2—ol°o2,

DUAUTY IN ORDERED I3ANACI-i SPACES

66

Proof.

Let F be a closed face and let c c P. Define = l.u.h.

= {a E F: a c}.

is directed in F and bounded in norm (by crpc exists by Proposition 3.2. Then the DP shows (TF has the required properties (i) and (ii) of Section 5 (see, for example, the proof of (iii) implies (iv) in 5.7). If b E G, a E F and 0< c a, b then c G F = {0} so that 1, a =0. Conversely, if b A a =0 for all a F then in particular b A (oFb) =0. But b b e G. For (ii), let F0={b: some A 0} and let Since

F = {c: c = l.u.b. A, A a directed subset of F0}. Clearly F c Fa. Let b E P be given and let

Ab={cEF: cb}. Then each c E Ab is of the form c = l.u.b.

where

is a directed subset of c Ab}. Then

F0. Let C be the set of suprema of finite subsets of U C c F0 and Lu.b. C = l.u.b. Ab F. This shows that £TFb = l.u.b.

Ab e F

and hence F isa complemented (equivalently, closed) face of P. Thus F = Fa, If b A a =0 then clearly b A C =0 for all c F0 and Proposition 6.8 applies to show b A C 0 for all c F = Fa. Finally, (iii) follows from (i) and Proposition 5.6. 6.16. THEOREM. Let (E, F) be an abstract L-space and let u P with = {a: —Au s a Au some A 0}. Then is an abstract lull = 1. Let M-space with order unit u. Proof. The set is a lattice ordered subspace of E. Furthermore, if ra u 1 for for all r 0 then (ra) v 0 = r(a v 0) = ra'
all r, so that order unit for

a 0. This implies that u is an Archimedean is an order unit space with unit so that, by Theorem 2.5,

ball

I={a: —ua_
67

ABSTRACT 1. AND M SPACES

It follows that

— ami

u/2m and therefore are both

am)

—

Let

=

(ak+1 — ak)

and

=

— ak).

The sequences (s,,1) and (t,,,) are increasing and bounded above by u. Thus

and (t,,,) are I-convergent in E Proposition 3.2 applies again to show and t = l.u.b. Since 0 s, t it both s and t to s and t with s = l.u.b. with are in (ak÷I —ak)=(s—sm)—(t—:m)

u/2" + u/2'" so that p[(s —t)—(am+i —aj)Jp(s Thus

Sm)+p(ttm) 1/2"''.

converges to ai + (s — t)

6.17. THEOREM order unit u, hull = 1, and

Let (E, F) be an abslractL-space with weak be as in Theorem 6.16. LetX = ext S, where S

is the slate space of is a TD compact Hausdorff space and the restriction + is a lattice-isometry. map 8: (ii) The continuous characteristic functions are lattice-isomorphic to the Boolean algebra 94(P) of closed faces in P under the correspondence (1)

F-* O(crFu).

(iii) There is a probability measure p on Xsuch that 0 U, open, implies p(U)>0, and there exists a lattice-isometry b of E onto with as a subspace of L'(/h)). =8 (regarding Proof.

The Representation Theorem (6.10) for M-spaces with unit,

together with Theorem 6.16, shows that X is compact Hausdorif and

8:

.+ CR(X)

is a lattice-isometry. The characteristic functions in CR(X) are identified by the property

so that they are precisely the images under 0 of

DUALITY IN ORDERED BANACH SPACES

68

E A for each F€ 96(P) and clearly F1 cF2

Theorem 6.15(1) shows

with projections (11, 02 implies 'YF1U UF2U. Moreover, if F1, F2 cr2U, then F1 cF2. To see this let b E F1 G2 (02 complesatisfying

mentary to F2). Since OIUEF2 we have bA(cr1u)=O. If F1 so that c = o1c OiU and hence C

Oc

u then

A(01U)=O.

0. Then

Thus b A u =0 and, since u is a weak order unit, b

F1+F2=F1

G1 czF2

so that F1 CF2. Next we observe that each a A equals where is the projection to the closed face Fa spanned by a. This follows from 6.15(u) since a n (u — a) = 0 implies u — a belongs to the complementary face Ga. Then 0=(Ta(U —a) gives O'aU = 0aa = a. This establishes (ii). We next show that X is 'ID. Let x U, open in X, and let D be a compact neighbourhood of x with D c U.

in CR(X) with

f(x) =1 and

0

on X\D,

and

on D,

on X\U.

Let Oc =1 and Ob = g. Let be the projection to Fe and let a = ou, h 0a. Now c u implies c = cc a so that h(x) 1. Since g Af =0, b n c =0. Thus (by 6.1 5(u)) and hence b n a =0. Thus b is in the complementary face gAh

0 on X\U. But then h is the characteristic function in

0 so that h

CR(X)of a clopen neighbourhood of x contained in U. This completes (1) and (ii). For (iii) we note first that is 11-dense in Esince, if is the closed face

spanned by u then a

if and only if a A u =0 if and only if a =0. Thus

= P.

But {c: c = I.u.b. A, A directed in

c

ci

and, since Next, define /L(Oa) = Hall for a 0 in extend uniquely to a linear functional on CR(X). Clearly belongs to the state space of CR(X) and hence is a probability measure, by the Riesz Representation Theorem. Define : -, by = 0(a) and note that

the positive cone in

11 is additive on

=

10(a)!

= L 0(Jal)

= lila Ii = jail.

C}IOQUET SIMPLEXES

69

with the subspace CR(X) c is Il-dense in E, and

Hence b is a lattice-isometry of (En, II) is dense (in the L' norm) in Since extends to a lattice isometry from E onto

We observe that (ii) says that the Stone Representation Space for homeomorphic to the space X of (i) and (iii).

is

Notes

In the last three sections we have merely touched on the theory of vector lattices, a subject that literally fills volumes. Some standard references are Schaefer [189] and Luxemberg and Zaanen [157]. The decomposition property and its relation to vector lattices goes back to Riesz [178,1791. The idea of complemented order ideals in lattices appears in [178] and the result of Theorem 5.5 in [179]. The reverse direction ((i) implies (iv) in Theorem 5.7) is due to Andô [18]. The fact that (iv) implies (i) in Theorem 5.7 was shown by Ellis [108] for the case E = A(K). We will have more to say about this in Section 7 on simplexes. The machinery set up in Section 4 appears in Asimow and Ellis [31]. The representation theorems for spaces of type L and type M are given by

Kakutani [139, 140]. Theorem 6.4 is due to Ellis [100]. The various characterizations of ordered Banach spaces whose duals are L-spaces was of some interest in the 1960s when Effros [93] termed them simplex spaces. A

somewhat different order-theoretic characterization from the one in Theorem 6.5 is due to Davies [72]. The directedness of E1 is the essential feature. This is developed somewhat differently in [72] and we discuss his approach in Section 11. The results of Theorem 6.10 and Corollary 6.11 go back to 1940—41 in papers by Stone [203], Kakutani [140], Krein and Krein [143], Yosida [218]. The representation of Boolean algebras in Theorem 6.14 is the well-known theorem of Stone [202].

7. CHOQUET SIMPLEXES An ordered Banach space (E, P) is order-isometric to an A(K) space if and only if E is an order unit Banach space, in which case K is the state space of E. As we saw in Section 6 if E is an order unit space and a lattice then E is order-isometric to CR(X), where X = ext K (and is compact). In this section

we study in more detail an important situation which is intermediate to these, namely, E is an order unit space with the RIP. Theorem 6.5 shows E* is then an abstract L-space and the state space is then a base for the lattice ordered cone

Definition. A compact convex set K (of a locally convex space) is called a Choquet simplex if A(K)* is a lattice.

I)UALITY !N ORDFRED BANACH SPACF.S

70

In view of Proposition 5.2 this is equivalent to the cone being a lattice. To summarize we state the following.

spanned by K

7.1. THEOREM. LeIK be a compact convex subset of a locally convex space. The following are equivalent:

(i) A(K) has the RIP; (ii) A(K)* is an abstract L-space; (iii) K is a C'hoquet simplex.

Theorem 5.7 shows that K inherits a rich facial-geometric structure from the general properties of dual lattice cones. In this regard we can simply identify the elements of A(K) with the positive homogeneous addi-

tive functionals on In addition each face of K spans a (by Theorem 2.6, Corollary 2.7) face of P* and conversely, the intersection of faces of P * with K are faces of K. It is conventional to call a face of Ksplit, or a split face, if it is the intersection of a complemented face of with K. Thus, from Theorem 5.7, we have the following. 7.2. THEOREM. The compact convex set K is a simplex if and only if each closed exposed face of K is split.

The notion of an infinite-dimensional simplex was originally formulated in connection with uniqueness of representing measures concentrated on the

extreme points, as formulated in Chapter 1, Section 6. This is clearly a property that finite-dimensional simplexes possess; in this case one says a point of the n-dimensional simplex has a unique set of n +1 barycentre coordinates with respect to the vertices. We recall the set-up of Chapter 1, Section 6, where (K) is the state space of consisting of the probability measures on K, and q:

is the quotient map and the quotient space is identified with A(K)*. Then q maps (K) onto K (as the state space of A(K)) with = x, where JK

=(/x)=f(x);

feA(K).

Let (K) denote the subset of Mt (K) consisting of niaximal measures. (aM (K) is never used to denote ext Mt (K).) Then Theorem 1.6.4 shows the restriction of q to (K) is onto K and Theorem 1.6.5 shows aM (K) is convex in Mt (K). Moreover, (K) is a (not necessarily closed) face of ii, Mt (K), for, if E aMt (K) and = (v ÷ eM (K) then we can > choose a maximal ii. Then

so that

=

and hence

= p. Similarly,

is maximal.

CHOQUIT SIMPLEXES

71

7.3. THEOREM (CROQUET—MEYER). Let K be a compact convex set. The following are equivalent: (1) K is a simplex; (ii) for each continuous concave function f on K, f is affine; (iii) for each x E K there is a unique element = x. (K) with

Proof. We identify K as the state space of A(K) so that (i) implies the cone p* spanned by K isa lattice and in particular has the DP. Givenf continuous and concave we can assume, by adding an appropriate multiple of 1, that f 0. If we consider I as a homogeneous super-additive function on (see the discussion preceding Theorem 4.3) then by Theorems 4.2(iii), 4.3 and 4.4(u), (in conjunction with Proposition 3.2) we see that f is additive on Pd', hence affine on K. If (ii) holds then, by Theorem I .6.l(ix), for —f€ Q(K)

{a EA(K): a

is directed. Hence, using Theorem 1.6. l(viii), for any

(K) with

x

a €A(K) and a
vE

(K) and

= x = qv then, by Theorem 1.6.5,

Thus p. = p on the linear sub-lattice Q(K) —0(K) of (R(K). But the Stone—Weierstrass Theorem (6.9) implies this subspace is dense so that p. = v, completing (ii) implies (iii). Jf (iii) holds and aM(K) denotes the the fact that (K) in subcone spanned by (K) is a face of But (K) shows aM (K) is a face of the (lattice ordered) cone in a one-to-one map of onto then in A(K)*. It follows from this that is lattice ordered, establishing (i). The comments at the beginning of this section indicate that the order unit spaces A(K) with K a simplex come very close to being abstract M-spaces with order unit and hence, CR(X) spaces. If A(K) is a lattice then X ext K We next show that and A(K) = (as in Theorem 6.10) is the converse is true. 7.4. PRoposii'iorsi. Let K be compact convex and let X = ext K be a closed (K) is maximal if and only if p.(X)= 1. subset of K. Then p. E

I

Proof. If p.(X) = 1 then for 1€ CR(K) we have, by Theorem 1.6.3 that I = on X SO that p.(f) = p.(f), implying by Theorem 1.6.5, that p.is maximal. If p.

is maximal then Theorem 1.6.8 shows p.(Fo) =0 for all compact G8 sets

DUALITY IN ORDERED BANACH SPACES

72

disjoint from X. If F is compact and F X = 0 then there is an f CR(K) Then Fo={x:f(x)= 1}isacompact I 1

with G6 with

FcF0cK\X.

Thus, the regularity of

shows

=0.

simplex with X = ext K. Then A(K) is Let K be a a lattice if and only if X is compact. In this case A(K) is lattice isometric to 7.5. THEOREM. CR(X).

with an order unit If A(K) is a lattice then it is an abstract and state space K. But then Theorem 6.10 shows X is compact. Conversely,

Proof.

let X be compact and let 0: A(K)-*

be the restriction map. Then

Proposition 7.4 shows (K) if and only if belongs to the state space of Thus the probability measures, Mt (X), on X as a subset of so (K) are precisely aMt (K). Thus (I, 0*,L) =SKf =(f, It follows that that 0* is one-to-one on Mt (X) and hence on 0(A(K)) is dense and therefore (since 0 is an isonietry) onto CR(X). Since 0 is order-preserving, one-to-one and onto it is a lattice-isometry.

Simplexes with the set of extreme points closed are called Bauer simplexes

and Theorem 7.5 shows they are precisely the state spaces (probability measures) on a compact Hausdorif space.

We will take up the theory of Choquet simplexes in greater detail in Chapter 3. We conclude here with a result that allows for a significant strengthening of the RIP in A(K). Let Khe a Qioquet simplex and let —f, g be convex u.s.c. functions on K with g f. Then there is 7.6. THEOREM (EDWARDS' SEPARATION THEOREM).

ana€A(K) with Assume first that strict inequality holds between g and f so that there is an >0 with g
{b€Q(K): b>g} is directed

down with infimum equal to g. Hence by compactness and the fact that f is l.s.c. we can choose hg 0(K) with

g
1,.. ., n}.

73

CHOQUET SIMPLEXES

Similarly we can find

A(K): j = 1,..

= inf {c1

.

,

n} with

g
Then the RIP in A(K) shows there is an a A(K) with

g

g—1/2" and a,1 —

For, if a1,.. .

,

1/2"

+

are chosen then gv

+


—

so that a,,÷1 can be chosen as required. Thus a = urn polates g and f.

and inter-

7.7. COROLLARY. LetFbe a closed face of the Choquet simplex K with —I, g convex u.s.c. functions on K with

gf. If a €A(F) and

glpa then there is a b A(K) such that

blp=a and Proof.

Let r>s be such that

K, x F}

r sup {g(y), a(x): y and

sinf{f(y),a(x): y€K,x€F}. Extend a to a,, a5 on K by taking a,. r on K\F and a5 s on K\F. Then —a,, a5 are convex and u.s.c. on K and hence so are (—ar) v (—f) = —(a,. A f)

and

Since a5 v g ar A

f

Theorem 7.6 yields a, A f.

a5 v g.

DUAI.ITY iN ORDERED BANACI-f SPACES

74

But (a5 v

=a

so that = a. Notes

Theorem 7.1 (the direction (iii) implies (I)) was established by work of Lindenstrauss [153), Semadeni [190] and Edwards [88] in 1964—65, although there seems to have been a lapse of some time before the relevance

of the 1962 paper of Andô [18] establishing the DP in the pre-dual of a lattice was recognized. The fact that the faces of a Choquet simplex are split was observed by Alfsen [2] in 1965. The terminology was introduced by

Alfsen and Andersen [6]. Theorem 7.2 is due to Ellis [108]. In this connection it is interesting to note that the property of each extreme point being split does not characterize infinite-dimensional simplexes. Indeed it is shown in Asimow [23] that the state space of a function algebra always has this property; however the uniqueness of maximal representing measures need not hold. Theorem 7.3 is found in Choquet and Meyer [63]. Theorem 7.5 is due to Bauer [34]. Theorem 7.6 and its corollary are due to Edwards [88].

8. DECOMPOSITION OF DUAL CONES

In this section and the next we will initiate a general study of ordergeometric properties which allow for various types of extension theorems. For example, if F is a closed face of a simplex K then the decomposition of K

by F and

together with the Edwards Separation Theorem, allows for extensions of continuous affine functions on F to elements of A(K) satisfying certain prescribed conditions. Typical questions concerning a compact convex set K and a closed convex subset F are the following: 1. When does each element of A(F) extend to an element of A(K)?

2. When can we find a e A(K) such that a is non-negative and F = 3.

If a e A(K) is non-negative on F can we find a non-negative b E A(K) such that b = a on F?

The first question is the key to various interpolation and approximation results in function spaces, dealt with in Chapter 4 and the second question relates to the problem of identifying peak sets for function spaces. The third question, concerning the positive extension property, turns out to be relevant to the problem of finding norm-preserving extensions. We return to these questions in detail in Chapter 4.

DECOMPOSILION OF

CONES

75

We first formulate these matters in terms of homogeneous functions on dual cones and then, in Section 10 apply the general results to A(K) spaces. We shall assume throughout that (E, F) is an ordered Banach space that is

a..normal and a-generated for some constants a and j3. Let Q* denote a subcone of We note that, by the Krein-SmuJyan Theorem, is if and only if is We let N denote the linear span of Q* in E*. Thus (Q* — Q*) N= — cI

Let Q={a€E: (a,x)O for allxEQ*}. Thus Q = _(Q*)O

and is a closed cone in E containing P. If M =0 —Q then M = (a e E: (a, x) =0 for all x E 0*1

(sometimes denoted (Q*)j and clearly

M=N°. Our point of view is to consider elements of E as functions, A0(P*) positive homogeneous and additive) on In fact we can represent (E, F) as the space the subspace of A(Pr) of functions vanishing at 0. We have from Corollary 1.4.8 that the restriction of E to A(Pfl is dense, and, since E* is positively generated, we have for some a, Ia

aIIa

allaH.

Thus the restriction map is onto and the restriction norm is equivalent to the

original. In similar fashion (as in Theorem 1.4.7) we can identify the linear space 0* Q* with unit ball co u as the dual space of A0(Qfl, with dual cone Q* The analogue to the first question above concerns the restriction map of F to A0(Qr). The answer is provided by Theorem 1.3.5, since E is dense in A0(Q?). Thus let

0 :E-+Ao(Qfl be the restriction map and note that 0" is the inclusion map taking Q* — Q* with unit ball co u —Qi') to itself with unit ball (Q* - Q*) The following are equivalent: (i) The restriction map 0 is onto,

8.1. THEOREM.

(ii) there is a number r such that for each s > r and a0 E A0(Q?) with 1 there is an a EE with Oa = a0 and llall<s, (iii)

Q*.Q* is w*_closed,

DUALiTY IN ORDERED I3ANACH SPACES

76

(iv) Q* — Q* is Il Il-closed, (v) (Q* — Q*) C r Co u —Qr).

To proceed we must be more precise about the relation between Q* and = M = (Q*) We say the cone Q* is self-determined if N Q* (N the Q*)• Thus Q* is self-determined if and only if it span of subspace with is the intersection of a By polarizing we can

characterize this property in E. 8.2. PROPOSITION.

subcone Q* of

The

and only if Given Q* C

is self-determined if

Q=(P+M). and a closed subspace I in Ewe say Q* is a section of

byl

if (i)

(ii) N = w*_cl (Q* — Q*)

JO

Property (i) is sometimes written as Q* = J1 so that (ii) says (by polarizing)

J=

Let J be a closed suhspace of E and let Q* JO Then Q* is a section of P* by Jif and only if 8.3. PROPOSITION.

Proof.

Q* is a section by J if and only if (ii) holds, which means JO

By polarizing, this in turn is equivalent to Such subspaces are automatically order ideals (c a b with c, b E J implies a I) since they are of the form (Q*)l• They are sometimes called almost Archimedean (the reason for the terminology will become apparent shortly) and characterize the subspaces I = there is an a e J such that J then for each x E (a, x) 0. If Q* is a subset of such that for each x E there is an (Q*) with (a, x)> 0, we say Q* is semi-exposed. This implies that Q* is a self-determined face of

aP

8.4. PRoPosITIoN.

The set Q* is semi-exposed in

if and only if

Q

where Q={aEE:(a,x)OforallxeQ*}andM=Qn_Q.

J)ECOMPOSTTION OF DUAl. CONES

Proof.

77

Q* is semi-exposed if and only if = {x E p*:

(a, x) =0 for all a M P}

=(M By taking polars, 0* is semi-exposed if and only if

1ff is a closed subspace of E then J is an ordered Banach space with positive

cone Jr P. Thus we say the subspace J of E is positively generated if 8.5. then

if Q* is a section of J3* by J and Jis positively generated is semi-exposed.

and J = M so that

Proof. We have Q* JO

8.6. PRoposiTioN. If Q* is semi-exposed in and X is a subset of p*\Q* then there is an a E with a 0 on Q*, a 0 on p* anda >0 on X.

For each x X there is an a neighbourhood of x in X such that >0 on Then, by compactness, we can choose x1,.. . , x,, + such that a = is the required functional. Proof.

8.7. COROLLARY. a E E with

If Q* is a semi-exposed

a0 on Proof.

Let Q* =

U,, open in F", and let X,, = Pr\U,,. Then

p*\Q*.(Jx and if a,, is chosen in (M

P)1 with a,, >0 on X,, then

a= n-=1

is

then there is an

Q*{xEp*:(a,x)0}.

and U,,,

in

the required function in (M P)1.

L)(JAIITY IN ORDERED EiANACH SPACES

78

Such a Q* is called exposed and Corollary 8.7 shows that Q* is exposed if is semi-exposed if and and only if Q* is a semi-exposed G8. Moreover, only if Q* is the intersection of exposed sets of P*. Next, let Q* be a section by f in and consider the property in question (3). We say Q* has the positive extension property (PEP) if for each a F with a C) on Q* there is a b E F with

b()

on

and h=a on

in terms of Q and J we can characterize this quite easily.

8.8. PRoPosrrloN. If only if

isa section by fin

then

Q* has the PEP if and

Q=f+P. Combining this with Proposition 8.3 and Corollary 8.5 we have the following.

Let J be a closed subspace of F. Then J gives rise to a with the PEP if and only if (i) J + P is closed in E,

8.9. COROLLARY.

section Q* JO (ii)

If in addition (iii)

then Q* is a semi-exposed face of P* with the PEP. If q : F Elf is the quotient map then the adjoint q* (E/J)* E* (using Theorem 1.1.2(v)) identifies (E/J)* with .1° and is an isometry (where J° has unit ball J?). Moreover, properties (i) and (ii) in Corollary 8.9 are equivalent

to the cone qP being closed and proper in Elf. Thus, the dual cone Q* —(qP)° = (q*) l(p*) = p*

that (E/J)* is order-isometric to (J°, Q*), In particular, the quotient space (E/J, qP) is Archimedean if (i) and (ii) hold since (ii) implies qP is proper and rq(a)q(h) if and only if b—ra €P+f. so

Thus rq(a)q(b) for all r

0 if and only if

b/r—aeP+J forall

r0.

Then, letting we get —a P+J so that qa 0. A closed subspace J of F satisfying (i), (ii) and (iii) is called an Arc himedean order ideal, if in addition (iv)

DECOMPOSITION OF DUAl. CONES

79

(so that the statements of 8.1 also hold) then I is called strongly Archimedean.

Let J be a closed subspace of E and let Q* The following are equivalent:

8.10. COROLLARY,

Jfl

(i) I is Archimedean; (ii) Q* is a section of.P* by I such that for each a E Q there is an m E Jsuch

that

a +rn a

E J.

a is a E Since I is positively generated, choose m2 El

with m2a—b,0. Then Conversely, (ii) clearly implies I is positively generated with the PEP. Verifying these conditions can, in practice, be quite difficult. What we seek next are sufficient conditions of a geometric nature (involving Q* in p*) for various of these properties to hold.

To motivate the next definition we consider the case where Q* is a complementary face of with projection ir. Then ir is a bounded (by normality and positive generation) projection of E* onto N = Q* — Q*. Let denote the complementary projection with range L. Then both N and L are norm-closed subspaces. Let q:

E*/N (quotient map)

and

withp oq=o., Then p is an isomorphism onto L with Ip(qx)II = IkrxII = Ikr(x

so that p is a Banach space isomorphism. The point here is that the induced quotient norm on L is equivalent to the restriction norm. The quotient norm, IIqzII, is the distance d(N, Er )(z) (Chapter 1, Section 3) of z from N, and if Z EL, llqz liz Ii iIoli IIqz Ii.

In particular if x E

then

xy+z;

YEQ*, ZEP*

with =

DUALITY IN ORDERED BANACH SPACES

$0

Thus x can be decomposed into an element of Q* and an element of F" far (relative to lixil) from N.

Now let Q* be a section of P* by .1, so that N = J°. We can measure the distance from N in E* by taking any norm-closed convex neighbourhood W of 0 in E* and letting

d(N, W)(x) = p(N + W)(x) = inf {r 0: x EN + rW}.

For convenience we let the N be understood and denote this by dw(x). We define Q* to be a decomposable section of P* by J if there is a positive homogeneous map ir : F"' Q* satisfying (1) ir 1 (the identity map) on Q*, J)*, (ii) for each X E E (1 (iii) 11(1 — ir)xjj for some (norm) neighbourhood W of 0 in E*. Thus each x P"' can be decomposed, as in the case where Q* is complemented, but the set of elements z = x — irx need not, in general, be convex. Indeed Q* may be "surrounded" by (1— iT)P*, and in particular, need not be a face of 8.11. THEOREM. Let Q* be a decomposable section of F"' by J. Then is ,v*_closed in E*, (i)

(ii) P +J is norm-closed in E.

Proof. We can assume that the neighbourhood W in (iii) is

r>0, so that x

dw(x)r (i) If

for some

implies

if and only if

x €J°+Er.

+J°)1 then

u=y+z+n;

yEQ*,

n€f°,

ZEP*

and IIzII

(y + z)

r.

Thus

(p* +J°)1 =

which is

Hence, by the

(ii) The decomposability is equivalent to

Theorem, F" +J° is

DECOMPOSrJ ION OF DUAL CONES

81

which yields

in E. We now apply the Gauge Lemma (1.3.2) with

B+'J,,

A+*P,

If a E Q then J(a)- = ?(P, Jr) and

so that d(P, Jr)(a) = d(a)< co. Hence a E P+J, for all s > r. This shows

(l+P) = Q =J+P. 8.i2. COROLLARY.

If 0" is a decomposable section of

by land

q

then

in E*/J° and

is

(E*/J°, qP*)

(I, I

(order-isometry).

From Theorem 8.11(i) we have qP* is in E*/J° with the usual identification of E*/J° and J* (q is the adjoint of the inclusion map i : Ic E). Moreover, Proof.

qP* = (qp*)OO = (i '(—P))° = (I

If (I, I P) is positively generated then (Corollary 8.5) Q* is semi-exposed. This leads us to the notion of positive decomposability: we say 0* is a positively decomposable section of P* by I if there is an h E** with h on 1° and the set W in (iii) can be taken as

0

W={zEE*: h(z)1}. Then

dw(x)=inf{r0:xEJ°+rW}=max{h(x),O}. Thus on

we have X

= y + z;

y E Q*

and

IlzII h(x).

8.13. THEOREM. Let Q* be a positively decomposable section of by I. is an approximately a-directed Banach space (a = lihil). Then (I, I Proof.

From Corollary 8.12 and Theorem 3.5 it suffices to show the

quotient norm on Let IIqx1II

is a-additive. Let (xj)7..i c

be given with x =

z = x — x. Then IIz.II

= h (x) = h (x + n)

(any n E

J0)•

x1.

I)UALLTY IN ORDERED BANACH SPACES

82

Thus JJhIIJJqxII.

8.14. COROLLARY. If Q* is a positively decomposable section of P* by J then Jis an Arc himedean order ideal and Q* is a semi-exposed face of with

the PEP. 8.15. COROLLARY.

of E with

Let (E, F) have the DP and let J be a closed subspace The following are equivalent:

(i) Q* is a face of and a section off; (ii) J° is a positively generated order ideal in E*; (iii) J is a strongly Archimedean order ideal in E; (iv) f is a positively generated order ideal in E. Proof.

Since (E, P) has the DP Q* is a face of

if and only if Q* is

complemented. In this case Q* — Q* is and Q* is a positively decomposable section of Thus, since Q* is a section by J, (I) implies that Jo =0* — Q* and, since Q* is a face, that J° is an order ideal. If (ii) holds it follows that 0* is a facial section by J so that Corollary 8.14 shows J is an Archimedean (hence strongly Archimedean) order ideal in E. Then (iii) implies (iv) by definition. If (iv) holds then we observe first that 0* is a face since 0* {x P*: (a, x) =0 for all a J} = {x P*: (a, x) =0 for all a J P} and hence is an intersection of faces. To conclude (i) then, we must show

or equivalently =

0* _Q*

Clearly Q* Q* cJ0. If x e J° and a f P

since Q* Q* is then, using Theorem 5.5,

(a, x vO)=sup{(b, x): Ob since J is an order ideal. Since J is positively generated, we have x

Hence J°= Q*...Q* Notes Theorem 8.1, in the context of A(K) spaces, was shown by Edwards [87] and Alfsen [3]. In the latter the number r is related to the bound on the norm of

the extension function. This is discussed in greater detail in Chapter 4 in

sIABILrFY OF SPLIT SETS

83

connection with interpolation in complex function spaces. The notion of seif-determinacy is introduced in Alfsen [1]. An example of a non-selfdetermining face is given by Asimow [25]. Proposition 8.3 (the A(K) case) is found in Ellis [101]. Other related results characterizing order ideals whose annihilators determine faces of the dual cone are found in Jameson [134], Ellis [100], Bonsall [44] and Asimow [26]. In the years 1968—70 much work on A(K) spaces centred on the various

extension properties of functions in A(F), F a face of K. The two main properties are incorporated in the condition of Corollary 8.10(11). These are the positive extension property (PEP) and the directedness in A(K) of the

null functions on F. As Corollary 8.10 shows these are tantamount to the conditions of Corollary 8.9 and hence the usefulness of designating the

Archimedean ideals. This terminology and the applicability to A(K) is found in Størmer [206] and, in the case of strongly Archimedean, Alfsen [3].

Geometric conditions guaranteeing these extension properties were first noticed for simplexes where each closed face is split (Alfsen [3]) and also discussed in Effros [93] and Lazar [147]. Asimow [23] gives geometric conditions on Kand F (in terms of gauges) that assure the directedness of .L

In Asimow [25] the more restrictive but geometrically more appealing notion of positive decomposability is introduced for compact convex sets as a sufficient condition for the Archimedean extension properties of Corollary 8.10.

9. STABILITY OF SPLIT SETS AND THE CHARACTERIZATION OF COMPLEMENTED SUBCONES Let E be a Banach space with bounded complementary projections ir and whose ranges are the closed complementary subspaces N and L. Let S be a closed convex set containing zero with Minkowski functional p = Ps. We say S is split with respect to ir if

p=p C The next two results will clarify the geometric content of this definition. 9.1. PROPOSITION.

Let A and B be closed convex subsets of N and L

respectively, each containing zero. Let

S=cö(A, B). Then

(i) S = Co (A, B) + PA + PR where

PA={aEN:pA(a)=O} and Pfl={aEL:pB(a)=0}, (ii) Ps = PA ° IT +

0

84

DUALITY IN ORDERED BANACH SPACES

Proof.

Let c,,

c

with

c,, = A,,a,, +

a,, E A,

E B and

By passing to a subsequence we can assume A,,

A

A,, +

1.

and p.,, -> p.. Now

-ftc = urn irc,, = urn A,,a,,

so that PA °

ir(c) slim inf pA(A,,a,,)

Hence 'ftc E AA and similarly

A.

E p.B. Therefore

c = irc +crc

where we mean 1 for 0/0. Then both (i) and (ii) follow, since A + p. 1.

9.2. PRoposiTioN.

The following are equivalent: (i) S is split with respect to IT,

(ii) (iii)

Proof.

if ps=psoir+psoo.then Ps = PA ° IT + PB ° U,

where A = S N, B = S L so, using Proposition 9.1, (ii) holds. If (ii) holds then Proposition 9.1(i) shows -irS c S N, oS c S L. Thus equality holds and (iii) follows. Clearly (iii) implies (ii) and hence, by Proposition 9.1 (ii), (1).

9.3. COROLLARY. IfS and Tare split with respect to IT then so is

R=

(S, T)

and

,rR =

(ITS, ITT).

We now consider a Banach space E with a closed subspace M for which the

polar N in E* is the range of a bounded projection ir. We let L and o• denoted the complementary subspace and projection. We take A, B as closed convex subsets containing zero in E such that both and B° are split. The next result shows the preservation of the "splitness" of polar sets

under the formation of intersections in E. We formulate this wi iout reference to order properties in E for application in Chapter 4, although our immediate goal is to determine the order properties of subspaces in (E, P) giving rise to complemented subcones of Pa'. Let A, B be closed convex sets containing zero in are split with respect to IT, whose range, the Banach space E, such thatA°, N, is the polar of the closed subspace M c B. Let C = A n B. Then 9.4. THEOREM (ANDO).

STABILITY OF SPLIT SETS

85

(i) A + M and B + Mare closed in E, (A°, B°) in E* then C° is split and (A + M) r's (ii) if C° =11.

(B+M)=C+M,

(iii) in general, if flcrfl 1, where o = 1—ir, then (A + M)

(B + M) C A

(B+E5)+Mforalls>O. Proof. We use the Gauge Lemma to show A +M is closed, with A *-+A, D -*(A + M). We need to show B the notation of I Section 3)

(A

kd(A, E1)(x)<

d(A,

and hence x E A +M. It suffices to show the polar of (*) holds. Thus consider the support functional p =

so that d(A, M1)(x) = d(A, M1)(x)< p(A +Mkr), where k > lull. Then p

Now, for any n EN, iio'xil =

liux +oizll = ilu(x + n)ii

kilx + nil.

Hence

II.1100.kp(N+Efl. Since A° is split we then have

pp(A°)o ir+p(A°)ou+rII.Ilocr = p(A°) ° ir + (p(A°) + ni' Ii) ° which says, by Proposition 9.1,

(A+Er)°),

(A+Micr)°

from which (*) follows by taking polars. Hence A + M is closed. If

C° =fl'

(A°, B°)

then, by Corollary 9.3, C° is split and so that

(C+M)=(C+M)=(A and hence (ii) is shown. Assume now that the complementary projection o' with range L in E* has 1. Let B' = B + E5 and CS = A B5. We shall apply the Gauge Lemma IIcrII

DUALITY IN ORDERED BANACH SPACES

with

As above, we require (**)

(A

To prove (**) we first evaluate the support functional, p(C5 +M,), of Cs+M,. Now p(Mr) =

o

(since 11o11 1).

Next p(C5) =

since (B')° is

(A°,

=

(Theorem 1.1.5). Thus,

=p(B)+slIi1p(B) ° ir+(p(B)+sII'II)oo since B° is split and 11o11

1. Hence

so that

=ëö

c

(D°, (C')° L),

where Hence

p(C5)p(D)°

o

and

p(CS +M,) = p(C5)+p(Mr) p(D) °ir +p(C5 + rh'

II) °

Therefore, (CS +

c

(D°, (C5 +

so that (**) holds, and hence (iii) is shown. complemented subcone of P* under the Assume now that Q* is a projection ir with N = Q* — Q*• Since N is the range of a projection N is Combining this with closed and hence, using Theorem 8.1, N is Corollary 8.9, we have that in particular 0* is a section of P* by the strongly Archimedean ideal J = N° in E. If Q = (_Q*)O then J =0 —Q, and

Q=J+P.

87

STABILITY OF SPLIT SETS

We have (E/J, qP) is order isomorphic to A0(Q*) and

a0b0 inAo(Q*) if and only if there are a, b e E with

P +J = Q, in Ao(Q*) if and only if a —b €J. Thus we say a b(Q) if qa = a0,

and a0 = b0

and

qb = b0

b— a

qa qb, or equivalently aIQabIQ*. We write a b + s if there is a z E Ee with a b + z.

Since E is positively generated we can, by renorming E, assume E* is i-normal, so that the norm in E* is monotonic on This guarantees that are less than or equal to I so the hypothesis of Theorem 9.4(iii) is flirfl and satisfied.

Let Q* be a let a, bj(i = 1,. . ., n ; j = 1,.. . (i) all 9.5. THEOREM.

,.

complemented subcone of m) c, c0 be elements of E such that

and

(ii)

Then for each g >0 there exists m e J such that

(In short, if the b's and a's can be interpolated in E and c0 interpolates the b's then c0 has an extension in E = A o(P1c ) that interpolates in E.) and a's on Proof.

By translating we can assume c =0 in E. let

A={c€E: Then the support functional p(A,) just equals a1 on

(since

0 a,) and

clearly

OnP*

so that

is split. If

then the support functional p(A) equals

inf{a1,. .. ,a,,} which we shall denote as p. But the definition of j5 (Section 4) shows that fl is

also the support functional of

{cEE**:cai,.

.

in(E**,P**)}

DUALITY IN ORDERED BANACH SPACES

88

and hence the Minkowski functional of A2).

This shows the w*..and fi. Il-closed convex hulls of A°1,. . ., Hence

coincide.

.

Thus

c0a1,. ..,a,,(Q) if and only if c0 "extends" to

c+ma1,. .

mEJ.

.,a,,,(P);

Similarly

bi,...,bpnco(Q) if and only if for some m'€J

b1,.. .,bmco+m'. Thus hypothesis (ii) says

c0E(A+M)tTh(B+M) which, by Theorem 9.4(iii) gives Co E

A (B

+M.

We now present properties in the ordered Banach space (E, F) that characterize the complemented subcones of P*. Let Q be a closed convex cone containing Pin E and let M = Q —Q be the annihilator of the

dual subcone Q* of P*. As above we write a b(Q) for the ordering induced by Q. In the next theorem, part (ii) characterizes complemented subcones in terms of the polar cones in E and part (iii) characterizes the annihilators of complemented subcones. The following are equivalent: complemented subcone of (ii) if a, b1, b2 are given in E with

9.6. THEOREM.

(i) Q* is a

Ob1, b2 then for

each e >0 there is an in E Msuch that

—e a+m

b2,

STABILITY OF SPLIT SETS

89

(iii) (a) if a1, a2 EM, b E E with a1, a2 b then for each 8>0 there is an

a €Msuch that

a1, a2a (b) if a EM, b1, b2 E P with a b1 + b2 then for each e>0 there exist

ai, a2EMsuch that

a=a1+a2 and a,b,+e

(i=1,2).

If J is a closed subspace of E satisfying (iii)(a), (b) then .1° Proof.

(i) implies (ii) is immediate from 9.5. To show (iii)(a) we have

0b—a1, so

is a

with annihilatorf.

complemented subcone of

b—a2

b—a2(Q)

and

that (i) gives a EM with b—a2

or —b—s <—a <—a1, —a2

which, multiplying by —1, gives (iii)(a). For (b) we have

0b1+b2—a and so that there is an

EM such that

+a1b1+b2—a. Thus —e—b2+a

so that, if a2 = a — a1 then a = a1 + a2 with

a1b1

and

This shows (iii) holds if M is the annihilator of Q*• Next let J be any subspace for which (iii) holds and let J1 =

We show

First, if x E p*\Ji then there is an I with (a1,x)>0. Then (iii)(a) shows there is an a €1 with a a1, 0, so that a E P land (a, x)>0. This shows is a semi-exposed face. We show next that I = Thus let

J=

and e>0 be given. Then

a=m1+p1+z1=m2—p2+z2;

mel,

p,€P and

IIziII<e.

DUALITY IN ORDERED BANACFI SPACES

90

Hence

m=rn2—m1=p1+p2+w,

flwll2e.

Since E is positively generated, for some constant a, we have

and

llpllae.

Hence

(11,2).

a1EJ and

rn=a1+a2; Now

rn—a1 —a2_
so that —a1 _<—p1—w+2ae.

Thus

pi+w—2ae and the normality of E gives lai

— Pill

13€

for some constant f3.

Therefore a

for some constant

depending only on a and f3, so that

Thus we have that J is. an Archimedean order ideal and we conclude by constructing the projection of P* onto 1'. Given a €P, x EP* let

p(a,x)=limsup{(m,x): m e-*O where

for fixed x,

Then 0 p(a, x) (a, x) and we show first that p(., x) is positive homogeneous and additive on F: if

= (a P)

= 1, 2) and s >0 then we have rn, (m,,x)>A,. Thus

ml+rn2EAe(al+a2) and (m1 + rn2, x) >A1 +A2.

This shows

px(ai +a2)px(ai)+px(a2).

a, + €/2, (1 = 1,2) and

APPLICATION TO A(K)

If

<

91

=1,2) then choose e >0 such that (m,, x) A,

for

all m, J Ae(a,)

(1= 1, 2).

For an appropriate constant a >0, m

implies, by (iii)(b),

(1=1,2).

m=ml+m2, Hence (m, x) A1 +A2. This shows

px(ai The fact that Px(ra) = rpx(a)(r 0) is straightforward. Next, if a is fixed then Pa = p(a,') is positively homogeneous and subadditive from the definition of p. The super-additivity follows from the property (iii)(a) in similar fashion to the above arguments. Thus we have that Px extends to an element of E* such that for each x 0 Px

X

and x E J implies jx =0. Conversely, if Lox =0 then (m, x) =0 for all m E .1 so that x 1° P's' = Q*• The additivity of each Pa shows the operator x Lox is additive on

and

so that id—p = ir, the complementing projection. Notes

The general definition of split sets and (the stability) Theorem 9.4 are due to Andô [191. Theorem 9.5 is a generalized version of the extension theorem for a split face of a compact convex set found in Alfsen and Andersen [6] and Asimow [25]. Various characterizations of the annihilators of complemen-

ted subcones have been given. Alfsen and Andersen [6] describe them in terms of "near-lattice ideals" in the case of split faces of A(K) and Perdrizet [167] gives a general condition which he calls hypostrict.

10. APPLICATION TO A(K) We now focus on the case where (E, F) is an order unit space with order unit u. Then £ = A(K) where K is the state space of E and so K is a w base for the dual cone F*. This allows us to describe properties of by restricting to K (and vice versa). Thus F is a closed face of K if and only if the

DUALITY IN ORDERED BANACH SPACES

92

(w*_closed) cone A

aO

and so forth. In particular, F is a section of K by the closed a face of subspace .1 c A(K) providing is

K and I = F1 = N°

F=

linear span of F given by

where N is the

N=

(Q* — Q*),

and Q* = U AF, AO

is

the linear span of F. We first identify the Archimedean ideals in A(K).

10.1. THEOREM. Let I be a closed subspace of A(K) and let F = 1° K. The following are equivalent:

(i) A(K)/J is Archimedean ordered with proper cone P = qP (q the quotient map), (ii) (A(K)/J, uA) is anormed order unit space, where a = q 1, (iii) I +P is closed and I (I +P) (I — P), (iv) F is a section of K by I with the PEP.

If (1) holds then, since a E A(K) implies ±a A I for some A >0, Ai2 and hence a is an Archimedean order unit for A(K)/J (as defined in Section 2). Thus (A(K)/J, z2) is a normed order unit space with unit ball Proof.

1={a:±aAa} and positive cone P = qP, by Theorem 2.5. Hence we have qE1

I

and so a + qE1

a + Ic P

so that mt P (quotient norm) is non-empty. Since P is linearly closed with non-empty interior, P is closed in the quotient norm, and hence I +P is closed. Then

I = (I +P) (I —P) of follows as well, since P is proper. This shows (iii). Now, the (iii) and (iv) follows from Corollary 8.9. If (iii) holds and b for all r 0 then

forallrO.

APPLICATION TO A(K)

93

Thus ra e 1+ J — P for all r 0

so that

aE(J—P)=J—P and hence

E —P. Then

shows P is proper so (i) holds.

This allows us to characterize the Archimedean and strongly Archimedean order ideals of A(K), using Corollary 8.10. K. 10.2. COROLLARY. Let J be a closed subspace of A(K) with F = (1) The following are equivalent: (i) A(K)/J is Archimedean ordered with proper cone P and .Tis positively generated, (ii) Fis a section ofKbyJsatisfying: foreach a E A(K), a 0 on F there is a b E A(K) such that

bIF=aIF and (iii) J is an Archimedean order ideal. (2) The following are equivalent: (i) .1 is a strongly Archimedean order ideal, (ii) property 10.2(1)(ii) is satisfied and a E A(F) implies there is b E A(K) with

We say F is a decomposable section of K by J if the cone Q* spanned by F is

a decomposable section of P* by J. The notion of positive decomposability is defined similarly.

10.3. THEOREM. LetFbe a section of Kby J. (1) The following are equivalent: (i) F is a decomposable section of K, (ii) there is a 8>0 such that each x e K can be written as a convex bination of y and z where

y F and

z e K with liz — J°II

8.

(2) The following are equivalent: (i) F is a positively decomposable section of K, (ii) there is an h E A(K)** such that h 0 on J° and each x K can be written as a convex combination of y and z where

y F and z K with h(z)> 1.

DUALiTY TN ORDERED I3ANACH SPACES

94

If (1)(i) holds then for some r y E Q* x = y ÷ z;

Proof.

1 each x e and

lizil

can be written nix —J°fl.

Then, using the additivity of the norm, (ii) follows easily (with ô = l/r). The

converse is also straightforward, as is the proof of (2), using the remark preceding Theorem 8.13. Let F be a positively decomposable section of K by J.

10.4. COROLLARY.

Then J is an Archimedean order ideal. Thus the extension property of Corollary 10.2(1)(ii) holds.

We now consider the case where F is a split face of K. Since the norm is additive on P" we can use 4.4(u) to obtain a strengthening of 9.5. 10.5. THEOREM. Let F be a split face of K and let —f, g be continuous convex functions on K.

(i)If and a E A(F) with

then for each s >0 there is a b A(K) such that

biF=a and (ii) If

g0
g[Fa then there is a b A(K) such that

bjp=a and Proof.

Since F is split the restriction map from A(K) to A(F) is onto.

Furthermore, it is readily verified, using Theorem 4.4(u) that the sets

Af={aEA(K):af Ag = {a E A(K): a

polars which split in A(K)* closed. If have

as

on

K}

g on K}

in Theorem 9.5. It follows that Af +J is

Af ={a A(K): a f on F}

APPLICATION TO A(K)

95

then

f(x) 1}

(Af)0 = {x E

and

(Af)O={xEQ*:flQ*(x)1},

Q* U AF. A

Since

Q* is a face of

>0

x E Q* implies

f(x) = (flo*Y SO

that = (Af)°.

(A1)°

By taking polars of this we have A1 + J = A1.

Thus (i) follows from Theorem 9.4(iii) and (ii) follows from Theorem 9.4(u) since in the case 0
Of course much more can be said if the complementary face of F is also closed. In this case we have a complete duality between the complementing projections and their adjoints in A(K).

10.6. THEOREM. Let K = co (F1 u F2), F closed and convex in K. Let A, = {a E A(K): a IF, 0}. Then the following are equivalent: (i) The F, are complementary split faces of K, (ii)

Proof.

Mt

In general if E = M1 EEM2 with projections ir, onto M,, then E* = with adjoint projections = ir? such that range = (ker irs) =

Mt. Thus if (1) holds and ir, is the corresponding projection in A(K)* then ir, so that u, = ir? is well-defined on A(K). Then (ii) is immediate. Conversely, (ii) yields the dual fact that is

= where N, = (Q7

Q7 the

—

cone spanned by F,. But then

K = Co (F1uF2) so that if x

x=x1+x2; Since

X,EQ7.

07 c N, this representation is unique and the 07 are complemented in A(K)*.

subcones of

It is convenient to have a characterization of split faces in terms of maximal ) then we is maximal (I,.'I = + + measures. If 0 E CR(K)* and

DUALITY IN ORDERED BANACH SPACES

96

say /h is a boundary measure. Let aM(K) denote the set of boundary measures on K. Let q : CR(K)*_, CR(K)*/A(K)J = A(K)*

be the quotient map, as usual, and MT (K) the probability measures on K. 10.7. LEMMA. LetFbe a closed face ofKand let G be a compact subset of K disjoint from F. Then given e(0 < e <1) and k 1, there exists a e A(K) such

that

a0 onK a<e onF ak onG. Proof. The set G is covered by a finite collection (Nj7,.1 of compact convex neighbourhoods in K, each disjoint from F. Then, since F is a face,

so that we can assume, by replacing G with co (U N1), that G is convex. Now consider the set H in A(K)* x R given by

H=co(Gx{k}, Kx{0}). Again, since F is a face, H can be strictly separated from F x {e} (0< e < 1) by a hyperplane in A(K)* x R which is the graph of an a E A(K) with

a0 onK,

a<e onF,

onG.

10.8. PRoposiTioN.

(i) If F is a closed face of K and

E MT(K) with

= x E F then

/.L C F

(ii) If F is a closed face of K and aMr(F).

E

9MT(K) with

F then

E

(iii) 1ff is a bounded 1.s.c. function on K and /h is maximal on K then Proof.

(i) Given G compact in K\F we can apply Lemma 10.7 with k =1

to obtain a A(K) such that

E>.a(x)=Ja so that supp c F. (ii) We have from (i) that supp c F and it suffices to show (by Theorem 1.6.5) that j2(f) for all f€ CR(F). If g E CR(K) and f= glp then

APPLICATION TO A(K)

97

Theorem 1.6. 1(vii) shows (since F is a face) 1(x) =

for each x E F. Hence i.t.th (iii) We have, as in Theorem 1 .6.1(viii), for example,

g f}.

g

Thus given 6>0 there is a g E CR(K) with and Then, using Theorem 1.6.5,

If F is a closed subset of K then p -. pIp is a complementing projection of CR(K)* onto We let A(K)1 denote the measures in A(K) representing 0 in A(K)* and let aA(K)1 denote the boundary annihilating measures. Then P* and Q* denote the cones in A(K) spanned by K and F respectively.

10.9. THEOREM. Let F be a closed face of K and let fOrI.4ECR(K)*. The following are equivalent:

(i) F is

a split face of K,

e

(ii)

(iii)

implies i3M(K)

which makes Q* complemen-

q = q °OFOfl

ted in

Proof. We show (i) implies (iii) first. Let F be split and let u be the complementing projection on A(K)* to A(F)*. (a) If p E then qp E F if and only if p(F) =1 (by Proposition 10.8(i)).

Let F' (ker o') n K be the complementary face. (b) If p oMt(K) then qp F' if and only if p(F) =0. To show (b) let Now, using Theorem 1.6.1(iii), f= 1

epi /= co (K x{1}, F x{0})+{0}x[0, oo). Now / is 1.s.c. and, since F is split, /

is

affine. Hence, using Proposition

DUALITY IN ORDERED BANACH SPACES

98

10.8(iii) and Theorem 1.6.1(ix),

if and only if (cr0

F' if and only if

=0. Thus (a) shows that = (q

0M(K)

for /h

(range (IF)

and (b) shows the same for aM(K) (ker 0p). Hence (iii) holds. That (ii) and (iii) are equivalent is straightforward and (iii) implies (i) since q maps the non-negative boundary measures onto Pd'.

Another geometric property, intermediate to "split face" and "decomposable face", is the notion of a parallelface. The closed convex subset F or K is called a parallelface if there is a convex subset F' of K such that

K = co (F u F') and the coefficients in the convex combination are uniquely determined. A trapezoid in R2 forms the most immediate model. Clearly a split face is parallel, and the fact that the function identically one on F and identically zero on F' has a unique affine extension to K shows that a parallel face is

positively decomposable. In fact, if a = XF then d is the (u.s.c.) affine function just described. There is a measure theoretic characterization of parallel faces similar to the one for split faces. 10.10. THEOREM. LetFbe a closed face of the compact convex setK. Then F is a parallel face if and only if

=0 for all E aA(K). Proof. Let F be a parallel face and let

where we can assume the representing the same x K. We write

are

aA(K)- be given. Then = maximal probability measures

/.4j

where

supp

cF

and

=0.

As in the split case, it is easy to see, using the a and a described above, that =0 if and only if F'. Hence the coefficients A, in the representation of are the same for i = 1,2. Thus g.c1(F) = so that p.(F) =0. Conversely, the measure condition guarantees that = A2 for any = Hence, if we let

F' ={z eK:

0

for some

/L

aMt(K) with

= z}

THE RIESZ NORM

99

then ,h(F) =0 for all

E aMt(K) representing z. It follows that

K=co(FuF') and the coefficients of the convex combinations are uniquely determined for each x E K. Notes

Theorem 10.5 with the stronger extension property (than that of Theorem 9.5) is due to Andersen [17]. The existence of an order unit in the pre-dual is essential and he uses a 1/2" argument which has been incorporated into

Theorem 9.4(iii) in our present proof. The measure theoretic terization of split faces is due to Alfsen and Andersen [6] and is also discussed for complex function spaces in Alfsen and Hirsberg [9]. This will

be dealt with in detail in Chapter 4. The idea of parallel faces appears in Hirsberg [125] and has been used by Briem [54] to give a characterization of simplexes analogous to that of Theorem 7.2.

11. THE RIESZ NORM Another way of employing the basic properties of normality and positive generation in ordered Banach spaces is through the notion of a Riesz norm.

An ordered Banach space is said to have a Riesz norm if the following two properties are satisfied: Definition.

(a) if —x (b)

y

then

Ilxli;

if IylI <1 then there exists x such that lix 11<1 and —x
It is easy to check that these properties imply 2-normality and approximately 2-generation. Our aim is to demonstrate that properties (a) and (b) are dual and that as a consequence (E, F) has a Riesz norm if and only if (E*, does. Let S be a convex subset of E(E*) containing the origin. Definition. We say S is (a)' absolutely order convex if x E S and —x y x implies y E 5, (b)' absolutely dominated if y E S implies there exists x e S with —x

yx.

Thus (E, P) has a Riesz norm if and only if E1 satisfies (a)' and mtE1

satisfies (b)'.

As previously we give set theoretic versions of these properties and position our polar calculus artillery. Let E = E XE and P = P x P.

DUALITY IN ORDERED BANACH SPACES

100

Define

SD={(x,x)EE: xES} = {(x, —x)

x E 5)

SA={(x,y)EE: x+yES} SB ={(x,y)eE: X'-y€S}. Note that (S)D = (SD),

= (SA), etc.

11.1. PROPOSITION.

(1) The following are equivalent: (i) S is absolutely dominated; (ii) SGCSD—P; (iii) SB c SA P+ ED. (2) The following are equivalent: (i) S is absolutely order convex; (ii) (SD fi) EG c

(iii) SA'IPCSB. These arc routine verifications. We simply note that, for example, x for some x S which is equivalent to —x
(1)(ii) says for y E 5, ±y

(x, y)=(w+x—y, w—x+y)/2+(x+y--w,x+y—w)/2 gives (1)(iii). We omit the rest. 11.2. PROPOSITION.

(i) (SD)° = (50); (ii) (SG)° = (S°)a: (iii) = (S°)p; (iv) =

(a, b) e (SD)° if and only if (x, a)+(x, b) = (x, a + b) 1 for all x E S if and only if (a, b) E (S°)A. For (iii), (SA)° = ((S°°)A)0 ((S°)D)°° = (S°)D. Similarly for the rest. Proof.

if and only if a + b E

It is immediate from these two propositions that (a)' and (b)' are dual order properties, except for the usual interjection of a closure in the pre-dual, leading to an approximate version of absolute domination. This is, of course, the reason for using mt E1 in property (b).

THE

NORM

11.3. THEOREM.

101

Let (E, F) be an ordered Banach space. Then E has a

Riesz norm if and only if E* has a Riesz norm.

If E has a Riesz norm then so does E* by direct application of Propositions 11.1 and 11.2 with S = mt E1. The converse follows similarly except that we only deduce from the order convexity of Er that Proof.

P+ Er].

= [(E1)A

But then EB is a closed subspace of E so that we can conclude the EB regularity of the restriction of the right-hand side to ED (from Corollary 1.3.3) to obtain (E1)B

for all a> 1. This gives (b)'. We recall from Section 6 that if E is a Banach lattice then E is regular if the norm is increasing on P and lxii = lix III. 11.4. PRoposiTioN. Let (E, F) be a Banach lattice. Then E is regular if and only if E has a Riesz norm. so that If E is regular and y E E1 then —Iyi y yj with = F1 is absolutely dominated. The monotonicity of the norm on P shows that if lxii so that absolute convexity holds. If E y x then lxi and

Proof.

has a Riesz norm and 0 y x then —x

y x so that x with lxii <1 then lxi

monotonicity. If flyfl< I and —x y 1<1. Regularity follows.

lxii gives

so

that

II

11.5. COROLLARY.

Let (E, F) be a Banach lattice that is positively

generated and normal. Then E can be given an equivalent Riesz norm. Proof.

Since E is normal we can renorm E with the unit ball (E1

to obtain a norm

(E1 +P)

satisfying 1-normality. Then define

p(x) = II

It is easily verified that p is an equivalent Riesz norm. Notes

The material in this section is developed in detail in the book of Wong and Ng [217]. We use their terminology here. Theorem 11.3 is due to Davies [73].

CHAPTER 3

Simplex Spaces

1. THE FACIAL TOPOLOGY FOR K, AND THE CENTRE FOR A(K). If K is any compact convex set then we will call a subset E of ext K = 9K F facially closed if E= is complemented in split 1.1. topology for Proof. co (F1

The facially closed subsets of aKform the closed sets for a is a compact space. under which

Proposition 2.5.6 shows that if F1 and F2 are split faces then so are F2) and F1 F2. Thus, if E1 = aF, (f = 1, 2) are facially closed then so

are

E1riE2=a(F1 and

E1 uE2=a(coF1uF2). Next consider an arbitrary intersection,

where

Fasplit. be the complementing projection of A(Fa)* in A(K)*. Since the Let finite intersections of split faces are split we can assume the 's are directed is a directed set, down by inclusion. Thus, for each x E bounded below, and hence 2.3.2 shows {crax} converges ifl flOflfl to

ax = g.l.b. Since

foralla, o

(IX 102

FACIAL TOPOLOGY FOR K AND CENTRE FOR A(K)

103

and

F then = x for all a, so that ax = x. Hence u is a complementing projection of A(K)1c onto and so F is split and

fl

= aF.

We have therefore shown that the facially closed sets form the closed sets for the facial topology of K. If {Ea} is a family of facially closed sets satisfying the finite intersection property, and with E(, = for some closed split faces is compact, then {F,} has the finite intersection property. Since each

is a non-empty closed split face with extreme boundary Therefore is compact for the facial topology. Recall that a simplex K is a Bauer simplex if X = is compact, in which case A(K) CR(X) and K = (X) (2.7.5). If F is a closed face then F is split and F = E, where E = F n X is a closed subset of X. Conversely, if E is a closed subset of X then using the Separation Theorem,

F

E

(X):

1 wheneverf€

1 on E}

The latter set is clearly split under the complementing map, A compact convex set K is a Bauer simplex if and only if the facial topology of aK is Hausdorff. 1.2.

Proof.

If K is a Bauer simplex then

is compact in the relative topology

and, by the above remark, every closed subset of &K is facially closed. Therefore the relative and facial topologies of 9K coincide. Conversely, suppose that the facial topology is Hausdorif. Assume that there exists a point x E c9K\0K, and hence distinct points y, z E 0F, where F is the smallest closed split face of K which contains x. Then, because the facial

topology is Hausdorif, there exist closed split faces G, H of K such that K = co (G H) and y G, z H. Since x E 3K, either x €0 or x H. But this contradicts the minimality property of F. Consequently 3K is closed for the relative topology and hence the two topologies for 3K coincide. If C is a closed face of K it now follows that 3C is facially closed, and hence C is a split face of K. Therefore, by Theorem 2.7.2, K is a simplex, and hence a Bauer simplex.

It will be of importance to determine the nature of the real-valued functions on 3K which are facially continuous.

SIMPLEX SPACES

104

We say that a function f A(K) belongs to the centre of A(K) if given g A(K) there exists an h A(K) such that h(x) g(x) for all x E Since functions in A(K) are determined by their values on äK the function h

will be uniquely determined by f and g. Moreover we will have h(x) = f(x) g(x) for all x It is evident that, when restricted to the central functions form a uniformly closed algebra of continuous real-valued functions on 3K containing the constants. If K is a Bauer simplex then A(K)IaK = CR(öK) and hence everyf E A(K) is central. On the other hand if, for example, K is a closed plane disc (or square) then it is easy to check that the centre of A(K) consists of precisely the constants. Our aim is to prove that the facially continuous functions are precisely the central functions restricted to We denote {fE A(K): 0 on F and f 0 on K} by 1.3. LEMMA.

(i) LetFand G be closed split faces of K withf A(F), g E A(G) such that f = g on F G. Then there exists h A(K) such that h =f on F and h = gonG. (ii) Let F,, G1 (j = 1, . . . , n) be closed spilt faces of K such that F, G1 = and K = Co G,). Then, if f€ and e >0, there exist with

fE (i) Consider graph fcz F x and graph g c G x Then 2.5.6 (ii) shows each x Co (F u G) has a unique representation in co (F G, F' Proof.

G, F G') where F', G' are the complementary faces. It follows that co (graph fu graph g) is a compact convex subset of K x R which equals graph h0, h0 a uniquely determined affine function on co (F u G). Since graph h0 is compact, h0 A (co(FuG)). Then, by 2.10.5, h0 extends to h €A(K). (ii) Let h1 A(co (F1 u satisfy h1 =0 on F1, while h1 =f on 01. Then, since co (F1 u G1) is a split face of K, we can find an extension fi of h1, c/n (2.10.5). In particular we have and and 11 fi

fi=f on G1.

Now suppose that we have constructed fj E (F,) for j = 1,. j=l

fl

onK

.. , r, such that

FACIAL TOPOLOGY FOR K AND CENTRE FOR A(K)

105

while

onco(CJG1). Put

n

E

f1EA(K)',

A(co (F,÷1 U G,+1)) satisfy hr+i 0 On Fr+i and = g on G,÷i. Then 0 hri g Ofl CO (Fr+j u G,+1), and so there exists an extension g + 8/n. We Clearly have fr+i E (Fp.ri)I Of hr+i such that and let hr+i

ri-I

f, /=1

r

f,+g+s/n

1=-i

(r+l)e n

onK,

and r—1

re fl

/=1

OflGr+i.

The proof is now completed by induction.

1.4. THEOREM. A function f: R is facially continuous if and only if 1= gIaK for some function gin the centre of A(K). First let f be facially continuous, and let u E A(K). If we prove that for each 8>0 there exists a v E A(K) such that Iv(x) --f(x)u(x)I <8 for all then it will follow that f has the required form. xE Proof.

We may assume that u 0 and that f is not a constant function. Then if each of length less than e, such 6>0 there exist open intervals U1,..., that f(aK) c U7=1 U1 while no collection of (n —1) of the U, covers f(8K). Choose compact sets C, c U,, such that U_1 C, f(aK). There exist closed (C,) = split faces F,, G1, = 1,. . . , n, such that = oF, and and hence F1 G1 = 0, K = co Lemma 1.3 now gives u E such that

j

For each j choose a1 E U1 and put v = while u,(x) = 0 when when x

If x then If(x) -- au <8 '(U1), and we can estimate

a1u1.

v(x)—f(x)u(x)I j-=1

j=1

u,(x)

Consequently the necessity of the condition in the statement is proved.

SIMPLEX SPACES

1

Conversely, let g be in the centre of A(K). Since the centre is a subspace containing constants, we can assume 0 g 1 and show E = — {x E äK: g(x) a} (0 a 1) is facially closed. Let L denote the space of restrictions of central functions to X = aK. Then L is isometrically isomorphic to a closed subalgebra of CR(X), containing constants. In particular, L is a (pointwise on X) lattice. Let = 1 A (g(x) + (1— a))" on X. Then if E EL be given by

{xEX: g(x)a},

forxEE Let F ={x E K: h1(x) =

1).

forxEX\E.

and

Then F is a closed face of K with

We use 2.10.9 to show F is split. Let let a E A(K). Let a,, E A(K) with

be a

boundary measure in A(K) and on X.

a,,(x) =

Then a,, a0, where a0 is u.s.c. affine and a0 = Thus, since supp c X, ,LIp(a)

=

a0

=

a

on F, ao(x) =0 for x E X\F.

= urn

Lao

J'

a,,

=0.

As we noted in the above proof, the centre of A(K) is always a sublattice containing the constants. It turns out that for simplexes the centre has a purely lattice-theoretic characterization. 1.5. THEOREM.

If K is a simplex then the centre of A(K) is the largest

sublattice of A(K) containing the constants.

Let L be a sublattice of A(K). If a, b L' and

Proof.

p=inf{a, b} then the definjtion of j5 in Chapter 2 Section 4 shows j5 = a A b and 2.4.4 shows j3 = p. The definition of j3 yields the fact that

forx€äK. Hence L is a (pointwise) sublattice of functions on äK. 1ff E L, and L contains the constants, let

B ={x E aK: f(x) a).

f

If g = v a then g a on K and

F={x€K: g(x)=a}

FACIAL TOPOLOGY FOR K AND CENTRE FOR A(K)

107

is a closed face (hence split) of K with E = OF. This shows f is l.s.c. in the facial topology. Similarly, —f is l.s.c. and hence continuous for the facial topology. Theorem 1.4 now shows that F belongs to the centre of A(K). To obtain a further characterization of the centre of A(K) when K is a simplex we need the following lemma. 1.6. LEMMA.

If F is a closed convex subset of a simplex K with

F is a face of K. Proof. We need to show that if

is a maximal probability measure on K with resultant y E F then supp is contained in F. Suppose that there is a point z E (supp Then we can find an f E A(K) such that f(x) <0 0, which gives the required = This leads to 0= g(y) = contradiction. 1.7. THEOREM. If K is a simplex then fe A(K) belongs to the centre of A(K) if and only if there exists a gin A(K) such thatg(x)=f(x2)forallx in OK.

Let F= {u E 3K: f(u)> 0}, and let x E OF. By Milman's Theorem we have x E 3K. Suppose that x = + where y, z E K. Then if and Proof.

denote

the maximal probability measures on K with resultants y and z

respectively we have 0 = (g — 2f(x)f+f(x)2)(x)

=

(g —2f(x)f+f(x)2) 1

2

1

J'

(g — 2f(x)f +f(x)2)

(f—f(x))

Therefore we have 1(v) =f(x) for ally in supp

But then u supp supp is contained in {u 13K: f(u) 0), and consequently y, z supp belong to F. Therefore x OK, and Lemma 1.6 shows that F is a face of K. It follows that the set {u 0K: f(u) 0} = OF is facially closed and, by adding constants to f, we see that {u OK: f(u) a} is facially closed for all a in

Hence f is upper semi-continuous for the facial topology. Similarly —f is upper semi-continuous for the facial topology, and the resuJt follows from Theorem 1.4. Notes

The notion of facial topology for a compact convex set K and the associated concept of centre for A(K) were first introduced by Alfsen and Andersen

108

SIMPLEX SPACES

[6,7]. The concepts have been extensively generalized by Alfsen and Effros [9] and others. We will look at some of these generalizations in Section 8. The motivation for the notion of centre comes from theory and this matter will be taken up in Chapter 5. In Chapter 4 the notion of centre in relation to complex function spaces and function algebras will be discussed. Theorem 1.5 is due independently to Fakhoury [111] and to Nagel [163]. If K is not a simplex then the conclusion of 1.5 may fail, even if a largest subiattice of A(K) containing constants does exist (see Ellis [106]). Lemma 1.6 is due to Mokobodzki (see Rogalski [180]), and Theorem 1.7 to Sternfeld [201].

Many other facial properties of simplexes are discussed by Goullet de Rugy [121].

2. FACIAL CHARACTERIZATIONS OF SIMPLEXES We have seen (2.7.2) that a compact convex set K is a simplex if and only if every closed face of K is split. Moreover every closed face of a simplex K isa simplex and E is a face whenever Eisa compact subset of aK (see Lemma 1.6). If K is a finite-dimensional compact convex set then it is easy to check that K is a simplex if and only if each x äK is a split face, or if and only if E is

a face of K whenever E c aK is compact. We now give some examples to show that neither of these properties characterizes simplexes in general. The first example arises naturally in the field of function algebras. Let D ={z E C: Izi 1), A ={f€ Cc(D): f is analytic on D°}, and let S = {4 E A*: 4(1) = 1 = 11411} endowed with the 2.1. EXAMPLE.

Then every proper closed face of S is a Bauer simplex and is strongly

Archimedean, and every x E aS is a split face of S. The setS is metrizable with 8S closed, but is not a simplex. Proof. The space A(S) is the uniform closure of (re A)Is. We identify the boundary circles 'y, F of radii I with their natural embedding X = u F in

S. Denote

h=(f,g)€C(X); Let

f=hI,,,

g—hlr.

denote the normalized linear Lebesgue measures on F, y. Since

for any non-zero integer n and A

on A(S). We observe next that CR(X) = (A(S)Ix, Iz I).

FACIAL CHARACTERIZATIONS OF SIMPLEXES

and let

To see this let h = (f, g) E n

F(z)=

109

It

k

,

k=()

k

k=O

be polynomials with a0, b0 real and lire F—f1l.,,<e

and lire G—gfli<e.

Let

H(z)=cizl+Eckzk where

ak=ck+2c.k

ao=(2)c+co = c + c0

bk

= Ck ÷ C-k.

Then re H = re F on 'y, re H = re G on F. = A.,(H) we conclude that Since c =0 if and only if

A(S)Ix = {(f, g) E CR(X):

= A1.(g)}.

be proper compact subsets of y, F with E = E., u Er Let BA, A.,(E.,), Ar(Er) <1 we have A(S)IE =

0. Since

and that F = E is a Bauer simplex and a strongly Archimedean face of S. In particular we see that = X. If = 'y and = 1' and A denotes the element of S given by

A(f, g) =A.,(f)Ar(g) are maximal and and the representing measures A,,, then A distinct. Hence S is not a simplex. Moreover, if A E F, a proper closed face, then there is an E, compact, in either 'y or I' (say 'y) such that E cF = 0 and 0. Since A,,=A,,IE+A,,I,,\E, A, isa Since EciF= convex combination of measures, one of which represents a point in E. But F is a face, so that this is a contradiction. Hence the smallest closed face containing A is S. Thus any proper closed face F = (E,, u Eç) where E.,, Er are proper subsets of 'y, F (E,, = F c 'y, = F c 1'). Finally, we invoke 2.10.9 to identify the split faces of S. Since A(S)1 r'M(X) = (A., Ar) we have F = (E) (E = E,, u c X) is split if and only if (A., — A CR(E)1 if and only if Ay(E'y)

0

In particular each x E as is a split face of S.

SIMPLEX SPACES

110

The facts in the following example are straightforward to verify using the same kind of techniques as for 2.!. 2.2. EXAMPLE.

Let

L = {fE

and let K =

E L*:

1]:

f(t) dt = f(t) dt, 1}

=

f(0) = with the w*_topology.

Then [—1, 1]\{0}, every x öK is a split face of K, and every proper closed face of K is strongly Archimedean and also a simplex. Moreover, if E c &K is compact then E is a face of K. However K is not a simplex.

It follows from 2.8.1 and 2.10.2 that for a strongly Archimedean face F in K

inf{s 0: a

A(F) implies 3b A(K), blF = a

and

sIIaIIp}<

lib IlK

This number is called the characteristic of F in K.

It is noticeable in the preceding example that the faces E are in general not split, and indeed their characteristics are not bounded. In this connection we will prove Theorem 2.3. By a standard compact convex set K we will mean that 9K is universally measurable for all regular Borel measures jL Ofl K, and that p.(K\i)K) =0 whenever is maximal. Of course every metrizable K is standard, and also any K for which c3K is closed. 2.3. THEOREM. Let K be a standard compact convex set. Then K is a simplex if either of the frllowing conditions hold. (i) For every compact .E c E is a simplex and is a strongly Archimedeaiz face of K. Moreover, the characteristics Eform a bounded set. (ii) For every compact E c E is a split face of K.

(i) If E aK is compact then, by hypothesis, A(K)IE = and has an extension g A(K) there is a constant a >0 such that eachf E

Proof.

with —

p

p be maximal probability measures on K with alifli. Let A (K)1. If p then, since K is standard, there is a compact set

F 9K and an e >0 such that

v(E')J> e ; say v(E')> e. Put = e/4a and choose a compact set E with E' E c i3K, /L(E)> 1— and

v(E)> I —13;letf€ 1 an extension off satisfying ajif II we obtain This contradiction gives ji = JEf dji f dv

v,

v(g) and and hence K

is a simplex. (ii) Let be a boundary measure in A (K)1 and let E be a compact subset we have FiFE A(K)', and so ji(E)=0. Since K is of aK. Then, if

standard and p. is regular it follows that p. =0, that is, K is a simplex. The next result shows that K being standard is necessary in 2.3. 2.4. THEOREM. There exists a compact convex setK which is nota simplex but which has the following properties:

Ill

FACIAL CHARACTERIZATIONS OF SIMPLEXES

(i) .3K is a Bore! subset of K; E is a split face of Kfor all compact subsets E of .3K.

(ii)

Let V = U { Va: a [0, 1]}, where the disjoint sets Ya each consist of three points se,, ta}. Topologize Y so that each and ta is isolated and such that each sa has a neighbourhood base consisting of the sets {sa} U 00. Then V is a compact Hausdorif space, each Ya is discrete in the relative topology and a s(1 defines a homeomorphism between [0, 1] and {sa: 0 a 1 }. We will identify [0, 1] and {sa: 0 a 1} in the remainder of the proof. Let A denote Lebesgue measure on [0, 1] and Jet A1, A2 be the restrictions 1] respectively. Define L to be the closed linear subspace of A to [0, and for all a [0, 1], of CR( Y) given by L = {f€ CR( Y): f(Sa) = dA2(a)}. Certainly L contains the constant functions. Sf(sa) dA 1(a) = To show that L separates the points of Y we first note that the functions =0 for all a, f,3(ra) = for a /3 and belong to L, where Therefore each and each is a peak point for L, and =1= so we need only show that L separates the points of {sa: 0 a 1}. But a /3 there exists anf in dA1 =5fdA2, and we can extend f to a function in L by putting f(rA) = f(s.,) = f(t.d,) for all 'y in Proof.

[0, 1].

L": 4(1) = = 1} of L then we may If K denotes the state space identify L with A(K) in the natural way (see 1.4). The Choquet boundary for L is identified with .3K, and this clearly equals U {ra, ta: 0 a 1}. Therefore the closure .3Kof .3K identifies with the Silov boundary V for L. In particular we observe that .3K, being the complement in V of the compact set {sa: 0 a 1}, is a Borel subset of Y. Define two measures on V by the equations v,(f) = ff(Sa) dA1(a), Y). Let B denote the family of functions of the form g + h, j = 1,2,1 E where g, h E Y), g = C) except at finitely many of the points ra, ta and = h(ta) for all a in [0, 1]. Then B forms a subalgebra of CR(Y) h(ra) =

containing constants and separating points of Y, so that B is dense CR( Y) by the Stone—Weierstrass Theorem. The mapping a

in

+ h)(ta) = + h)(ra) g takes only finitely many values. Consequently the mapping a is also measurable for f€ CR(Y), using the density result which has just been established. We therefore obtain, for / = 1, 2,

J

+h)(ra) + (g

and hence for all f in

dA1(a)

=J

dA,(a)

=

J (g + h)

Y)

j'

(1)

SIMPLEX SPACES

112

Clearly the measure

We now wish to find all measures in

to A(K)1 M(Y), and the definition of L implies that is the linear span of and N, where N is the 0 a 1} in M( Y). Suppose linear span of the measures — to in N, where that {'q a)} is a net =

—

belongs

=

in A(K), Then for each a [0, 1], we can find we see that converges pointwise to = 0. Using the fact that there is a g(, A(K) with ga(ra)= 1 = we see that = discrete part fld of rj has the form

1,

such that fl({sa}) +

=0 except for and hence the

00

has no — qd, belong to A(K)1. Since — 1} and {ta: 0 a 1} are discrete it atoms and since the sets 0 a follows from regularity that — is concentrated on {sa: 0 a 1}. But then — rld)(h) =0 for all h E CR[0, 1], and so q = This description of N shows that the only possible measures in aM(K) A(K)1 are the multiples of because no maximal measure can have an atom lying outside then E must be Now if E is a compact subset of

Consequently fld, and also

finite, and it is easy to construct a function in A(K) which peaks on a face F

with aF = E. Since p(E) = 0 for a finite set E we have p.'jE E whenever and hence isa split face of K. To complete the proof we need to show that K is not a simplex, and this will be the case if we show that is a boundary measure. We first note that + is the only maximal probability measure on K with resultant se,.

Indeed, if p.' is another such measure then

+ Eta) — p.' E t9M(K)

+ + bp. for some real number b. Since p.' is a A(K)1 and so p.' = probability measure and p. has no atoms we can deduce that b =0. The measures p1 will be maximal measures if and only if v,(f) = v1(f) for

each continuous convex function f. Using (i) we have =

f

= j'

dAj(a) dA1(a) =1/1(/),

+

where the second equality follows from 1.6.2 and the fact that

the only maximal measure representing sa. Therefore p. = boundary measure.

—

is is a

FACIAL CHARACTERIZATIONS OF SIMPLEXES

113

If F is a face of K then the complementary set F' consists of the union of all faces of K which are disjoint from F. Each x in K may be decomposed F, z E F' (see [2, Proposition 2.6.5]). If x Ày +(1 —A)z, with 0 A 1, y F' is itself a face of K and if the constant A is uniquely determined by x then F is called a parallel face of K (see 2.10). A closed face F of K is parallel if 8M(K), or equivalently if is an affine =0 whenever E

function, where for any bounded real-valued function f on K we write gf}, x€K. f(x)=inf{g(x): A closed subset D of K is said to be dilated if whenever is a maximal probability measure on K with resultant x E D then is supported by D. Evidently every closed face of K is dilated, and every compact subset of 8K

is dilated. If D is a dilated subset of K then a continuous real-valued function f on D is called affine if p(f) =f(x) whenever is a maximal probability measure on K with resultant x €D. If D is a compact subset of 8K then it is clear that every continuous function on D is affine. It is easy to verify that if D is a closed face of K then a continuous function on D is affine in the preceding sense precisely if f E A(D). In the case when K is standard, Theorem 2.3(i) shows that K is a simplex if there is a constant C such that whenever E is a compact subset of 8K then

E is a simplex and a face of K and each f This result can be rephrased to say that K is a with simplex if and only if every continuous positive function f on E has an with extension g Cu/h, where C is a constant independent of E. Theorem 2.4 shows that the hypothesis that K is standard may not be dropped in this result of Rogalski. E

We present below an analogue of Theorem 2.3(1) which is valid for all sets

K. In the case when K is standard the compact subsets of 8K play a role similar to that played by the dilated sets when K is not standard; this is because when K is standard the supports of the maximal measures on K can be approximated by compact subsets of 8K. A"(K) stands for the space of bounded affine functions on K. 2.5. THEOREM. vex sets K.

The following statements are equivalent for compact con-

(i) K is a simplex.

(ii) If D is a dilated subset of K and if f0 is continuous and affine moreover for each closed on D then f has an extension g e face F of K, inf {a(x): a E Ab(K), 0 a 1, aIF = 1} =0 for all x inF'. (lii) The convex hull of any finite number of closed faces of K is a face of K, and for each closed faceFof K, inf {a(x): a A"(K), 0 a 1, alP =

1}=Oforallx inF'. (iv) Every closed face of K is parallel.

SIMPLEX SPACES

114

The implication (1) implies (iv) holds because every split face is parallel. (i) implies (ii). The second statement in (ii) holds trivially for split faces, so it is sufficient to prove the first statement of (ii). D) then If D is a dilated set then D is a face of K. Indeed if x e x eD, so that the maximal representing measure for x is supported by D. But then = by the extremity of x, so that x 9K. Lemma 1.6 now shows that ë6 D is a face of K. To complete the proof it will be sufficient to show that every function f which is continuous and affine on D has an extension in whenever and are probThis will follow if = D. ability measures on D with the same resultant y D let denote the unique maximal probability measure For each x D representing x. Given 6 >0 there exists a continuous concave on + 6 = A.,(f) + 6. Since f is affine on D function u f such that = ü(x) (see 1.6.2), and because ü is we have, for x in D, f(x) = convex we obtain Proof.

/Li(Ü)

It follows that obtain =

i(f)

=

AV(f) + 6.

and applying the same argument to —f we

= /h2(/) as required. (ii) implies (iii). It is sufficient to prove the first part of (iii) for two closed

faces F and G of K. If x belongs to aK\(F G) then it is evident that F u G u{x} is a dilated subset of K and that the function is affine and = 1 and =0 on F u G. Therecontinuous on F u G u {x}, where The set H = fore there exists on extension A(KY of

fl

(0): x E

u G)} is a closed face of K containing F u G, whose

extreme points belong to F u G. Therefore H = co (F G) and (iii)

is

proved. (iii) implies (i). To prove that K is a simplex we will show that A(K)* is a vector lattice. Using the Bishop—Phelps Theorem it will be sufficient to prove that p exists whenever p A(K)* attains its norm at some f E A(K) with exists in A(K)* for n 1, and if p in norm, then 11tH = 1. (Indeed if p in norm (see [108]).) If is a representing measure for p on A(K) with , and so it = Dpil then Il/Lu = V ÷D,L1 =P(f) =ffd —ft follows that supp + F, supp /h c G where F and G are the closed faces of K given by F = {x K: f(x) = 1}, G {x K: f(x) = —1}. Without loss of 0. Define o-(g) =f g 0, for generality we can assume that g A(K), so that clearly if p, 0. To show that o' = p + we need to prove that

f if whenever / p, 0 in A(K)*. By the Bishop—Phelps Theorem, for A(K)* which attains its norm and is each natural number n there exists with such that jifr,1 — (/ — o')lJ < 1/n. If A?, is a representing measure for of K such that then there are disjoint closed faces H, = EA(K)* is defined by If suppA

OF SJMPLEXES

FACiAL

for g E A(K), then we can decompose EG

H, y,, E (G

115

+ (1---

=

)'. We will show that

where

-.0 as n

If (1— 0 then E H, and hence belongs to the complementary set for the face co u G). Therefore, by (iii), we can find an A"(K) with 0 on co u G) and i—i/n. Identifying A(K)** with A"(K) as usual, we have 0 (4)

= 1

1

I

1\ 1 ——) +.

— It follows that in general we have A"(K) with 0 =0 Ofl Co (F u 1,

—*0. SimiLarly we can find

1—1/n, and

hence

-÷0, and consequently

We obtain

/1

1

1

——

-.0. It now follows that if

h A(KY then = urn { J h dA —

(4) —u)(h) = urn

0,

and hence 4) o as required. (iv) implies (iii). Let F and G be closed, and hence parallel, faces of K and let H = co (F u G). To show that H is a face of K we need to show that if A is a maximal probability measure on K with resultant x E H then supp A c H. v be We have x = ty + (1— t)z, where 0 t 1, y F and z E G. Let maximal measures on K with resultant y, z respectively. We have = A —

faces

— (1— t)v

=

E A(K)1 aM(K), and since F, G and F G are parallel =

G) = 0. Therefore

A(FuG)=A(F)+A(G)—A(FrG)=

1,

because and ii are supported by the faces F and G respectively, so that supp A is contained in H. The second condition of (iii) is satisfied since, for any closed parallel face F, the function with =1 on F and XF =0 on F'. belongs to The following result gives two ways in which facial structure distinguishes l3auer simplexes amongst simplexes. We will say that a compact convex set

SIMPLEX SPACES

116

is a closed split face of K whenever K satisfies Størmer's axiom if (Ua {F4j is any family of closed split faces of K.

2.6. THEOREM. Let K be a compact simplex. Then K is a Bauer simplex if and only if either of the two following conditions hold. (i) The closure of each face of K is a face of K; (ii) K satisfies Størmer's axiom. Proof.

The necessity of the two conditions is trivial. (i) implies (ii): Let {Fa}

be closed faces of K, and hence split faces of K. Since the convex hull of finitely many of the is a face of K it is easy to check that co (Ua Fa) is a face of K. Consequently Fa) is a face of K; and so K satisfies StØrmer's axiom. Now assume that K satisfies Størmer's axiom, and let E c i.iK be relatively closed. Then there exists a compact set G K such that E = G and Milman's Theorem shows that E) c G. Now each x E is a (split) face of K, and hence E is a face of K because of Størmer's axiom. But then E) G aK, so that E = E). Therefore E is facially closed and it follows from the proof of 1.2 that K is a Bauer simplex. Although faces of simplexes are simplexes it is far from true that crosssections of simplexes are simplexes, as the following result amply illustrates.

2.7. THEOREM. Let S be the w*(l', c)-compact simplex {y 11: IlyIl = 1, y,, 0, for all n}, and let K be any metrizable compact convex subset of a locally convex Hausdorff space. Then there exists a linear subspace M of 11 such that M S is affinely homeomorphic to K. Proof. Let be a sequence which is dense in 0 for all n, and let n 1, and go = 1 g,,. Define a linear operator T: c -+ A(K) by Tx = n 0, belongs to c. Let M be x,..g,, where x =

the range of T* in 1'. Since T(c) is dense in A(K), T* is one-to-one; also T* is positive since T is positive. If K and if e = {1}E c, then we have = (Te)(4)

= 1.

=

Therefore T* maps K into M S. Suppose that eM S. Then we have 'I'(l) = 'V(Te) =

= 1,

and =

=

0,

n 1,

E A(K)* and that

IDEALS (N SIMPLEX SPACES

117

0 for all n, where is the nth coordinate vector in c. Hence we have ensures that 'I' 0 and consequently that and the density of {f,,} in 'If

K. Therefore T* is an affine homeomorphism of K onto M S.

Notes

Examples with similar properties to those of 2.1 and 2.2 have been given by Rogalski [181], to whom Theorem 2.3 is due. It appears to be unknown whether, in the standard case, the condition that E is a simplex can be removed from 2.3(i). Theorem 2.4 is due to Ellis and Roy [110] and it involves the Bishop—de

Leeuw porcupine topology construction [40]; it is related to examples studied by Alfsen [5, 1.4, 2.3]. The fact that condition (i) of 2.5 implies the first statement of (ii) is due to Lifros [94,3.3; 95,2.3]. The equivalence of (i) and (iv) of 2.5 is due to Briem [54]. Theorem 2.5, in the form given here, is due to Ellis and Roy [110]. The first statement of 2.5(u) certainly does not characterize simplexes since it

does not imply uniform boundedness of the affine extensions involved. Similarly the first statement of 2.5(iii) does not characterize simplexes; an example of this phenomenon is given by McDonald [159, 1.9]. In the duality between A(K)* and A"(K) the set F4 = 1} = 0} is called the quasi-complement {x E K: inf{a(x): a E A"(K), XF a of the face F (see Alfsen and Shultz [10, p. 15] and also Section 5.6 (below). The second statement of 2.5(u) and 2.5(iii) can be re-written F* = F'. This condition does not imply that F' is convex, but it does imply that for each x E K the constant A E [0, 1] in the decomposition x = Ày + (1— A)z, y F, z e F' is unique. Theorem 2.6 is due to Størmer [206]. Of course every finite-dimensional compact convex set satisfies condition 2.6(i) since every face is closed. It is not know precisely which infinite-dimensional compact convex sets satisfy

this condition. Some results of a local nature on this topic have been obtained by Roy [184]. Theorem 2.7 is due to Lazar [146]. In the same paper he shows that every infinite-dimensional Banach space contains a compact set which is contained in no bounded simplex lying in the space.

3. IDEALS IN SIMPLEX SPACES

Recall that a subspace I of an ordered space (E, P) is an order ideal c El implies b El) if and only if P is a face of P. It is (a b convenient in the context of A(K) spaces to also require that I be positively and generated. Thus we will say I is an ideal in A(K) if is a face of 1= The first result is a direct reformulation of 2.8.15.

SIMPLEX SPACES

118

3.1. THEOREM.

Let K be a compact simplex and let I be a closed linear

subspace of A(K). Then I is an ideal in A(K) if and only if I = FL {fE A(K): f(x) =0, Vx E F} for some closed face Fof K. It is worth noting that if I is a (not necessarily closed) ideal that I' = j0 K is still a closed face F and hence F1 = J00 = I is a closed ideal. If {4} is a family of closed ideals in A(K) then, if Fa = we have fla 'a Ua Fcc)1. Hence if K is a Bauer simplex, so that Størmer's axiom holds, then fla is a closed ideal in A(K). This fails in general, as the following example shows.

L = {fE ca[—1, 1]: f(0) = 3.2. EXAMPLE. Let K {4i E L*: 4(1) = = 1} with the w *topology, and let 1,, = {f E A(K): f(1/n) = 0}. Then A(K) is a simplex space and is closed ideal for each n, while fl1 I,, is not positively generated.

An interesting problem is whether Bauer simplexes are characterized among simplexes K by the property that the intersection of an arbitrary family of closed ideals in A(K) is again an ideal. In the metrizable case, at least, the answer is affirmative, and we now discuss this result. We first consider a theorem which leads to the construction of many interesting examples of simplex spaces. c3Mq (K) will denote the maximal probability measures on K with resultant q. 3.3. THEOREM. Let K be a compact metrizable simplex and let q 8K\8K. Suppose that Y is a closed subset of 9K such that Y (aK\aK) = {q} and Then A = {f€A(K): f(y)= 0, where {ir}=

ir(f), Vy E Y} is a simplex space A(H) where aH can be identified with aK\ Y. Proof.

If a = ir(Y) then 0< ir(ÔK\ Y)

1—a. Let 7r(f), Vfe Y}

L ={f€ and put

X is homeomorphic to Y. To each naturally a measure A M( Y) so that (A

M(X) there corresponds

—A(Y)ir)(f)

for each f CR(ik). Therefore, if N={A —A(Y)ir: A €M(Y)},

IDEALS IN SiMPLEX SPACES

119

by approximating A by point measures we have N un X). If i9K) we have A — A (Y)ir E N then (for each Borel set B (B

A

=

( Y)ir(B), so that for each Borel set A c Y we obtain

A(A)= ,1(A)+A(Y)ir(A) Hence A is uniquely determined by hAil

—a) and also

p. 1(Y) 4 ir( Y)Ip.l( Y)(1

—

a)

1

It follows easily now that N is norm-closed, and hence generated) so that N = L1. since N is We can write

(1.3.5

p.(f)

={p.

VfE

some v E

=J

whcrc (i9K) for x E K; in fact = 0 when fE A(K)IaK, — satisfiesf h dv = the function h(x) = while if p. E A(K)' and f o (cf. Alfsen [2] Proposition 2.3.14). Since A =

L A(K), A' equals the

of A(K)'+L', and A(K) A EM(Y), Vf€ CR(aK)};

+(A

(*)

if and only if it is norm-closed (1.3.5). In the represen-

this set is

tation of p. E A(K)' + L' we may, by transferring an atom if necessary, assume that A =0; hence for any Borel subset C of Y\{q0} we have

p.(C)= v(C p.

and consequently

=

aK\oK

and

We obtain the inequality IIAIIlIpll(1 +2a/(1 —a)),

and as before, it follows that A(K) + L' is norm-closed.

120

SIMPLEX

Let r denote the quotient map of into M(aK)/A ',the dual space of A. Define, for p E M(aK), A (ii) E M( Y) by A (v)({qo}) = 0,

A (v)(A)

dv + v(A) + ir(A)A (v)( Y)

= JaKVIK

and let be as in (*) so that (v)(B) = v(B) for each Bore! subset of

for each Borel subset A of Y\{qo}, r(JL (ii)) =0. It is easily verified that (aK\aK) u Y. Therefore

Y) = 0,

—

and

r(v —p(v)) =

If r(ii1)

and

T(v),

Y) =0

lvi

then

Y)=0} and so there is a A v2(B)

M(Y) with

ii2(B

dv2+ A (B

Y)— A(Y)ir(B)

aK\aK

for each Bore! subset B of obtain

(reasoning as above). Since v2(aK\aK) =0 we

v2(B)

A(B

Y)—A(Y)ir(B)

putting B = Y we have A (Y) =0; this together with the fact that P21 Y =0 gives A =0 and consequently P2=O. This argument shows that the A* -. S, such that /(i(v)) = p — map is well defined. Clearly is linear and onto, and if çb(r(vj)) = #('T(V2)) then i.'1 —p.(vi) = and

—

so that r(vi)=r(v2).

Letv 0 belong to M(9K) B c (aK\aK) u V then

and let B be r(p)(B) =0, and if B

a Borel subset of

If

V we have

(v)( Y)ir(B) 0, ôKVIK

so that b(r(v))0. We have, for r(v)EA*, lir(v)ll = 'nf {liv

= liv —

For p M(aK) and B a Borel subset of

II:

r(v ) =

V we see from the definition of

A(v)(Y) that =

-f

aK\iiK

—v(Y\qo)ir(B)(1 _a)_1.

dv

IDEALS IN SIMPLEX SPACES

121

Therefore Y)

kb(r(v))li = I

ivj(aK\ Y) +

Y)j'

Y) +

uK \uK

Y) ( Y))d

Y) + ivl(OK\aK) + l'H( Y\q0) so

+ vl( Y\q0) I

lvii,

that and 4' is an isometry. We have proved that A * is isometrically order isomorphic to S. It is

evident that S is an L-space and that the extreme points of the state space of S can be identified with 3K\(aK\aK) Y) = 3K\ Y. 3.4. COROLLARY. Let K be a compact metrizahie simplex such that whenever is a sequence of closed ideals in A(K) their intersection is also

an ideal. Then K is a Bauer simplex. Proof.

if K is not a Bauer simplex then there is a q e aK\aK such that

supp (ir) has at least two distinct points

and P2, where fir) = Mq (3K). Let

be a sequence in 3K\{p,, P2) which converges to q, and let Y u{q}. If A is defined as in 3.3 then A is a simplex space A(H) with = c)K\Y. Since co (p3. P2) is a closed face of H we can find g,, g2 E A(H) i, j = 1, 2 and gj 0. We must have g1(q)> 0 since Pi, P2 C with gj(pj) supp (ir) and sowe can definef = —(g,(q)/g2(q))g2 c A. Sincef(q) = 0 and fc A we have f(x,1) =0 for all n. Put 4 = in A(K) so that is a closed

ideal and fE I,, g f, 0, and so g(q) =

I. Because I is an ideal there exists a g €1 with =0. Therefore Pi belongs to the closed

face g '(0) of K which contains q, so that 0= g(p1)f(p1) =

1.

This

contradiction shows that K is a Bauer simplex. if K is a compact simplex with 0 c aK then A(K): f(0) = 0) is a = partially ordered Banach space with normal, generating cone and such that Ao(K)* is a vector lattice. The space A0(K) is also called a simplex space (without a unit) and Theorem 3.1 has a complete analogue for A0(K).

Notes Theorem 3.1 is due to Efiros [93]. Examples of simplex spaces for which the family of closed ideals is not closed under arbitrary intersections have been given by Bunce [58] and Perdrizet [167]. The example presented here in 3.2 is due to A. Gleit. Gleit [116, 117] proved Theorem 3.3, and also proved Corollary 3.4 for simplex space A0(K), with K metrizable, showing that A0(K) is an M-space

SIMPLEX SPACES

122

if and only if the intersection of an arbitrary family of closed ideals in A0(K) is a closed ideal. are in one-to-one correspondence with the The maximal ideals in points of aK\{O}; in fact an ideal I is maximal if and only if I = {X}j for some x E aK\{0}. Effros [93] developed a hull-kernel topology for the maximal

ideal space of A0(K) and also of A(K) which is essentially the facial topology for aK\{O}.

4. STONE-WEIERSTRASS THEOREMS FOR SIMPLEX SPACES Let L be a simplex space and a closed linear subspace of A(K) containing the constants, with Choquet boundary äK. Then for each 4.1. THEOREM. x E öK the sets

Q(x) —={y c K: f(x) =f(y), Vf€L} are closed faces of K, and

L ={fE A(K): f is constant on Q(x), Vx aK}. Consequently, if L separates the points of Proof.

then L = A(K).

Let x E ÔK and suppose that Ày +(1 —A)z E Q(x) for some y, z e

K and O0 a g e L with = L; by the hypothesis that g f,O and g(x) <e. Hence we have

e >g(x)=g(Ay+(1—A)z)Af(y)A, and since e >0 was arbitrary we obtain a contradiction; so Q(x) is a closed

face of K.

Let fE A(K) be constant on 0(x) for all x in inf {g(y): g L, g f} for each y

and define = K. We evidently have and also for

x€K

(by 1.6.2)

1(x=sup{,i(f): where P. is the set of probability measures on K with L-

resultant x. Therefore if x E

and

we have supp

contained in the face Q(x),

so that f(x)=f(x). Let F denote the family

{g€L: g>f}, F and write u for the pointwise minimum of g1 and g2. Since L is a let g, simplex space the family

F1={v€L: v
STONE—WEIERSTRASS THEOREMS FOR SIMPLF.X SPACES

123

is directed upwards with supremum ü such that ü(x) = u(x) for all x E aK, by arguing as above. Therefore ü ---1 is lower semi-continuous and affine on K and hence u ü >f on K. Consequently for and so attains its infimum on F1 with >f(x). A simple compactness each x E K there is a

argument together with the fact that L has the Riesz separation property now shows that there is a v E L with u > v >1 on K, and hence F is directed downwards. It follows that / is affine and upper semi-continuous and hence that f = on K. Therefore f is a uniform limit of a sequence of functions in F, so that

/

f E L.

Finally, if L separates the points of =

then, since ={x},

for each x OK, we obtain L = A(K). The classical Stone—Weierstrass Theorem corresponds to the special case of

Theorem 4.1 where K is a Bauer simplex and L is a closed sublattice of A(K) containing the constants; in the case L is a lattice ideal in L for each x E OK, and hence the Choquet boundary O,,K of L equals OK. Simple finite-dimensional examples show that the conditions that L be a simplex space and that OLK = oK cannot be dropped from the hypotheses of

Theorem 4.1. The next result is a generalization of the separable case of the Stone— Weierstrass Theorem, in the same spirit as Theorem 4.1. 4.2. THEOREM. Let 1 E L c M, where L and Mare closed linear subspaces which separate points of the compact metric space X, and such that of = OMX. Then L = M if and only if whenever 1,
s=1,..., m, where 11€L and f€M, there exists an i€L with

Note that L can be identified with a closed linear subspace of M = A(K), separating the points of OK and containing the constants, and that 9K = 3MX = OLK. Therefore, if I A(K) and 1ff and fare defined as in the proof of 4.1, we have f(x) =f(x) =f(x) for each x OK. Since X is Proof.

metrizable there exists a dense sequence {im} in L, and hence we can choose of functions which are pointwise increasing sequences {g,1} and suprema of finitely many of the 4,,, and such that g,,
SIMPlEX SPACES

124

Notes

Theorem 4.1 is due to Edwards and Vincent-Smith [91]. Generalizations to

the case of simplex spaces without constants have been given by Edwards [89]. Applications of Theorem 4.! to uniform approximation of harmonic functions have been given by Vincent-Smith [212]. For example, let D be an

open subset of W' with closure D and boundary D*, and let M = {fE f is harmonic in D}, L ={fE f extends to a function harmonic in a neighbourhood U, of D}. Then M can be identified with a identifies with the set of regular points of space A(K), K a simplex, and D*. The Choquet boundary for L coincides with the set of stable points of

D*. In this set up Theorem 4.1 gives the result of Deny [77] that L is uniformly dense in M if and only if every regular point of D* is stable. Further applications in potential theory have been given by Vincent-Smith [212] and de la Pradelle [174, 175]. Theorem 4.2 was proved by Alfsen [4].

5. PRIME SIMPLEXES AND ANTILATTICES A compact convex set K is said to be prime if whenever F and G are (closed) semi-exposed faces of K such that co (F u 0) = K, then either F = K or G = K. A(K) is called an antilattice if wheneverf and g belong to A(K) and

fAg

exists

in A(K) then eitherfg or

5.1. THEOREM. A compact convex set K is antilattice. Proof.

if and only if A(K) is an

Suppose first that A(K) is an antilattice and that K co (F u G),

where F and G are proper semi-exposed faces of K. Then there exist f, g 0

in A(K), / g 0, such that f(x) =0= g(y) for all x F and y €0. It is evident that 0=f A g in A(K), so that either f g or g But then either f =0 or g =0, which gives a contradiction. Hence K is prime. Conversely, suppose that K is prime and that f, g A(K) such that fA g exists in A(K). Without loss of generality we can take g =0, and we note that if h(x) = mm {f(x), 0} for x E K, then (f A O)(x) = h(x) for each x E 0K (cf. 1.6.1(ix)). Therefore if F = {x €K : (f A 0)(x) =f(x)} and G = {x K: (fn 0) = 0} then F and 0 are exposed faces of K such that OK F u 0. Hence K = co (F u G) so that either F = K or G = K, that is either fA 0=f orfn 0 = 0. Thus orf 0 so that A(K) is an antilattice.

5.2. COROLLARY. JfK is a compact convex set such that OK= K then K is prime.

PRIME SIMPLEXF.S AND ANTILATTICES

Proof.

125

Let f, g€A(K) such that fng exists in A(K). Then (fAg)(x)=

f(x) A g(x) for each x E 3K, and hence for each x E OK= K by continuity. so that A(K) is an Sincef A g is affine this implies that either f g or g antilattice and K is prime.

An important example of a metrizable simplex K with K =0K will be given in Section 7. We now give an example of a naturally occurring prime

simplex S with 35 S. By the state space S of A we mean S {c/

A*:

=

5.3. EXAMPLE.

1= 1} with

the relative

The state space S of the function algebra A={f€

C(s): f(O) =f(l), f analytic on

={z E C: fzj 1), is a prime

where

simplex. Proof.

extreme points of S coincide with the peak points for A (cf.

[114]), and these are the points of r\{l}, where F = {z

C: Izi = 1}. The space

A(S) can be realized, when restricted to F as 2ir

1

f(e'6) dO.

CR(F): f(l) = — I

Evidently S is a simplex. Suppose that fE A(S) and that f A 0 exists in A(S);

then (f A 0)(z) =f(z) A 0 for all z C F. Now A(S) can be identified with the uniform closure of re A (see Chapter 4) and hence g = A 0 has a continuous In particular we have extension to which is harmonic in

f

f(1)A0=g(1)=g(0) 1

=—J

1

1,2ir

g(&°)dO=_J

2iro

'1

=f(0) A 0 (— Ijo f(e'8) do) A 0. = 2ir

If f(0) 0 we have that is f 0. If f(O)O then

—f(e") A 0) dO =0, which implies that f = g,

(f(e'°) A 0) dO = 0, which implies that f 0. Therefore A(S) is an antilattice and S is a prime simplex.

Notes

The notion of prime simplex K was introduced by Effros and Kazdan [98],

and they showed that K is a prime simplex if and only if A(K) is an antilattice. Amongst many other interesting results they prove that if A(K) represents the solutions of Laplace's equation for a bounded region in then K is either a Baiier simplex or a prime simplex.

SIMPLEX SPACES

126

The definition of a prime set for non-simplexes, and Theorem 5.1, is due to Chu [651. The application of these results to will be discussed

in Chapter 5. If, in the definition of a prime set K, we replace the exposed faces Fand G by closed split faces we obtain the notion of a weakly prime set; for example a plane rectangle is weakly prime but not prime. The is studied in relevance of this notion to function algebras and Ellis [107].

6.

TOPOLOGICAL CHARACTERIZATION OF THE EXTREME BOUNDARY OF A COMPACT SIMPLEX

Bauer simplexes are, by 2.7.5, in one-to-one correspondence with compact Hausdorif spaces. The extreme points .3K of a compact simplex K, endowed with the relative is a much more general topological space. The precise classification of topoLogical spaces which have the form .3K with

K a compact metrizable simplex is given below. We recall that a separable complete metrizable space is called a Polish space. 6.1. THEOREM. A topological space Xis homeomorphic to .3K, where Kis a compact metrizable simplex, if and only if X is a Polish space. Proof. If X is homeomorphic to .3K, where K is a compact metrizable convex set, then since .3K is a of the Polish space K (1.6.6) X must

be Polish (see [48, 9, 6.1]). Conversely, let X be a Polish space and let (T, d) be a compact metric space in which X can be embedded topologically as a dense Ga-subset (cf. [48,6.1]). Hence we can write X an, where the sets are open and decreasing, with G.. = T. Let U be an open subset of T and for each positive integer n put ={x

T: d(T\U, x) l/n}.

of {x: d(T\U, x)> 1 /(n + 1)} covering Choose open subsets and let {Vk} be the sequence of all sets U a partition of unity {gk}c CR(U), subordinate to {Vk} (cf.

[48, 4.3]). Since each Vk has a positive distance from U there exists an with hk =0 on T\U and hk = gk on U. Therefore we have hkE hk

Applying this result to each in place of U we obtain functions CR (T) and constant E (0, 1), with h

-

C)

as k

00,

h

EX'Ii(EMI3 BOUNDARY OF A COMPACT SIMPlEX

127

and diam (supp

for each n. Putting p"k

=

we obtain (*)

n=fl {

If h', is not identically 0 on T\X then it is different from 0 on an open subset of T\X and since X is dense in T, the set K

hZ has infinitely many points. We can therefore choose for each n and k with $0 distinct points Now define y: A'f(T) -.M(T) by on X and n=O k0 and let

EM(T\X)}. in M(T) or equivalently that The next step is to prove that M is If eA'f(T\X) then we have the unit ball M3 is 111.411

because

is a sequence in M1 we can find

E X, and so if

in

A'f(T\X)1

For each n we can, by of = subsequence of find a j, and hence a diagonal of such that is, for each n, argument gives a subsequence w*.convergent to some if E Defining such

00

0C)

P

'-'

L

11

n=O

we see that IIvIl

00

,$

n—Ok—fl

1; in fact if fE CR(T),

PkP

Ii

1, then

n-=Ok—O

and since

= in the

for the

it follows that Iv(f)l then M1 will be

.pflfl 1. If we can show that y(v1)-+

compact and hence

SIMPLEX SPACES

128

M(T\X), we have

For fECR(T), p.

Ifd'y(p.)=

J

>

/

n=Ok=O\J

J

where

=

dp.

J

=

J

-* we have for the a for all n, k. Given I and r >0 there exists a 8>0 such that If(s) —f(t)I < r whenever <5, and for each ii N choose K(n) d(s, t) <5. Choose N satisfying such that <ô for k >K(n). If n > N then diam (supp h so that t) <8 whenever 0; if k > K(n) and n N then the t), same inequalities hold. For such values of n and k we have

Since

and therefore

e f

N K(n)

f /

sej

fdy(p.)—

J

&=O

Therefore the series

converges uniformly on M(T\X)1 to

for all n and k we obtain ff

Jf dy(p.), and since a

d'y(v). This completes the proof that M is

in M(T).

Let /: Mt (T) M(T)/M be the quotient map and let K = 4(MT (T)). If p. E Mt (T) satisfies 4(p.) = p.

for some x E X, then p. = E M, and so if we can find A E MT (T) with A — E Al and A ({x}) =0. However, if

p.1 EM(T\X) with = )'(fL ')= p.' —

A—

n0

((f

+

can have at most of the mass at any single point of X. This A— contradiction shows that the map f ° 'I': X K, where 'I'(x) =8? for x E X, is then

a continuous injection. Moreover, if E is closed in X then E = S X for some compact set S T so that ° =4) ° 'V(S) ri 4) ° 'V(X), that is 4) ° \I'(E) is closed in 4) o 'I'(X). Therefore X is homeomorphic to 4) ° We finally show that 9K =4) o and that K is a simplex. If c then is a closed face of (T) and hence intersects ext Mt (T), so that OK is contained in 4) o 'I'(T). However if t E T\X then t E for some n so that =

n=() k=O

((J

where A k-SO

= s,—A eM,

EXTREME BOUNDARY OF A COMPACT SIMPLEX

129

so that A EM: (T). Since k=fl

4)(A)

can only belong to 8K

Ic -0

if

—

€M,

k—I)

which

is impossible since this measure has no mass in T\X. Therefore it

follows that OK c Let

4)

°

/hEA(X)' and

o'P(X)),

put

Since i' is concentrated on X it will follow that ii =0, and hence that K is a simplex, if we show that v c M = (M0)°. If g E M0 then the function h : K given by h(k) = J7.g d(4 1(k)), is well defined and belongs to A(K). Therefore we have

0= J'K h

= s,q'(x)

h(çb

o

'1['(x))

°

'P(x))

= X

g(x) dv(x)

and so v M. Finally, if 4) ° 'P(x) = w does not belong to OK, for some x E X, then there is a p' E (K) with resultant w so that Sw

1E

M(4) o

But then s,.., = /L1 by the preceding argument, which gives a contradiction. Therefore we obtain OK =4)0 'V(X). Notes

The topological classification of OK, where K is a compact metrizable simplex is given by Choquet in [62, p. 183]. The proof given here is due to Haydon [124]. Some of the techniques used in the proof of Theorem 6.1 have been applied elsewhere. For example, Lazar [148] has shown that for any uncountable Polish space X there exists a bounded closed convex body K in 12 with OK homeomorphic to X.

Andenaes [15] has extended the result of 6.1 by showing that every complete, metrizable locally separable space is homeomorphic to OK for some compact simplex K. A characterization of OK, where K is a standard simplex, has been given by Stacey [199].

SiMPLEX SPACES

130

7. POULSEN'S SIMPLEX—ITS UNIQUENESS AND UNIVERSALITY The following important example was given by Poulsen [173], and is called Poulsen 's simplex. 7.1. THEOREM.

There exists a compact simplex K in j2 with 3K= K.

Proof. We construct simplexes Sn in Euclidean n-space R", embedded in 12 in the natural way. Let

and let

S. = co (S1 u ((0,

R2.

forming a 22-net for S2, and let

Next choose points p3,.. .,

z,=y1+2'e1€R', where

denotes the lth coordinate vector in R'; define = co (S2 U {Z3,.

c R' for 3 I k.

. - ,

Now choose a for Sk and define Sk4.1,... by the same procedure. Evidently each is an n-dimensional simplex and P,tSm = Sn for m n, where If we define is the projection of 12 onto

K = ci U

I

then the construction gives aK= K, and K is compact in 12.

For each x E K, Consequently we have

x and hence f °

-*1 weakly for each f A(K).

A(K)= UA,1 U

=I

where ={foP,1:f€A(K)}. Therefore

to show that A(K)* is a Banach lattice, and hence that K is a

simplex (see 2.5.4, 5.5) it is sufficient to show that An has the Riesz Separation Property, and that is clearly the case because each is a simplex. Recently some remarkable results concerning the uniqueness and universality of Poulsen's simplex have been proved. To present some of these results we need first to consider representing matrices for simplex spaces

A(K).

POULSEN'S SIMPLEX

131

7.2. THEOREM. If K is a compact metrizable simplex there exists an increasing sequence {E,,} of finite-dimensional subspaces of A(K) such that,

for each n, E,, is isometrically isomorphic to some

space, and with

E,,.

It will be sufficient to show that, given a g e A(K) and a there exists for any e >0 a linear subspacc E of A(K) containing L space with d(g, E) < e and such that E is isometrically isomorphic to a space Proof.

Indeed if such a process can be carried out we can inductively take g to be a member of a countable dense set in the positive part of the unit ball of A(K) and a sequence of c's converging to 0, obtaining an increasing sequence of subspaces of A(K) with E1 = lin {1 } and d(g,,, < and such that E,1 is

isometrically isomorphic to a space Clearly we will then have cl U1 =A(K). Suppose then that L, g and c >0 are given, with ugh = 1. Since L is

and 1 is an extreme point of the unit ball S of

isometrically isomorphic to L with

S+1

IIfII2},

it is easy to see that there exists a basis for L with f1 0 for each .1 . . fi,. , and . +ffl =1. Define

byTx=(g(x),f1(x),...,f,,(x)), and let W be the range of T. If q denotes the projection of n coordinates, then

'onto the last

qW={tER': tjo,> Each t E a(q W) is of the form t = qs for some s E W and we can choose a i W such that qW = qW', where W' = finite collection s, Sm co {s1,. . . , Sm}, and such that for each (a, y) W there is a point (a', y) W'

with a — a'I < c. (In fact we can decompose qW into finitely many nonoverlapping polyhedra F1, . . , F', such that and /i2 have oscillations on each P, less than e/2, where h1, qW R are defined by .

= sup {a: (a, y)

W},

/22(y) = inf {a:

(a, y) W}.

Let E be the finite set of points in W of the form (h1(y), y), i = 1, 2, y . . , r. If E' is a finite subset of a W such that co E' can define W' = co B'.) Let A1,.. . , W' [0, 1] such that

SF1, / = 1,.

A(w)s1

co E then we

SIMPLEX SPACL3S

132

and Aj(w)=1

for each w E W', while A1(s1) = 1, 1 f < m. For each w E W\W' define A1(w) = A(w') for some w' E W' such that qw' = qw and p(w') —p(w)I where p denotes the projection of R"1 onto the first coordinate. For each

x€K we will have A.(Tx)=1, and 7"

g(x)--

Aj(Tx)s

<e.

1

We will show, in the lemma below, that these facts imply the existence of functions

lt,...,1mEA(K) such that rn

PU

4=1,

IIljII=1,

j1

and ljs,'O<6.

If L is the linear span of {l,} then it is easy to see that L will satisfy the requirements. The proof of 7.2 will therefore be completed by applying the such that Tx, = s1. following lemma, with u1 = A1 ° T and x E 7.3. LEMMA.

I r

Let/i, g1 E A(K),

c R and

E aKfor 1 I k, I

s and let u1 : K — R be functions such that g1

aquj

each I.

Then there exist functions 4 e A(K) such that g1

Proof. co({xr:

cr4l1

and lj(x,)

Uj(Xr)

for each j and r.

= 1. Since F = If m = 1 we may as well assume that each is a closed face of K we can find an hEA(F) with

h(x,) = u3(x,), 1 r s, and hence the result follows in this case from the Edwards Separation Theorem (2.7.6).

POULSEN'S SIMPLEX

133

Now suppose that m> 1 and that the lemma is true with m —1 replacing m. We can clearly assume that each is either 0 or 1. If aim = 1 we have ni-i —

rn-I

a11u1

—

>

1=1

and hence, if

= 1 also, we have rn—i

g•

g,;

(a11 — cv,'1)u1 I— 1

if a0m =0 we have rn-i

g,,,

The induction hypothesis now gives the existence of li,...

, 4,,

-i E A(K)

satisfying 'n—I 1—1

and tn—i

g10

a,011,_f,0,

lj(Xr)=Uj(X,),

/=1

We can apply Edwards Separation Theorem again to obtain an 4,, A(K) such that, for each i and i' with aim = aim = 1 we have rn—i

rn

a1',!j.

g1— j•'•'l

while lrn(Xr) = Urn(X,) for

1 r s. The functions la,...

,

in, now satisfy the

requirements of the lemma.

In the course of the proof of the last theorem we constructed positive isometries T: with T(1, 1,..., 1) = (1, 1,..., 1). It is easy to see that we can re-order the coordinate basis of to obtain a basis v1,. . , such that, if u1,. . , ii,, is the coordinate basis for we have .

.

'U

Tu, = v, +

Ij

a1, v1,

i='n—l

where a1, 0 and

n,

a1, = 1 for each i = n +1,..., m. If m > n +1 then we

can define, for 1 j

w = v, +

apv,, i—n--2

and Vn-..i.

SIMPLEX SPACES

134

It is easy to check that un {w1,.. . , W,,. d is isometrically isomorphic to and contains Therefore in Theorem 7.2 we can actually obtain spaces E,, isometrically isomorphic to Moreover the isometric embedding T of in takes the form

where a1O, a

and

finite set (f',. . . , f,j in A(K)' a partition of unity for A(K) if = 1 for each i then we refer to

for all x E K; if, in addition, a peaked partition of unity. f1(x) =

1,

7.4. LEMMA. LetKbe a compact ,netrizable simplex, Fa proper closed face of K and {e1,. . . , a peaked partition of unity for A(K). (i) If IIC,IFII = 1, i = 1,. . . , ii, there exists an affine continuous projection 0 from Konto Fsuch that e1 =e1 ° Qforj= 1,..., n.

(ii) Let {f',.

be a partition of unity for A(F) and let a1,... , a,, E I I n. Then if?iK a, =1 and cup =f, K, and if g >0, there exists a peaked partition of unity {gi,.. , gn+ i}for A(K)such that gulF =f,fori = 1,.. . , n + 1, and lie, —(g,+a,g,t. i)Ii<e . . ,

such that

.

fori=1,... ,n. Proof.

(1)

Let 5, E aF be peak points for e,, 1 j P:K-

F by Px

=

e,(x)s,,

n,

and define

x E K;

so P is an affine continuous projection from K into F which satisfies e, = e, ° P for I = 1,.. . , n. Since A(F) is separable there exists a dense sequence {u,,} in A(F)1 (the closed unit ball of A(F)), and hence 2 defines a metric in A(F)* which induces the

on Let E be the p-completion of A(F)* and define 'I' : K (E) (the non-empty closed convex subsets of E) by = P '(Px) F. It is evident that is affine, and also that P : K -* PK is an open map. Consequently, if U c E is open then {x

EK

is also open, so that 'I' is lower semi-continuous. By Lazar's Selection Theorem [145, Corollary 3.4] there exists a continuous affine map Q : K —' E such that Qx for each x in K, while Qy = y for each y in F. Therefore o is a projection from K onto F, and (P ° Q)(x) = Px for each x E K. Finally

we have e1oQ=(e1oP)oQ=e,oP=e1 for

1jn.

SIMPLEX

135

c aK such that e,(s1) = 1. Since F is a proper face of (ii) Choose s,. . K, 3K\F is dense in K and so there exists a point t,,+1 E 9K\F such that .

<e and if we putt1 while if s1€Fwe choose that e,(tj) — e,(s,)1 <. for I I n. Let F' = co (F, . , so that F' is a closed face of K containing By part (i) there exists a projection 0: K F' such that e, ° 0 = for I I n. on F, g,(t,) = e5j1, and g,(x) = We define g1, . . . , gn+1 E A(K) by putting g1 g,(Qx)foreachx inK.Thenforeach g = 1, g,1i=fi, for I I9K\F t1

If 1 such

.

and finally lie,— (g, +

= sup {Ie,(x) — (g,(x) + xcK

=sup X (2K

= max

e,(:,)—(g,+a,g,,.1 I

7.5. THEOREM.

Let K1, K2 be compact metrizable simplexes with

= K1,

/ = 1, 2, and let F1, F2 be closed faces of K1, K2 respectively and : F2 F1 a surjective affine homeomorphism. Then there exists a surjective affine homeomorphism '4' : K2 K1 such that = d'. Proof. Let E A(K1), E A(K2) such that Iu,,II = = 1 for each n = 1, 2, . . . , and such that un ({u,,}) and un are dense in A(K1) and A(K2)

respectively. We will construct inductively peaked partitions of unity for for A(K2),/ m, m = 1, 2,..., and non-negative A(K1) and numbers such that a,,,, = 1 for m = 1, 2,..., with the following properties: I — — ti S e,,,,, —e,,m+i -'-aj.me,n+i,m+i, (V €1 Ji,m Ji.PPT+1 -I-aj,mjm.i,m...i, tor — 1

m=1,2,...;

(ii) (iii)

x E F2, —

1,2,...;

I 1 m, / m, m = 1, 2,... 1 I m, / m, m =

11<2 ', (iii)' fIfi,m

(iv) if = lin (er,,, then, for each n >0, there exists an m >0 such that d(Uk, Em) < 2" for 1 k n, (iv)' if Fm un then, for each ii >0, there exists an m >0 such that

d(Vk,Fm)<2" for =I Firstly we define

Now assume that

have been constructed for k = I,.. , m, k .

and

m so that (i), (i)',

SPACES

136

(ii), (iii), and (iii)' hoLd, and also so that (iv) and (iv)' hold for some integer n. By Theorem 7.2 and the remarks following it, there exists a subspace c such that of A(K1) isometrically isomorphic to d(uk, Em.,-r) <2

numbers partitions of unity

for 1 k n +1. Moreover there exist non-negative = 1, and peaked k = m,. . ., in + r— 1 with for A(K1), k = m,.. . , m + r— 1, such that +

= ef.k

1

/71

1

= Ci.k for k = in +1,... , m + r —1, 1 i k, for k in, and k j m + r.Then(i) and (ii) holdup torn 4-r —1 and (iv) holds for En,+r and Put eLk =

n+l.

is a partition of unity for A(F2) and for 1 in, By 7.4(u), for each €>O there exists a peaked partition of unity for A(K2) such that

Now we have

° cfr}

x EF2,

1,

and —

Hence if e <2_2(m t i.. J4m+i Ji, Ji,m

+ a,,,,tfm+j)II < e

for

and if we define, for I i tm II

tin+i

1 i

rn.

k and 1 k rn +1, —

tm+1 tm--i J1,2 T-al,lJ2,2 ,

it for 1 i k and I k in. follows that < for 1 i k, k j m + r and In precisely the same way we construct k = m + 2,. . . , rn + r —1. Again, by Theorem 7.2 there exists a subspace Fm+r.,., of A(K2), containing Fn,+r which is isometrically isomorphic to <2 for 1 k n + 1. Now construct and satisfies d(Vk, above —

. .

procedure, interchanging the roles of A(K1) and A(K2), and then define +r+s, rn +r+ 1 k +r+s—1. In using k this way we ensure that (1), (i)', (ii), (iii), (iii)' hold up to in + r + s — 1, while (iv) and (iv)' hold for n +1. This completes the inductive construction. 1 I in and m 1. Let e,rn = and = it follows easily from (iv) that the linear span of Because — ej.mlI<

is dense in is dense in A(K1); similarly, the linear span of A(K2). Moreover, using (i), (i)' and (ii), we obtain for I I in and

in = 1,2,.. .

and , e1,,,, +aj,,,iem+i,,n+i, fi,m fi.m(X) for all x E F2. is a peaked partition of unity for A(K1) Since, for each j m,

e,,m(c1'(x))

are linearly independent. and in particular For each m and 1 i in we define Te1,,, = fi,m. The equations obtained above show that T is well-defined on a dense subspace of A(K1). Since for so is

POULSEN'S SIMPLEX

each fixed m un (e1,,,,,.

137 . .

,

is isometric to

it is easy to see that T is an

isometry on this subspacc. Consequently T extends to an isometry from A(K1) onto A(K2), and satisfies Tg(x) = g(q5(x)) for each x e F2 and g A(K1). Since TI = 1 we can now define '1': K2 -÷ by = T*1K2. It is clear that '1' is a surjective afline homeomorphism with = An immediate consequence of the above theorem, taking F1 and F2 to be singletons, is that K1 and K2 are affinely homeomorphic to each other, and

also to the Poulsen simplex. We will denote the Poulsen simplex by 7.6. THEOREM. If K is any compact metrizable simplex then K is affinely homeomorphic to a closed face of Se,.

If we can construct a compact metrizable simplex S with = S and such that K is affinely homeomorphic to a closed face of S, then the result will follow from Theorem 7,5. We may assume that K is a compact convex subset of a Banach space I?. (We can take E to be the completion of A(K)* for the metric p defined in the proof of Lemma 7.4.) We identify K with the subset {(x, 0): x E K} of E x E E x 12, where denotes the nth coordinate vector in and write (0, j2 Using the method of proof of Theorem 7.1 we can easily construct a and simplexes sequence of points in E x j2 with the properties: Proof.

(i) S0=K; for 11=0, 1,2,. .. =co (Zn+i (iii) PnSm = Sn for m n, where I-',, denotes the projection of B x 12 onto E x R"; (ii)

(iv)

c aSm for m

(v) for each e >0 there exists an n such that is an c-net for then, as in the proof of Theorem 7.1, it S = co follows that S is a compact simplex with of S.

=5, and clearly K is a closed face

A consequence of Theorem 7.5 is that is homogeneous. Moreover, every Polish space is, by Theorems 7.1 and 7.6, homeomorphic to a closed subset of aSp.

In particular [0, 1] is homeomorphic to an arc in aS,,. Using Theorem 7.5 it now follows easily that aS,, is arcwise-connected.

Notes

Theorem 7.2 was proved by Lazar and Lindenstrauss [1491. As we noted above, the spaces appearing in 7.2 can be chosen to be isometrically

SIMPLEX SPACES

138

isomorphic to

and the isometric embedding of +

=

and {u1,,,}, {Vk,,,÷1}, 1

are suitable bases of

and

j

into

where

a1,,,

0,

n, 1

k

n+1

takes the form a1,,,

1

respectively. The triangular matrix {a,,,,},

1 j n, n = 1, 2,..., is called a representing matrix for A(K). The

representing matrices for A(K) are not unique, hut in some cases they do determine certain properties of K. For example, Lazar and Lindenstrauss [149] show that K is a Bauer simplex with 0K totally disconnected if and only if A(K) can he represented by a matrix {a1.,,} where, for each n, a1,,,,, = a1,,, = 0 for j for some/n with 1 j,, n.

1,

Sternfeld [201] gives characterizations of Bauer simplexes, and of the Poulsen simplex, in terms of representing matrices. The Uniqueness Theorem 7.5 and the Universality Theorem 7.6, for the Poulsen simplexes, are due to Lindenstrauss, Olsen and Sternfeld [154]. They also prove that OS,, is homeomorphic to 12. The proof of 7.5 uses ideas similar to those of Lusky [155] in proving that the Gurarij space is unique. (A Gurarif space is a separable predual of an L1-space such that for any 8>0

and for any finite-dimensional Banach spaces E F and linear isometry T:E-*G, there exists a linear extension T:F-*G of T with (1—e)IIxII IITxJI (1 + e)IIxII for all x in F.) Some analogies between A(S,,) and G have

been given by Lusky [156].

8. NON-COMPACT SIMPLEXES AND CONVEX SETS

Let K be a convex set in a real vector space E such that K is linearly compact, that is every line in E meets K in a (possibly empty) compact subset of the line. Without loss of generality we can suppose that K is contained in a hyperplane of the form e'(l), and also that un K =E. Let C be the cone generated in E by K, with vertex 0, and partially order E so that

C={xE.E: x0}. In these circumstances we say that K is a simplex if E is a vector lattice. A result of Choquet and Kendall (see Kendall [139]) shows that a linearly compact set K is a simplex if and only if K has the property that, whenever

A 0 and x

E, there exists ,a 0 and y E E such that K (AK + x) = K be a linearly compact simplex embedded in the

partially ordered vector space (E, C).

NON-COMPACT SIMPLEXES AND CONVEX SETS

139

(i) The set = CO (K u —K) is the closed unit balifor a lattice norm in E, such that the norm is additive on C. (ii) If —f, g are bounded convex functions on K with f g, then there exists a bounded affine function h on K with f h g. (iii) If E is complete for the norm in (i) then every closed face of K is split. (i) If p is the gauge for then p is evidently a semi-norm, and since E C: e(x) 1} it follows that p and e coincide on C; in particular p is additive on C. If x E E then we have Proof.

')+p(x) If p(x) <1 then x E Therefore

that x = Ày —(1— A)z with 0 A I and y, z K. and so We have + therefore shown that p(x) = = e(IxI)> 0 if x 0, and that = p(x). The statement (i) now follows If we define f(ax) = af(x), g(ax) = ag(x) for a 0, x E K, then —f, g are subadditive positive homogeneous functionals on C. The methods of so

Theorem 2.4.2. can now be used to obtain the required h. (iii) If E is complete for the norm in (i) then E is an L-space (2.6.2) and hence is an order-complete vector lattice. If F is a closed face of K then the continuity of

the lattice operations in E implies that un F is an order-complete lattice ideal in E, and consequently there exists an ideal L in E such that E is the order-direct sum of un F and L (see Peressini [168, p. 39]). If F' = L K it is now easy to check that F and F' are complementary split faces of K. Statement (i) of Theorem 8.1 does not hold in general for non-simplexes, as the example (2.1.6) shows. Statement (ii) is a simple analogue of Edwards

Separation Theorem for compact simplexes (Theorem 2.7.6), but this property does not characterize linearly compact simplexes. In fact if E consists of the real polynomials on [0, 1] and if K = {fe f(t) dt = then E possesses the Riesz Decomposition Property and so statement (ii)

holds; however since E is not a vector lattice K is not a simplex. We now give an example to show that statement (iii) may fail if E is not a Banach space. 8.2. EXAMPLE.

Let E = CR[0, 1], K ={fe E':

F=jf€K: IJo f(t)dt=0

f(t) dt =

1)

and let

-

Then Kis a linearly compact simplex andFis a closed face ofKfor the induced norm in E. However, F is not a split face of K.

All the statements, except the final one, are evident. Suppose that F =0 for all is a split face of K with complementary face F'. Then, since Proof.

SPACES

140

f€ F there must exist a u E F' with u(4)> 0. But then we can easily decompose u = Ag + (1 — A )h with 0
face we conclude that g E F F'. This contradiction shows that F is not a split face of K.

Let K be a linearly compact convex set, not necessarily a simplex, but embedded in E such that>: = co (K u —K) is linearly bounded, that is every line in E intersects>: in a bounded (possibly empty) subset of that line. Then the gauge of>: is a norm forE which is additive on C, and the dual space E* is

order and isometrically isomorphic to the Banach space Ab(K) of all bounded affine real-valued functions on K, with the supremum norm. If>: is the closed unit ball for E then the proof of Theorem 2.7.2 shows that K is a simplex if every closed face of K is split.

We now illustrate, in a very special case, some results concerning the structure of Banach spaces. Let K be a linearly compact convex set such that>: is the closed unit ball in E for a complete norm in E. A linear projection P E E such that o P I, for the pointwise ordering, will be called an L-projection, and the range of an L-projection will be called an L-ideal. 8.3. THEOREM.

(i) 1 is an L-ideal in E if and only if I = un Ffor some split face Fof K.

(ii) The L-projections on E form a commutative subset of B(E) the bounded linearoperators on F, and they are precisely the extreme points

of the set {T€B(E): 0 (iii) If {I,,} is a family of L-ideals in F then fla ía and Ua are also L-ideals in E. and (iv) If is a family of split faces of K then fl,, Fa are also split faces of K.

(i) Let F be a split face of K with complementary face F', so that F is the direct sum of un F and lin F'. If P is the projection of E onto un F then Proof.

it is clear that 0 P 1, so that un F is an L-ideal.

Conversely, let P be an L-projection and let F = K P(E) and F' = K (I — P)(E). Then since 0 P I and since un F nun F' = {0}; it is easy to check that F and F' are complementary split faces of K.

(ii) Let P1 and P2 be L-projections on E, and let F, = K n P1(E) for j = 1,2. Now F1 nF2 is a split face of K (see the proof of 1.1) and so there is an associated L-projection Q. If x K then x can be uniquely decomposed as

x=

4

>: A,y,,

4

whereA10,

>: A,=1

j-=1

NON-COMPACT SIMPLEXES AND CONVEX SETS

141

and

But then Qx =A1y1 =P1P2x

P2P1x,

so that the linear operators Q, P1P2 and P2P1 agree on K, and hence are equal. Therefore, since clearly IIP1I 1 for any L-projection P 0, the L-projections form a commutative subset of B(E). Let A ={T E B(E): 0 T I} and suppose that P is an extreme point of A. Then P2 E A and also

2P—P2——P+P(I---P)EA.

The equation P =

P

P2

P=

P for some T1, T2 E A. Then we have

P=

+

and 0=

—

P)T1 +

—

P)T2,

and because PT, P and (l—P)T1 0, forj = 1, 2, it follows that T, = P7 = P for / = 1, 2. Consequently P E (iii) Let {Ia} be a family of L-ideals. To prove that fla Ia is an L-ideal we can assume that the family is decreasing, because the intersection of finitely many 4 is an L-ideal (using (i)). If {Pa} is the corresponding family of decreasing in C for each x C; since E is L-projections we have is norm-convergent to a limit complete and the norm is additive on C, converges strongly to an L-projection P such Px E C'. Therefore the net that P(E) = fla Ia. Moreover the complementary projections The final part of I P and (I — P)(E) = cl Ua (I— (iii) now follows. (iv) The required facts follow by direct verification using the results in U) and (iii).

By the above result the L-projections on E generate a commutative Banach subalgebra of B(E), with identity I. This Banach algebra is denoted by and is called the C'unningham algebra for E.

An operator T E B(E) is said to be order-bounded if —Al T Al for some constant A >0, and we denote infimum of such A by 1T111. It is easy to

see that

is a norm and that = sup

We

x K} 11Th1.

can identify A"(K) with A(K1) where K1 is the state space of the

complete order unit normed space A"(K). Suppose that T E B(E) satisfies

of T (1— a)I, where 0
SIMPLEX SPACES

142

x€3K1 we have a

—a). Putting A =(1'*l)(x) we obtain where y, z K1 satisfy

x = Ày + (1 — A )z Af(y) = (T*f)(x),

T*)f)(x) for all fE A (K1), and hence x = y = z, that is (T*f)(x) = (T*1)(x)f(x). Consequently for any order-bounded T E B(E) we see that (T*f)(x) = (T* I)(x)f(x) for all x E fE A(K1). (1 — A

)f(z)

((1—

Therefore I1TII= IIT*II= sup {I(T*1)(x)j; x

=inf{j9>O: =inf{f3>O:

It is now clear that every T E

is order-bounded. The converse

inclusion and a representation of 't9'(E) is given in the next theorem.

coincides with the order-bounded operators on E, and is order and isometrically isomorphic to C'R(fl) for some hyperstonean space fl. 8.4. THEOREM.

Let 0(E) denote the order bounded operators on E. Then the above reasoning shows 0(E) is a complete Banach algebra with identity 1, and that I is an order unit defining the norm in 0(E). Moreover, if S, T E 0(E) then T*S* are order bounded on A(K1), so that for fE A(K1), x E aK1 Proof.

(S*T*f)(x)=(S*1)(x). (T*f)(x) = (S* I

(T*1)(x) . 1(x) = (T*S*f)(x).

Therefore S*T* =T*S*, so that ST TS and 0(E) is commutative. It follows that 0(E) is order and isometrically isomorphic to GR(11) for some compact Hausdorif space (1 (cf. Kadison [136]). Arguing as in the proof of 8.3(iii) we see that 0(E) is boundedly order

complete, and hence Ills extremally disconnected. The L-projections in 0(E) correspond to functions where G is open and closed in Il, and linear combinations of these functions will separate points of (I, and will belong to Therefore the Stone—Weierstrass Theorem gives 'e(E) 0(E). For each x K we can define a probability measure on Il such that, identifying 'e(E) with Ck(fl), 1(Tx),

If {Ta} is an increasing net in

TE

with supremum T then Tax converges

NON-COMPACT SIMPLEXF.S AND CONVEX

143

strongly to Tx for each x E K, and so = 1(Tx

—

T4,,x)

IITx

0.

—

Thus each is a normal measure on Ii and, since these measures separate the functions in CR(fl), the space (I is hyperstonean.

The algebra C(E*) of order bounded operators on = A"(K)= A(K1) is a commutative Banach algebra with identity I and, for TE C(E*), 11Th

= 11Th1 = inf

{A >0: —Al T Al].

We can also deduce that the map 4) : A(K1) defined by 4)(T)(x) for xc is an algebraic, order and isometric isomorphism of C(E*) on to the centre of A(K1). For this reason (Y(E*) = C(Ab(K)) is called the centralizer of Ab(K).

is an algebraic, order and isometric The mapping isomorphism of the cunningham algebra onto the centralizer C(A"(K)). The centre of A(K1) = A"(K) is w*_closed. 8.5. THEOREM.

If T E 'e(E) then —A! T Al for some A >0, and hence —Al T* Al so that T* E C(E*). If we can show that every $ E C(E*) is of the form S = T* for some T then the first statement of the Theorem will Proof.

follow. It is obvious that A b(K) is boundedly order-complete, and consequently

C(A"(K)) is boundedly order-complete. Therefore C(A"(K)) is the closed linear span of the projections P E C(A"(K)) such that 0 P < !, and so it will

be sufficient for the first statement to show each such projection P is on A"(K). Hence it will be sufficient to prove that and (I —P)(A"(K)) are

Let Q€C(A"(K)) be a projection such that 0 Q

and let L

Q(A"(K)) ri A, where A is the closed unit ball of A"(K). If is a net in L which is to fEA then —1 f. 1, so that Q(fa) Q(1) and consequently —Q( I) Q(1) since (Ab(K))+ is Therefore we have 0=(I — Q)(—Q(1)) f—Qf (I— Q)(Q(1)) = 0,

so that Qf =f and L is that C(A"(K)) is statement is proved.

The

Theorem now implies

Putting Q=P and 0 = (I—F), in turn, the first

SPACES

144

Theorem 8.4 and its proof show now that U(A"(K)) can be identified with CR(ft) for some hyperstonean space fl, and that the measure on f, given by ,LX(T*) =

(T*1)(x),

is a normal measure for each x E K; thus functional on for xEE, then

is a linear for every x E K, and hence for every x E. If ir(x) = ,Lx(T*) = (T*1 )(x)

for T*

so that T*1. Since IIT*lI

= UT* III for all T* E C(A1'(K))

we obtain {T*1: T* E C(A1'(K)), I1T*

1

}

= ir*({T* E C(A"(K)): IIT*II

I }.

the is Theorem implies that the centre of A"(K), that is {T*1: T* C C(A"(K))}, is

Finally, because

We conclude this chapter with a brief mention of some important recent work concerning versions of Choquet's Theorem. A Banach space B is said to have the Radon—Nikodym property (RNP) if whenever (X, p.) is a probability space and m : Si-> B is a countably additive measure (convergence in norm) which is absolutely continuous with respect to p. and has bounded variation, then there is a Borel-measurable function F: X B such that m(E) = SE F dp. for each E E where the

integral is a Bochner integral. For example, every weakly compactly generated Banach dual space has the RNP. Every Banach space B with the RNP has the Krein—Milman property, that is every bounded closed convex subset of B is the closed convex hull of its extreme points. The converse is true for Banach dual spaces but is unknown in general. If K is a bounded closed convex subset of a separable Banach space then ÔK is universally measurable, but need not be a Borel set.

8.6. THEOREM. Let K be a bounded closed separable convex set in a Banach space B having the RNP. The following statements hold. (i) For each a K there is a probability measure p. on K with p.(äK) = 1 and JKX dp.(x) = a (as a Bochner integral). (ii) K is a simplex if and only if for each a K there is a unique probability measure p. on K with p.(3K)= 1 and SKX dp.(x)= a.

NON-COMPACT SIMPLEXES AND CONVEX SETS

145

Notes

The first part of Theorem 8.1 was proved by D. A. Edwards [87]. Parts (ii) and (iii) appear in Asimow and Ellis [32]. Theorems 8.3 and 8.4 are due to Alfsen and Effros [8] and to Wils [216]. in their fundamental paper [8] Alfsen and Effros develop a structure theory for real Banach spaces. There is a very extensive literature extending the results of Alfsen and Efiros, in particular to complex Banach spaces and to Lindenstrauss spaces. We mention only the paper of Lima [152] which develops some connections between L-ideals, M-ideals and intersection properties of balls in Banach spaces.

Some applications of the structure theory to function algebras and to will be discussed in Chapters 4 and 5. Some applications to other Banach algebras have recently been given by Smith and Ward [196, 197]. Theorem 8.5 is due to Alfsen and Effros [8] and to D. A. Edwards [90];

see also Cunningham et a!. [69].

Interpretation and application of the above material in quantum mechanics has been given by C. M. Edwards [85,86]. Much more informalion concerning the RNP in Banach spaces can be found in the books of

Diestel [78] and Diestel and Uhl [79]. An example of a bounded closed convex subset K of a separable Banach space such that äK is not Borel has been given by Jaync and Rogers [135]. Theorem 8.6 is due to Bourgin and Edgar [49, 83, 84]. The same authors have also obtained more technical results in the non-separable case. In this context Mankiewicz [158] has shown that if K is a closed, bounded convex subset of a Banach space X with the RN P, then each y K is the resultant of a separably-supported complete Borel probability measure a on K with the property that p.(ä(K Y)) = 1 for every closed separable subspace Y of X containing supp

CHAPTER 4

Complex Function Spaces

0. THE COMPLEX STATE SPACE Let A be a complex Banach space and let S be a

convex subset

of A?. Let

Z=co(Su—iS)

B=.co(Zu-Z) N = U nB

(the real linear span of Z)

and consider A0(B), the space of real affine homogeneous functions on B, with the supremum norm II/1i=sup{If(x)I: x EB}=Sup{If(x)I: x EZ}.

Let 0: A

Ao(B);

(Oa)(x) = re (a, x).

identifies the real dual Then 0 is a real linear map with dense range and A0(B)* with (N, B) (cf. Chapter 2 Section 8, for example). Now, Theorems 1.3.5 and 2.8.1 yield the following result. The following are equivalent: (i) 0 is a real isomorphism with (ii) NiccB (Ni={x€N:jIxjI1}). a E A then A0(B) is isometrically If S is contained in

0.1. THEOREM.

isomorphic to A (Z) since S c a '(l) implies Z [0 (a + ia

(1).

Example 1. Let 1 E A be a closed separating subspace of C(X), X compact Hausdorif, and let S be the state space of A. Then, by a simple computation,

ball

ha jj v'2110a11 146

TUE COMPLEX STATE SPACE

147

that Theorem 0.1 shows A is real isomorphic to A(Z), Z co (S u —iS),

so

the complex state space.

example 2. Let A be a complex unital Banach algebra and again, take S to be the state space arising from the unit u of A. A theorem of Bohnenblust and Karlin [43] shows that (i) holds with c = '/2e. The fact that (i) implies (ii) in this case was proved by Moore [162].

To illustrate the utility of the complex state space we demonstrate the algebraic significance of S being a split face in Z. Thus, as in the above examples, let A be real isomorphic to A(Z). We denote

H(A) = {a A: (a, s) real for all s

S}.

These are the real functions in A in Example 1, and, in Example 2, are called the Hermitian elements of A. The following are equivalent: (i) S is a split face of Z; (ii) A

0.2. THEOREM.

Proof. Since Z = co (S u —iS) and both S and —iS are closed we see that S is split in Z if and only if S and —iS are complementary split faces. Thus, we can apply Theorem 2,10.5(1) which states that S and —iS are complementary split faces if and only if

A1 ={Oa:

A2 ={Oa: OaI_js

But Oa E A2 if and only if im (a, s) =0 if and only if a E H(A), and similarly Oa E A1 if and only if ia E H(A). Thus the equivalence of (i) and (ii) follows. In the case of Example 1 we say A is seif-adjoint if Theorem 0.2(u) holds, in which case 11(A) = re A.

In Example 2 this provides a geometric version of the Vidav—Palmer if and Theorem [46] which states that the Banach algebra A is a only if A = H(A)Ei?iH(A). One is sometimes only interested in the real linear span of the state space 5c. A*. This is denoted H(A*) and is identified as usual with A(S)*. This induces the norm on lix us

= inf {a +

x = as1 — /3s2; a,

0, s1

S}

in Theorem 2.2.3). If H(A*) is a real dual L-space the representation of —5) (for lix Ii = 1) is unique. Grothendieck [123] showed that this is also the case if A is a C*_algebra, But Sinclair [195] noted an instance (as

x E co (S

COMPLFX FUNCTiON

148

where the uniqueness failed fo hold with A a unital Banach algebra. Thjs however. In the uniqueness property does not characterize commutative case any Dirichiet algebra A (one in which re A is dense in gives rise to a simplicial state space and hence has H(A*) a dual L-space. Of course, A need not he seif-adjoint, considering, for example, the disc algebra. Notes The complex state space was first introduced by Asimow [24] in the context of Example 1 in order to discuss geometric peak-point conditions related to the Bishop Theorem for function algebras. We discuss this in greater detail in Section 4. The formulation of Theorem 0.1 and its application to unital Banach algebras appears in Asimow and Ellis [32].

The theorem of Bohnenblust and Karlin as well as the developments leading to what has come to be called the Vidav—Palmer Theorem are discussed in Bonsall and Duncan [46].

1. COMPLEX REPRESENTING MEASURES

Let X be a compact Hausdorif space and let C(X) denote the space of continuous complex-valued functions on X. We let M(X) denote the dual space of regular (complex) Borel measures on X. As we have seen, the real subspace of M(X) of real measures is an L-space. The order structure also applies to a complex measure by means of the total variation measure, 1i'given by

(for f0).

If E is a Borel set then a Borel partition of E}

=sup

(see Rudin [185]). It follows that 11i4 =

= p4(X). Also

where h is a Borel function on X such that =

1

(a.e. p4).

Finally, if p is a Borel map from X to Y then for each

PEM(Y) given by

and

for all bounded Borel functions g on V.

M(X) there is a

COMPLEX REPRESENTiNG

149

Let A be a closed separating subspace of C(X) containing the constants. We identify X with its embedding in the state space and say E M(X) is a boundary measure if either = 0 or is a maximal probability We denote boundary measures by measure on aM(X) if A is understood. We let S denote the state space of C(X), in other words, the probability measures on X, so that aAS (not to be confused with ext S) denotes the maximal measures on Xc SA. The Choquet—Bishop--de Leeuw Theorem asserts that for each L E Sn there is a (maximal) aAS such that

L(f)= for all fE A. This measure is unique if and only if SA is a simplex. In this section we derive the complex version of this result: for each L E there is a (complex) E aM(X) such that = IILII

and

L(f) = for all fE A. We also discuss the uniqueness question. This is accomplished by identifying complex measures on X c SA with real measures on TX c where

T={tEC:ItI=1} and TX={tx: t€ T,XEXCSA}. To do this we identify C(X) with the subspace M in C(X x T) given by Let ./

E

M = {f E C(X x T): f(x, t) = f(x, 1) = f(x), for all t e T}. C(X x T) be given by 4(x, t) = t and define

by'l'f=4f. Since

1, 'P is an isometric-isomorphism and so is

M(X x

M(X x T);

= ji

with

=

d1.Z.

Clearly We note that the restriction of ji E M(X x T) to the subspace M defines an element E M(X). In fact it is easy to see that X x T -* X

We now define, for /2 E M(X x T), H12

with ir(x, t)

x.

150

COMPI..EX FUNCTION SPACES

Thus

d/h=4d(4oTr I). It follows that for f a bounded Borel function on X,

Jf(x)

=

tf(x) d12 (x, t).

f

Let 4 be a probability measure on X x T and let

1.1. Ff4.

(x)

If

Proof.

represents L c A* with IILII = 1 then

It is straightforward that

1= IILII = sup

161 °

=4

Now

°

{jL(f)1: fEAr} =sup

1.

Hence I,.LI(X) =4 a ,r'(X) so that equality must hold for any measurable

subset E of X.

if

= h dIp! (Ihi = 1) we can invert Ii as follows: let

E M(X) with

(Ih(X)(X,h(X)) so

that the graph G of h, given by G ={(x, t):

is a Borel set. We consider

t=h(x)}{(x, t): Ih(x)= 1} M(X X T).

°

LetL e A* with IILI1 = 1 and E M(X)1 representL. = H4 for 4 a probability measure on X x T if and only if

1.2. PROPOSITION.

Then

/L

Proof.

e M(X)1

Given

—I

/.4

0Ø'i,

representing L and 6 =

we have, for

I

fE M,

(H4)(f) = J'

XxT

tf(x) d4(x, t) =

=

Thus H4 =

XxT

(4'f)

o

)=

('I'f) ° X

j(x) =

If H4 =

representing L, then Proposition 1.1 shows —1

Hence

1 =I/4(X)=J'

t)

dIpI

COMPLEX REPRESENTING MEASURES

151

sothatth(x)= 1 a.e. Bore! function on X x T

1. Butthenifg isabounded = fg(x, h(x))

Io

f

= XxT

Let U denote

g(x, t) djZ(x, t)

= XxT

g

and observe that

ext U=T' (extSA), a subset of the compact set TX which is homeomorphic to X x T. An element fE C(U) is called invariant if f(tx) = f(x) for all t E T. dt, where dt denotes normalized linear Lebesgue Define mv f(x) = measure on T. Recall (Theorem 1.6.5) that a probability measure on U is

maximal if and only if /h(Bf) = functions on U), where

1

for all fE Q(U) (continuous convex

B1=B(/)={xe U: 1(x) =f(x)}. denote the measure on U given by

Let

= 1.3. PRoPosITIoN.

Letf E C(U) and /h E MT(U).

(i) mf is invariant and if f is invariant, mt = f. (ii) 1ff is convex then so is invf. (iii) If f is invariant then so is f. (iv) represents x then represents tx.

(v) If/is convex then B(inv f) c B(f). Proof.

(i)—(iv) are straightforward. For (v) we recall (Theorem 1 .6.1(vii)) /h

eMI(U) represents x}

and, since f is convex, we have (jh

Let 1(x) —f(x)> 0 and choose /h representing x such that

/h(f)—f(x)>0. If g: T -, R is given by g(t)

COMPLEX FUNCTION SPACES

152

then g is continuous with g 0 and g(1) >0. Hence

dt = L

0

d/h(y) dt —invf(x)

= /h(inv f) — mv f(x) (mv f)A(x) — mv f(x).

is a probability measure on X x T we can regard ji as a measure on the Since ext U c TX, any maximal measure on U is supported by TX. If

homeomorphic image TX c U =

1.4. PRoPosmoN. Let j.Z be a probability measure on TX and let = H1.Z. Then (i) (ii)

Proof.

ft

is maximal on U if and only

oir' is maximal on SA.

IffEA then d4 =

ff(tx) d12 (x, t) =

tf(x) d/Z (x, t) = ft d(Hj2)

= Li d/.L.

If f is an invariant Borel function on U then

ffd1z = ff(x) djZ(x, t) gives

f is continuous and convex on U then Proposition 1 .3(u) and (iii) 4(B(invf))=4 °

Hence by Proposition 1.3(v) 4 is maximal on U if and only if ji oir'

is

supported by X c SA, a face of U, so that 4 olr' is maximal on U if and only if it is maximal on SA. maximal with support in SA on U. But 4

is

1.5. THEOREM. LCtLEA* with 11L11 1. (i) There is an element/h E 0AM(X)1 such that /h represents L. (ii) The measure in (i) is unique in the boundary measures of norm one if and only if L is represented by a unique maximal probability measure 4 on Given L E A* with IIL1I = 1 we can choose, by the real Choquet-Bishop—dc Leeuw Theorem a maximal 4 on TX C U representing L. Let = H4. Then p = M(X)1 and represents L by Proposition 1.4(i). Since is maximal on SA by Proposition 1.4(u) it follows from Proposition 4° 1.1 that is maximal. Hence j.i. is a boundary measure. The =4 uniqueness equivalence of statement (ii) now follows from Proposition 1.2. Proof.

COMPLEX REPRESENTING MEASURES

153

We say the subspace A c C(X) has the uniqueness property if each L E A* represented by a unique boundary measure with We charac.. = terize this in terms of the geometry of in the following corollary. Further

applications of uniqueness to Lindenstrauss spaces will be discussed in Section 9. We call a (not necessarily closed) proper face F of a simplex if the cone spanned by F in A* is a lattice. This is of course consistent with the usual definition of simplex in case F is (Chapter 2, Section 7). Since F is proper,

Fc{x eAr: IIxIi= l}. 1.6. COROLLARY. The subspace A c C(X) has the uniqueness property if and only if each proper face of A1c is a simplex.

Proof. Let U = Ar and let F be a proper face of U. If A has the uniqueness property then Theorem 1.5 shows the resultant map is one-to-one from a face of the maximal measures in MT(U) onto F. But the maximal measures themselves form a face of the simplex MT(U) (see the remark preceding

Theorem 2.7.3) so that F is affinely equivalent to a simplex. For the converse let lix Ii = 1 in U and let F be the (algebraic) face of U generated by x. Then

Fc{yc U: IIyiI= l} is a simplex. We show that maximal measures on U representing x must agree on the cone Q(U) of continuous convex functions in Since the linear span of Q(U) is dense in CR(U) this will establish the uniqueness property, by Theorem 1.5(u). Thus let fE Q(U) and let be a maximal measure representing x. By Theorem 1.6. 1(viii) there is a —g E Q(U) with

fg and

f gd/h0. Since the cone spanned by F has the Riesz decomposition property it follows from Chapter 2, Section 4 (cf. Theorem 2.7.3) that flF, given by sup { fA1f(y1): y

is affine on F. Similarly,

F

is affine and

onF.

COMPLEX FUNCTION SPACES

I j4

Thus, using Proposition 1.6.2 and Theorem 1.6.5 we have

1(x)Jfdi2 Since >0 is arbitrary we have

f

=/(x).

Notes

The Representation Theorem (Theorem 1.5(i)) is referred to as the Hustad Theorem. Hustad [132] constructed the map H and used it to obtain the representing measure Hirsberg [127] showed that in general H11 is a boundary measure if Z is maximal on The construction of the inverse map in Proposition 1.2 is due to Fuhr and Phelps [112] who proved the uniqueness statement Theorem 1 .5(u). They also proved the geometric version of the uniqueness property in Corollary 1.6. Unit balls with that property are referred to as simplexoids. Many of the techniques herein employed, and in particular those of Proposition 1.3, are due to Effros [96]. For the analogous theory for subspaces A not containing constants we refer to Phelps [172].

2. INTERPOLATION SETS Let A be a closed subspace of Cc(X) and let E be a closed subset of X. If '1' is

the restriction map from C(X) to C(E) then the Tietze Extension Theorem shows 'I' is onto and in fact maps the unit ball U of C(X) onto the unit ball V of C(E). Then embeds (M(E), V°) onto the subspace (N, are (M(X), U°). If then In, = = complementing projections with N = range in. Let 9 = 'VIA and denote range 0 by AlE. We say E is an interpolation set for A if is closed in C(E). Since 'VU = V, we have a subset B of C(E) is closed if and only if \V '(B) is closed in C(X). In particular A is closed if and only if A +M is closed in C(X), where

M = ker 'V = {fE C(X): lIE If qx,

0}.

denote the quotient maps qx :M(X)-+M(X)/A1 =A* = :

then

=

0

INTERPOLATION SETS

Let U =qx(U°),

155

= qxN. Then

)Q

'IE(V) with

identifies I

We can now formulate Theorem 1.3.5 in this context.

2.1. THEOREM. Let E be a closed subset of X and let A be a closed separating subspace of C(X) with constants. The following are equivalent: (i) E is an interpolation set for A; (ii) (A + M) (U + M) c r(A U + M) for some r 1;

r 1;

(iii) (A + M) (mt U + M) c r(A (iv) Nis w*_closed in A*; (v) N is norm-closed in A*;

forsomer1.

(vi)

In particular, if E is an interpolation set then for each g E and a> 1 (using (iii)) there is an f E A such that flE = g and HflIx Another formulation of 2. 1(vi) is useful. By taking inverse images under qx we see that (vi) is equivalent to

(U°+A or

N Since 0*0

=

+A-

a

Nil =

the last statement is the same as + (AIE)111

+A111

for all

E M(E).

The following are equivalent: (i) E is an interpolation set for A;

2.2. COROLLARY. (ii)

r 1 and all

+

(iii) lIirim + (AIE)'ll Proof.

EN;

rllir2tnll for all m E A'.

As noted above, (ii) is a reformulation of 2.1(vi). If (ii) holds and

m€A1 then llirim + (AIE)11 rllirim +A'll

= rIl—ir2m +A111

rum — ir2m +A—II

rjI'n.2m11.

If (iii) holds and €N, m E A, then +mfl=fl'7r2m +(irim + (AIE)111 + (1/r)llirim + + irim) — (grim + (AlE) ')ll = Hence

+A111Il,z +(AIE)ii.

+ (AlE)' II.

COMPLEX FUNCTION SPACES

156

We say E is a full interpolation set for A if AlE in fact equals C(E). This is equivalent to E being an interpolation set for which (AlE)1 = {O} in

N=M(E). The following are equivalent: (i) E is a full interpolation set for A;

2.3. COROLLARy.

EN;

(ii) +A111 for some r 1 and (iii) Iliri mu rIIir2mII for all m E A1.

Proof.

The above remark shows (i) implies (ii). The proof of (ii) implies (iii)

is the same as in Corollary 2.2. If (iii) holds then, in particular, E is an interpolation set. If E (AlE)1 then E A1 N and IIILII

so

rlllr2Ihll = 0

=

that (AlE)1 = {0} and E is a full interpolation set.

We will say a subset B of M(X) is A-stable

if by &AB so that

qx(B

aAM(X)). We will denote B if and only if

= qx(B) B is A-stable

Hustad's Theorem shows U° is A-stable. If, in addition, N is A-stable then

we can modify the measure theoretic conditions of Corollary 2.2 and Corollary 2.3 by restricting attention to the boundary annihilating measures, aA1.

2.4. COROLLARY. Let N = M(E) be A-stable. Then the following are equivalent: (i) E is an interpolation set for A; r 1 and all m E (ii) llirim + (AIE)111 Proof.

If N is A-stable then for

E aM(X), =

÷aAil

since

/.L=sv+m;

v€U°,

mEA'

v'EW,

m'EA.

implies, by Hustad's Theorem, that

p.=sv'+m'; Since

ii' E

t9M(X), it follows that m' ÔA1. Thus, if E is an interpolation

INTERPOLATION SETS

157

set then Corollary 2.2 shows + (A IE)II

+

for all E Thus (ii) follows form E since ir1m E oA also. Conversely, if (ii) holds and x E N U, choose p, E oN with = x and let m be any element of M1. Then the computation of Corollary 2.2(iii) implies 2.2(u) shows + mu

+ (AIE)'91

and hence

<

+

Thus x

r(U°

=

+

so that

Cc and E is an interpolation set.

If N = M(E) is A-stable then the following are

2.5. COROLLARY.

equivalent: (i) E is a full interpolation set for A; (ii) IIirimlI rjIir2mII for some r 1 and all m E OA'.

Let E' = M A, where M = (fE C(X):

= O}. Then is the (complex) subspace of A* spanned by E as a subset of SA c A*. We define the A-hull h(E) to be

E X:f(x)

= 0 for all f€

Similarly, the A-convex hull k(E) is given by

k(E) c h(E). 2.6. PRoPosrrloN. Proof.

for allf€A}.

The set k(E)={x EX:

The Separation Theorem shows (using rotation by t T)

k(E)={x c-X: ref(x) sup ref(y) for allf€A} y€E

={x €X:f(x)€ëöf(E)forallf€A}.

Thus x k(E)andf€A implies f(x)IIIfIIE. If

k(E) then

for

COMPLEX FUNCTION SPACES

158

some fE A. Hence there is a disc D(a; r) about a C of radius r such that

f(E)cD(a;r)

and

If t T is chosen so that t(f(x) — a) 0 and g = t(f— a) then g(E) c D(0; r) with Thus Ig(x)I > IIgIlE.

We close this section by observing some instances where N =

M(E)

is

A-stable. 2.7. PROPOSITION. If E is a closed interpolation set forA such that either (i) E aAX, or (ii) E = F X, Pa closed face of SA, then M(E) is A-stable.

Also, (ii) implies E = k(E).

If (i) holds then each element of MT(E) is maximal (cf. Theorems 1.6.3 and 1.6.5) so that M(E) C aM(X). If (ii) holds and is an clement of

Proof.

representing x F the M(E) (Proposition 2.10.6). Since (M(E)) is spanned by F we have M(E) is A-stable. Notes

As indicated in Chapter 2 the closed range conditions of Theorem 2.1 have been recognized in various forms by many researchers. The seminal work in this regard is contained in the 1962 paper by Glicksberg [118]. In particular he gives the equivalences of Corollary 2.2(i) and (ii) as well as Corollary 2.3. The equivalence of Theorem 2.1(i), (iv)—(vi) appears in Edwards [87] for the face of K. Alfsen [3] also realA(K) case where N is spanned by a

discusses this situation and exhibits the constant r in statement (vi) as a bound on the norm of extensions. General formulations of this principle for A(K) and for complex function spaces are given in Asimow [23,26].

The formulation of Corollary 2.2(iii) in the general case is given by Gamelin [113]. The use of boundary measures of course awaited the maturation of Choquet theory and first appears in Alfsen—Hirsberg [9] where stability is guaranteed by the hypothesis that E is a compact subset of aAX (cf. Proposition 2.7(i)). They prove (ii) implies (i) in Corollary 2.4 for the case r =0. This very important case will be discussed in detail in Section 4. Corollary 2.4 is proved in its present form by Briem [50] where the same stability condition (E c aAX) is assumed. The alternate stability condition (Proposition 2.7(u)) appears in Hirsberg [126]. Further abstract versions of the closed range conditions can be found in

Andô [19], Roth [182] and Asimow [28, 29]. Roth [182] also discusses conditions involving boundary measures in a quite general setting.

GAUGE DOMINATED EXTENSIONS AND COMPLEX STATE SPACE

159

The equivalence of the geometric definition of k(E) and the statement of Proposition 2.6 has been well known in the context of function algebras. It is shown in Alfsen and Hirsberg [9] for the case &i E a parallel face of SA and in general by Briem [50]. Examples of closed subsets that are not interpolation sets are plentiful.

We show, for example, in Section 6 that for A the disc algebra and E a proper closed arc of T we have AJE dense in, but not equal to, C(E).

3. GAUGE DOMINATED EXTENSIONS AND COMPLEX STATE SPACE

If E is a closed interpolation set in X for a subspace A c C(X) then Theorem 2.1 shows there is an r 1 such that each g E is the restriction of an fe A with
pr:X*IR

(lonE

+ ;

tronX\E

,

(AIR) with

< 1 on

r1

then Pr is a strictly positive bounded l.s.c. function on X such that if IgI
on

called the extension constant, e(A, E), which we will discuss in detail in Section 6.

In this section we introduce machinery for dealing with questions of extending gauge-dominated functions on E to X and take a closer look at complex state spaces in this framework. To obtain results of greater generality we can require the dominator, p, to be sensitive to argument as well as modulus in the complex plane. Thus, let

(T={zEC:IzI=1}) be a strictly positive l.s.c. function and let

U ={fE C(X): re tf (x)p(x, t); (x, t)EXx T}. We will denote the Minkowski functional p(U) by JJ

Thus

= sup {re tf(x)/p(x, t): (x, r) E X x T}. Since p is strictly positive and l.s.c. p is bounded away from 0. Hence there is

a constant k such that where JJ. J denotes the usual uniform norm in C(X). We shall refer to a generated in this way as a gauge on C(X).

1

If in addition p is bounded above then is equivalent to 1. II. In general U is a closed convex neighbourhood of 0 so that is sub-additive and positive homogeneous.

COMPLEX FUNCTION SPACES

160

We say the p-interpolation problem has approximate solutions for A, E if for each g E (AIE) and e >0 there is anfE A with fIE = g and + Here refers to PJEXT. We say E is an approximate p-interpolation set for

A if approximate solutions exist. In particular E is an interpolation set for A. We say the p-interpolation problem has exact solutions if the f can be chosen

Thus, if E is an interpolation set for A then Theorem 2.1 with Il/iL says that for a minimum r < E is an approximate pr-interpolation set. We take up some special cases in the remainder of this section and return to interpolation problems in general in Section 5. First, we indicate how results for real spaces, in particular A(K) spaces, can be derived in this context. If A0 is a subspace of then we consider

Ao+iA0c C(X). Let

be real gauges on X and define

forxEXx{1}

p(x,t)= p(x) forxEXx{—1} +00

elsewhere.

Then

U={fc-C(X):—p so that the p-interpolation problem for A, E reduces to interpolating by functions in A0 dominated below and above by —p and respectively. We next elaborate on the complex state space discussed in Section 0 by constructing gauges which induce a natural identification of A with real order unit spaces. We give a partial ordering to the complex numbers C (considered as a 2-dimensional real space) by letting P be any proper closed convex cone with non-empty interior. Then the dual cone {z E C: re az 0 for all a E P}

is also proper with non-empty interior. Let I denote a compact base for with extreme points a and $ of modulus 1. Clearly a and $ may be any two elements of T (depending on F) such that Ire aPI <1. Then each z e C has a unique representation as a real linear combination of a and $ with

p*{zEC: Let u denote the element in mt P such that re au = re $u = 1. We note that is given by the base norm (L-norm) on (C,

GAUGE DOMINATED EXTENSIONS AND COMPLEX STATE SPACE

161

,

One sees from elementary calculus that the minimum of Izi for occurs midway between a and j3 so that

=1

c = mm {Ire (ã/3)1"21, urn

Then the order unit (M-norm) on (C, u) is given by I

= max {Ire yzl: 'y

a,

and IzIu

Now let

(1

fort=±y

We denote the gauge induced on C(X) by Il' =sup{Ire yf(x)I: x EX,

a, /3. so

that XEX}.

We show next that induces an order unit structure on C(X) which allows an identification with real functions on

Xo=Xx{a, f3}.

COMPLEX FUNCTION SPACES

162

We shall think of Xo as a disjoint union of X7's;

= a, j3 where

K,, = X x {'y}

is identified with X.

The choice a = 1, /3 = —i, yields the structure introduced in Section 0. Let A be any separating subspace of with constants and define

'I': C(X)

'l'f(x, y) = re 'yf(x);

CR(XO);

y = a, /3

let 9= Let u denote the constant function identically u. Let = a, /3) be measures on X and define on X0 by 1Z = with Then = and

re (f,

= (4'f,

= re

J'

=

f

re af(x)

(af d/La ÷ 131

+J'

re /3f(x)

= re

so that = a correspondence between measures on Xo and complex measures on X analogous to the Hustad map H of Section 1. 3.1. THEOREM. The map 'I! is a real Banach space isomorphism into norm given by CR(Xo) such that if is the II! IL = II'IP/li = sup {Ire

yf(x)I: x X; y = a, /3}

and

P={f: 'I'fO}={f:f(X)cP} then each (A, P, IL, u) is real isometrically order isomorphic (under 9) to a closed (separating with constants) subspace B of CR(Xo), with Ou = 1. Further, (1) The state space, denoted ZA, of (A, u)* is given by

ZA=co(aSAU/3SA) with aSA, 13SA closed disjoint faces of ZA. (ii) In case A = C(X) then the state space

Z = co (aS U /35)

(S =

where yS ('y = a, /3) are closed complementary split faces of Z. (y = a, /3) be probability measures on X and ji =

(iii) Let

e

GAUGE

AND COMPLEX STATE SPACE

(1

163

—

is a

boundary measure forA if and only if the are maximal measures if and only if ji on Xo c aX u /3X is maximal on ZA. Furthermore, if ji = MR(X) then j2 if and only if/2 = (iv) For each L c Zn there is a boundary measure Z representing L and for each L EA* there is a boundary measure /L such that =

on X c

Proof.

Clearly 'I' is a real linear map, and in view of the preceding remarks,

c is as above. Since 'l'u(x, 'y)=re yu = 1, 'IPu= 1. Also 'l'fO if and only if re yf(x) 0 (y = a, $) for each x, if and only if f(x) E P. Now let B be the image of A under 0 in Then B separates points

in each X,, and the function Oã is identically one on X. andre ã/3 on X0. This shows the evaluation map on X() is one-to-one and that S8

where

=

is a closed face of

Since, for f A,

re (f, 6*(x, y)) = (Of, (x, y)) = re (/ yx) we have

O*S1=ySn, and (i) is shown.

Let A = C(X) and let M(X). Then, since each z C is a unique real linear combination of a and the real measures given by

f0 are uniquely defined. In particular, for u Z, the expression A 1.

=

S)

is split with the other as complement. is unique. Hence each E A* and For (iii) and (iv) we note first that if M(X) and p.

OnXQaXUI3XcZA) then, using the remark preceding the theorem,

=ref fdp. =re(f, L)

COMPLEX FUNCTION SPACES

164

so that ji represents L with respect to A(ZA) = ji if and only if A1.

OA,

and conversely. Thus

L

If L E ZA then

L = 'yaLa + (1—

L,3 E SA•

Then the Choquet—Bishop--de Leeuw Theorem yields maximal

Then

=

+ (1—

we have p.

E S with

L and

C(A/La

is maximal. Since so that p. is a boundary measure if and only if each is maximal on SA' Since is maximal on ZA if and only if is a face of unit ball is the

co(ZAU—ZA) the rest of (iv) follows. We observe that Theorem 3.1(iv) can be phrased as saying ZA and (A, u)r = Co (ZA u —ZA) are stable in A*.

4. PEAK SETS AND M-SETS The realization of the closed subspace 1 A c C(X) as A(Z4) enables us to apply the techniques of ordered Banach space theory in Chapter 2, Section 8. This allows for a geometrically appealing development of peak set and M-set (defined below) criteria. A closed subset E of the compact Hausdorfi space X is called a peak set for the (closed, separating, with constants) subspace A of C(X) if there exists an f€ A with

onE and Ifkl

onX\E.

In particular peak sets are G5's. We say E is a generalized peak set for A if E is an intersection of peak sets. Thus E is a generalized peak set if and only if for each compact K disjoint from E there is an fK E A with HfKII=l,

onE and

fKI<1 onK.

If E is a generalized peak set and a G6 then E is a peak set (use

I

In the following we identify (A, u) with the real affine function space A(ZA) in accordance with Theorem 3.1(i) where, forf A, z E ZA, we have Of(z) = re (f, z).

AND M.SETS

PEAK

165

Let E be a closed subset of X and let J be the image under 0 of E1 so that

aEUI3ECZA}. Let

J={aEA(ZA):

Z(E) is a section of

by J.

If for some a, j3 T with re <1, Z(E) is a positively decomposable section of ZA by J then h(E) is a generalized peak set. 4.1. THEOREM.

X c so that if y X\h(E) and h(E) = then yg E1' mX and hence ay, f3y ZA\Z(E). Let K be a compact subset of X disjoint from h (E). Now Theorem 2.8.13 shows there is an a E A(Z) such that and a >0 on aK u (3K in ZA. a 0, a 0 on Z(E) Proof. We have J° =

Now a = ref, with f€ P. Thus f(X)c P. Since for XE h(E), re (f, ax) =0 = on h(E). For y K, re (f, ay), re (f, j3y)>0 so that re (f, 13x) we have

f(y) P\{0}. Since P is proper there is an s>O such that u

C D(0, ui).

By multiplying f by a positive scalar, we can assume f(X) c P., 0< r <S.

Thus, if g = 1 —(1/u)f, then

onh(E),

gI1 onX

and onK. We now consider two types of peak set conditions that have arisen in the study of function spaces, and show that both are instances of Theorem 4.1. Further measure theoretic conditions will be discussed in Section 6. For the first, we define E to be a strong hull for A if there is a constant

k >0 such that for each neighbourhood V of E there exists fv E A with k and re fv> I on X\ V. We say E is a weak hull if for some

fv E F1, iifvIl

constant k >0, for each neighbourhood V of F and e >0 there exists such that e

Ofl F,

IIfv,eIf k

and

refv,8

1

on X\V.

If E is a weak or strong hull with constant k then choose P symmetric about the positive real axis such that

D(0; k) r'i{z EC: re z 1}c++P. Then u is real and we let a, a denote the generators of

Let

F=ëGECSA and E=co(aFuãF). 4.2. THEOREM. Let E be either a strong hull or a weak hull and interpolation set and let P be as above. Then Z(E) = £ and is a positively

COMPLEX FUNCTION SPACES

166

decomposable section of ZA by J. Furthermore, E = h(E) is a generalized peak set.

Assume first that E is a strong hull and let h E (A**)k be a point of the net v. Since each Iv E1, Iv 0 on E'1 and hence so is h. Now let z E ZA\E and let /L be a probability measure on ZA representing z Proof.

with supp p c äZc aX u oX. We show first that if p.(aE U äE) =0 then Let W be a neighbourhood of aEuäE in ZA with p(W)
Iv ( y)

Hence reafv(y), (aX U then Ofv(z)

Thus,

f Ofv

+

if B=aVuãV and C=

Of v

—r)—ke

Thus re Now, for any x and representing measure we can, by restricting /L to aE u iiE and its complement in aX u aX, write as a convex combination of where supp

c aE u

and

/L2(aE u

=0.

Hence x is a convex combination of y E and z with re h(z) Thus on Theorem 2.10.3 shows, since re h (z) that F is a positively decomposable section by J. Since y X\E implies re h(y) E = h(E) is a generalized peak set. y If F is a weak hull and interpolation set then the same argument shows there exists h (A**)k with ZA = Co (E u {z

Z: re /i(z)

is the algebraic span of But the fact that E is an interpolation set means F. Hence h 0 on F and F is a section of Z by E'.

As an application we consider the case where A is a function algebra (also called a uniform algebra), that is a closed subalgebra of C(X) containing constants and separating points. 4.3. COROLLARY (BisHol' PEAK POINT THEOREM). Let A be a function algebra and let Then Xo is a generalized peak point. aAX = ext

PEAK SETS ANI) M-SF:rs

167

Proof. We take a = 1 and p = —i. Since x0 E ext SA, x0 E ext ZA. Let V be a Now Lemma 2.10.7 shows there neighbourhood of xo and let K = X\ V c is a Of€ A(ZA) (fE A) such that

—6Of onZA Of(xo) =0

kOf onK where 6>0 and k large are chosen as follows: let g = I

g(x)i =

E A so that

and hence g(xo)i = 1

IIglIe°

gie"

onK.

Let h=1—g(xo)g.Then 0

h(x0)

IIhII

1

onK. Thus k and 6 can be chosen so that, say, 2h = fv as in the definition of strong hull. It follows that {Xo} is a strong hull and hence Xo is a peak point. If Z(E) is a split face of ZA then it is in particular a positively decomposable section so that Theorem 4.1 again applies. This will be the case precisely when the natural splitting projection of M(X) onto M(E) (by restriction) can be carried down to A* by the quotient map -, 4. This is the basis for the second type of peak set criteria, based on properties of A1 and 3A In general a closed subspace N of a Banach space B is called an L-ideal if the unit ball B1 is split by N. Recall from Chapter 2, Section 9 this means there is a projection ir of B onto N with complementary projection o• and IlxII=IIirxII+IIoxII,

x €B.

Thus we have that M(E) is an L-ideal in M(X) under IT1/L

We denote the complementary projection

=

We say E is an M-set for A if (i) M(E) is A-stable (see Proposition 2.7, for example) (ii) m E implies 'rr1m = mIE E A1. It follows from Corollary 2.4 (where (ii) holds with r =0) that if B is an M-set then E is an interpolation set. Choose any a, j3 T with Ire ã/31 < 1.

COMPLEX FUNCTION SPACES

168

A

We continue with the representation of A as A(ZA) and let denote images in A* under the quotient map from M(X). As previously we denote

F=ii?iEcSA, = M(E)t and ZE = co

Let

ZA=2 We

and Q* be

let

4.4. THEOREM.

E=co(aFLIf3F). u f3SH) C Z M(X). Then and

the cones in M(X) spanned by Z and ZE.

The following are equivalent, where N denotes M(E):

(i) E is an M-setforA; (ii) ZF is a split face of ZA in A*; (iii) Q* = N is a complemented subcone in

(iv) N is an L-ideal in A* with N SA Proof.

If

A

c A*;

F.

(i) holds then the map

=

oq1 for

is a well-defined projection on A* with contained in i1. The stability of N shows the range of equals N. If x E N SA and v c 85 with = x then choose any aA so that (irs ii) = x and ON with 4 = x. Then v — ir1v(1)=x(1)= 1. Hence ii = F = é?i E. Since ZE = co (aF u j3F) we have ZE N ZA and Q* = N If x c pK then Theorem 3.1 (iv) shows there is a with 4 = x. Hence OQ" and so lriiL Q'. This shows (1) implies (ii) and (iii), the latter two being clearly equivalent. We show next (ii) implies (i).

is a face of ZA, F is a face of SA, so that

Since

Proposition 2.7 shows M(E) is A-stable. If m E

and

(ya,f3) and

OflaXu$XCZA then Theorem 3. 1(iii) shows gives A(ZA)'. But

E OA(ZA)'. Since 2,. is split, Theorem 2.10.7

= (irimf

= and hence

A

Thus (i), (ii), (iii) are equivalent. If (i) holds then, as

above, the map =

for

isa well-defined projection onto )Q. The fact that 0 (U = M(X)1) is split by N is then immediate from the A-stability of U° (Theorem 1.5(i)). Hence (1)

PFAK SETS AND

169

implies (iv). If (iv) holds then let denote the splitting projection onto iQ We show takes to Q*, establishing (iii). If x E SA then, since At is split by N, 1

flxfl =

x(1)

+ 17'2X(1)

1.

Thus IIfr,xII.

Hence belongs to the cone spanned by SA = x E YSA then belongs to Q* so that

=

F It follows that if

Let E be an M-set for A. Then (i) E is an interpolation set for A, and (ii) k(E) is an M-set and generalized peak set for A with (M(k(E)))

4.5. COROLLARY.

Proof.

N.

As noted above it follows from Corollary 2.4 that E (and hence

k(E)) is an interpolation set. Thus N is and Z(E) = N ZA = ZE. It now follows that h(E) k(E) is a generalized peak set from Theorems 4.4(u) and 4.1. Notes

The idea of gauge-dominated extensions introduced in Section 3 goes back to Bishop [39] in connection with his generalization of a theorem of Rudin

[186] and Carleson [59]. We shall have more to say on this in the next sections. The idea of T-dependent dominators is due to Roth [180]. The extension constant briefly alluded to in Section 3 was introduced in Gamelin [112]. This will also be developed further in Section 6. The Peak Point Theorem for function algebras (Corollary 4.3) was given in 1959 by Bishop [37] where the notion of strong hull is introduced in terms condition". The 2" iteration used by Bishop and of the celebrated which is generic to results of this type is, in our development, incorporated in

the geometric condition of positive decomposability via the Extension Theorem of Chapter 2, Section 10 and ultimately, the Gauge Lemma of Chapter 1, Section 3. The geometric proof of peak point theorems for complex functions spaces appears in Asimow [24] and the general work of Theorems 3.1, 4.1 and 4.2 in Asimow [26]. Various hull conditions have been given in connection with peak-set criteria. Their relation to approximate identities in general Banach algebras is discussed in Curtis and Figa-Talamanca [70].

The measure condition (ii) in the definition of M-set, but without reference to boundary measures, (i.e. condition (iii) of Corollary 2.2 with r =0) appears in 1962 in Bishop [39] and Glicksberg [118]. The latter shows

COMPLEX FUNCTION SPACES

170

that for A a function algebra (details in Section 7) the measure condition

implies E is a strong hull and hence (using a version of Bishop's argument in [37]) a generalized peak set. This is essentially Corollary 4.5 with the more restrictive hypotheses. The Bishop Peak Point Theorem can also be viewed

as an instance of Theorem 4.4 since we show in Section 7 that x0 E aAX implies the line segment co ({xo, —ix0}) is a split face of ZA. As discussed in the notes of Section 2, the definition of M-set originates in

Alfsen—Hirsberg [9] and Hirsberg [125] where it is introduced in analogy with the real A(K) characterization of split faces in Alfsen and Andersen [6] (Chapter 2.10.7). The notion of L-ideals (as dual to M-ideals) is introduced by Alfsen and Effros [IS] in the real case and Hirsberg [126] in the present context. The results of Theorem 4.4 are in [126]. Similar results for function algebras are given in Ellis [102].

5. DOMINATED INTERPOLATION Consider now the gauge interpolation problem introduced in Section 3, with

U = {f C(X): re tf(x) p(x, t); (x, t) E X x T}. We wish to interpret conditions for solutions in terms of measures in öAA and

so the properties of the dual U° become important. In particular the

A-stability of U° is crucial. For this, we require that p be A-superharmonic, meaning that for each x E X and probability measure on X representing x E SA we have p(x,

t)f

p(x, t)d/L(x).

We shall assume now that the gauge p is bounded above. This minimizes some technical complications in Theorem 5.1(v) below and does not entail a serious loss since if E is an interpolation set p can be truncated in accordance with the extension constant e(A, E) and the particular g E AlE in question. Thus we take

and let

Xu={(x, t,s)€Xx Then

p(x,

is compact since p is l.s.c. and Y0 = graph p = {(x, t, s):

s = p(x, t)}

is a G8 set in X0.

Take E to be a closed set in X and

N=M(E).

=

IE, the projection of M(X) to

DOMINATED INTERPOLATION

171

Let V = {g e C(E): re tg(x)

p(x, t); (x, t) E E x T}.

Finally, we defIne the gauge evaluation X0 -. M(X);

(f, 4'0(x, t,s)) = tf(x)/s.

5.1.

(i) U°= (ii) If h is a bounded Borelfunction on X wit/i re th(x) p(x, t) on X x T then Ii E U°° in C(X)**, that is, re

Ii

1

for

E U0.

by N = M(E).

(iii) U° is split in

(iv) UIE=V. (v) If p is A-superharmonic then U° is A-stable. Proof.

(I) Iff€Uthen re (f, 40(x, t, s)) = re tf(x)/s re tf(x)/p(x, 1) s 1 so Y0)) c U0.

If f€ W° then re tf(x)/p(x, t) I for all (x, t) E X x T so that fE U. Hence U°c W°°= W. (ii) Let E U° and s >0 be given. By Lusin's Theorem (see Rudin [185, Theorem 2.23]) there is a g C(X) and F compact in X such that

<

= hip, For each x X\F choose o <'IPX

1,

Ih!

Ofl X.

CR(X) such that 'IPX

on F

1

and

=0.

t)+e for all t€ T}. Let V, ={y €X: re Then and hence there exists is an open neighbourhood of F , . and corresponding 'J!1,. . and V1,..., such that X is , Xi,.. . Let covered by V1,..., =

n

and f = g.

Then either re

t)

or, for x E o

re tf(x) = 'V(x) re tg(x)

'IP(x) re tg(x)
172

COMPLEX FUNCTION SPACES

for all teT. Hencefe(l+e)U and

reith(h)=rep(f)+reJ Thus h

U°°.

then choose /;€ U (i = 1,2) such

and

(iii) If IT1/.L

that re (ft,

il1r,/Lilp

—

Let

h =IJXE Then, using (ii) li.' IL

re (h,

+ iilT2P, IL — 2e.

V and hence

(iv) Clearly

V°c

U° N and g V let h be the extension of g identically zero on X\E. By (ii)

ref so that

V0. But

V°= implies

(UiE) =V. But (iii), together with Andô's Theorem (2.9.4) shows U +M is closed in C(X), where

M={f€

onE}.

Hence UIE is closed in C(E) and equals V. To prove(v) weJirst demonstrate a special case and then combine this with Hustad's Theorem for the general result. For this we take p to be l.s.c. on X with 0<8 p k and A a closed separating (do not assume 1 A) subspace of C(X), such that Xc {x A*: lix ii = 1} (under evaluation). Let

U={f€ C(X): refp} X0={(x,s)€Xx01:

Y0{(x, s)€X0: s =p(x)}.

J)OMINATED INTERPOLATION

173

We now take

'I'f(x, s) =f(x)/s

'1': C(X)-+ C(X0);

with closed range B c C(X0). Let (xi, The (x2, s2) in Separation Theorem shows there is an f€ A such that (/ x1/s1) (f, in (A, A*). Hence B separates points in X0 so that we can consider X0 embedded by evaluation in B1C and we denote and 0 =

5.2. PRoPosI'uJoN. In the above and u. represents x with respect to A then

if p(x)

c d/2 whenever ji. E S

(i) 0*V=U; (ii) If ii

is

a maximal probability measure on V then v( Y0) =

jh

d/2

=J

and if

(h(x)/p(x)) dv(x, s)

for any hounded Borel function h on X. In particular /2 E U0. =(x, p(x)) and ir Let :X-+X0; a probability measure on X representing

(iii)

1

then

/L

ir(x, s)= s. If /2o is X c SA and

d(/2() 0.01)

dz.'0

On

X0

then

0 v0 1,

v0 represents (Xe, p(x0)) in V with respect to B,

and

= /.Lo/p(xu). (iv) If v is maximal on

then

(v) If L U then there is a

= v is maximal on K U° maximal on K such that

K A*. represents L

with respect to A. Proof.

(i) Since 0*(x, s) = xis and U =

{x/s: (x, s) E X0} we have 0* V =

U.

(ii) Let p = I — X{o} on so that the lower envelope functional of V. Since i' is maximal, supp v c X0 and 1=

s): p(x, s) =

s)}] = v[{(x, s):

where the last equality follows from the fact that 0

is the Minkowski s) = l}] C

and f€A then Of(x, s) =f(x)/s =

(f(x)/p(x))(p(x)/s) = (p(x)/s)Of (x, p(x))

174

COMPLEX FUNCTION

so that p(s)).

(x, p(s))

j5(x, s) =

Hence jY(x, s) = I implies p(x) = s and the statements in (ii) follow.

(iii) The definition of v0 shows that for any bounded Borel function g on

xu g

= (1/p(xo)) L p(x)g(x, p(x)) dp.0.

Thus the super-harmonic property of p gives 0 vu I on Xo. If f€ C(X)

then = ('lit,

(f,

=

(f(x)/s) dv0(x, s)

f

(*)

= (l/p(xo)) I p(x)f(x)/p(x) dpo(x) = (l/p(xo))(f, p.o). Jx Thus 'lt*p0 = /.Lo/p(xu) and if fE A

f(xu)/p(xo) = (Of)(x, p(Xo)) = (Of,

(f,

Vu)

that v0 represents (Xo, p(x0)). (iv) Letf be a continuous convex function on K and let g = 'I'(flx). Then g e C(X0) and so is well-defined on V. Now if Xo X then from Theorem so

1.6.2

f(xo) = sup {f fdp.o: p.u€S

and

and so by (iii),

1(xo, p(xo)) sup where v0 is related to

{J'X()

g dii0: p.o E S and

XO}

as in (iii).

But then, by (*), g dvo

=

('lit) dv0 = (1/p(xo))

dp.0

so that p(xo)) f(xo)/p(xu). Hence, using (ii), dp. j' (f—f) =

((f—f)/p) dv f (1— g) dv =0

since ii is maximal. Thus p. is maximal.

DOMINATED INTERPOLATiON

175

such that O*L = L. Then let v be a maximal (v) Given L e U we let L probability measure on V representing L and let /L Then p E U° by (ii) and is maximal on K by (iv).

To conclude the proof of Theorem 5.1(v) we consider C(X) embedded as the T-invariant functions in C(X X T) and take

4(x,t)=t. let

U = {f€ C(X x T): re tf(x, t)/p(x, t)

1}

U1 ={f€ C(Xx T): ref(x, t)/p(x, :)s 1} and note that 'I' U = U'. Thus if B =

with range B,

We will apply the set-up of Proposition 5.2 to the subspace B with respect to the set U1. Clearly X x T is embedded in the surface of Br We observe next that the by evaluation such that K = (X x T) A-superharmonicity of p yields the hypothesis of Proposition 5.2. For, if g E B then g(x, t) tg(x, 1) and SA

=

a face of Br. Thus, if p is a non-negative measure on X x T representing (x, 1) then p(X x {1}) =1. If a, denotes the homeomorphism of X x T given by

is

a1(y, s)=(y, ts) and jz' is given by

= oaj- then for g = 4f, fE A,

p '(g) =

J

g°a, dp

=tf

= JXx{1}

g(y, t) dp(y, 1)

f( y, 1) dp ( y, 1) = g(x, t)

Xx{1}

so

that p.' represents (x, t) and has support in X x {t}. Conversely, by

applying a,, each p.1 representing (x, t) is of this form. Hence j'XxT

Xx{1}

p(y,t)dp.(y,1)p(x,t).

given L U in A* let L = O*L, L€

B*. Proposition 5.2(v) yields ,Z E (U')° such that 4 is maximal on Br and represents L with respect to B. It follows from Theorem 1.5 that p. = H4 = is a boundary Now,

measure for A on X and by the above p represents L and belongs to U°.

COMPLEX FUNCTION SPACES

176

We now characterize, in measure-theoretic terms, approximate pinterpolation sets for a closed, separating, with constants, subspace A. The results here are refinements of conditions in Section 2, using the gauges 11 in place of the standard norms in C(X), M(X). We take E to be a closed

subset of X with

M={feC(X):fJE=O} and N = M(E) c M(X). We continue to denote quotient map images in A*

by , and U = {fe C(X):

re

tf (x)/p(x, t) 1}.

The following are equivalent for a bounded dominator, p, and imply that E is an interpolation set: (i) E is an approximate p-interpolation set for A; (ii) A +M is closed in C(X) and 5.3. THEOREM.

(iii)

(iv)

+A

all

E

A is closed in E C(X). Since U° is split (Theorem 5. 1(iii)) by N, Andô's Theorem shows U + M is closed. Thus, the fact that E is an approximate p-interpolation set translates into (ii). The polar of the statement in (ii) is (*)

or, equivalently, Then applying the quotient map shows this is equivalent to (iii) and also (iv). The equivalence of (iv) and (v) is the same computation as in Corollary 2.2 since and IPIP

keeping

lip,

in mind the presence of the minus sign in (v), reflecting the

non-homogeneity of

.

Finally, if (*) holds then the polar statement is

U+M). p is bounded the equivalence of and IL shows, using Corollary 2.2, that (v) implies E is an interpolation set so that A + M is closed. If

5.4. COROLLARY. Let N = M(E) and Uu be A-stable sets. Then the following are equivalent for bounded dominators, p:

DOMINATED INTERPOLATION

177

(i) B is an approximate p-interpolation set for A; for all oN; = fl1z for all m E OA—. (iii) iiirim +A1 (ii)

Note that if Corollary 2.4) Proof.

is a boundary measure and U° is A-stable then (as in +A

for

= Il,L + OA

(*)

e ON.

Thus (i) implies (ii) implies and implied by (iii) follows as in Theorem 5.3 and Corollary 2.2. To show (ii) implies (i) it suffices, by 5.3, to show

If x 0

then choose

E ON with 4= x and /L E

Then (ii), together with (*) shows ÷

=

+

= ijz + AlL

1

so that

jj.=v+m;

i'EU° and

Hence V EN and

5.5. COROLLARY.

Let E be a closed subset of X such that either

(a) EcOAX, or Pa closed face of SA, (b) and let p be a bounded A-superharmonic dominator. Then the following are equivalent: (i) E is an approximate p-interpolation set for A; all (ii) liirim Proof.

If (a) or (b) and (i) or (ii) holds for p bounded then B is an

interpolation set and Proposition 2.7 shows N is A-stable. Theorem 5.1(v) shows U° is A-stable so that Corollary 5.4 applies.

We show next that if B is an M-set for A then we can conclude exact solutions for all A-superharmonic dominators. This is a special case of more general criteria presented in the next section. Exactness is tantamount to the

closedness of the set A U + M in C(X).

COMPLEX FUNCI'ION SPACES

178

The following are equivalent for a bounded dominator, p: (I) E is an exact p-interpolation set for A;

5.6. THEOREM.

(ii) A +M is closed, A U + M is closed, and

U+M; =M

(iii) A U + E1 is closed in A, where

Proof.

A, and

If (1) holds then E is an interpolation set and A +M is closed. If

1 so that g E (A U +M) then g e AlE and Theorem 5.1(iv) shows U+M)=A (i) implies g EA U+M If (ii) holds then A is closed and hence A (U +M) = A U+E follows from the analogous

equality in (ii). If (iii) holds with p bounded and

fE(A+M) then given e >0,

f=a+m+z=u+m';

aEA,rn,m'EM,

llzlkZs,

and

uEU.

in C(X) so that

Now

a EA

Hence / E (A U +M). Therefore

U+M) and the equivalence of ll• and ii shows, by Theorem 2.1, that E is an interpolation set so that A + M is closed. The set equality in (iii) then easily yields

U+M We can extend Theorem 4.4 as follows. 5.7. THEOREM.

The following are equivalent: (i) E is an M-set for A; (ii) For any bounded A-superharmonic dominatorp,

U is split by Proof.

in A* with U

= (U° N)*.

The fact that E is an M-set shows that = q ° in

°q

for

is well-defined (Theorem 4.4)so that the A-stability of U° (Theorem 5.1(v)) ° q = q ° 'ira, U N [o(U° N)]. shows U is split by under Since

DOMINATED INTERPOLATiON

If (ii) holds, then the case P M-set. 5.8. COROLLARY.

set for any Proof.

179

1 gives Theorem 4.4(iv) and hence £ is an

If E is an M-setforA then E is an exact dominator p.

If p is bounded then Theorem 5.7(u) shows

so that But Andô's Theorem shows A U + E is closed so that Theorem 5.6 gives

exact interpolation. Since p A k is A-supcrharmonic for any k 0 the conclusion holds in general. The conditions for exact solutions in these results depend on splitness in A*. We give some more general criteria for exact solutions based on decom posability in the next section. The utility of boundary measures and superharmonic dominators is made most apparent by considering the situation for real A(K) spaces. The closer

the measurcs come to having their support on the boundary (ext K) the more accurately they portray the geometry of K. To take the simplest possible case let K be the unit interval [0, 1] and let E = {O}. Then the condition

m E AL implies ir1m A' + fails (take m = — 81,2) unless one restricts attention to boundary measures. This phenomena is also reflected in less trivial settings. For

example if A is the disc algebra (details in Section 6) on the unit disc C then normalized Lebesgue measure u on T represents 0 so that o• — E A1. One gets around this by restricting A to T since T equals aAIX. Again, considering the case A(K), we noted in Section 3 that

U={f: —p Then p is A-superharmonic if and only if (Proposition 1.6.2) p = (p and are concave. For extending affine = (p) , that is, if and only if p' and functions this is clearly a relevant hypothesis (as the above example shows). Notes

Theorem 5.1 is due to W. Roth [182, 183]. Abstract versions of the (approximate solution) Theorem 5.3 using set inclusion formulations are given in Roth [182], Andô [19], Asimow [28]. These are outgrowths of

COMPLEX FUNCTION

180

conditions based on the gauge Pr (introduced in Section 3) developed in , Briem [SO]. We discuss this in more Gamelin [13] and, in the case of

detail in connection with exact solutions in Section 6. Theorem 5.7 relating M-sets and exact interpolation for superharmonjc

gauges is given by Roth [182]. Corollary 5.8 is essentially the Alfsen.Hirsberg Theorem [9].

6. DECOMPOSABILITY AND EXACT INTERPOLATION In Theorem 4.1 the positive decomposability of ZA by was used, via Theorem 2.8.13, to conclude the positive extension property (PEP) for the = subcone E This in turn is equivalent to P + E being a closed subset in A. In Corollary 5.8 we used the splitting of polar sets U to conclude

the closedness of A U + E. It is convenient to introduce now a broader concept of decomposability that leads to a more widely applicable closedness condition. Let S be a closed convex subset of the (real or complex) Banach space E and let M be a closed subspace of E with N = M° in E*. Following the definition for decomposability of cones in Chapter 2, Section 8, we let W be a norm-closed convex neighbourhood of 0 in E* and define the distance, 4,, to N via W by

dw(x)—p(N+ W)(x) = inf {r 0: XE N + rW}=p( W)(x +N) where the notation p(A) means the Minkowski functional of A. = M in E. Then we Let S° N be a section of S° by M, i.e. (50

say S° N is a decomposable section of S° by M if there is a positive homogeneous map :5° -+5° N with IT2X

such that (i) IT1X

X if X E

=XITIX

N

(ii) p(S°) =p(S°) o +p(S°) ° 1T2 (iii) lIir2xIi dw(x) for x E S°. If S° is a cone then this agrees with the previous definition since p(S°)(x) = 0 for x E S° and +00 otherwise. If is a bounded linear projection such that (ii) holds, ((i) and (iii) being automatic in this case) then is split by N. 6.1. THEOREM. If 5° S + M is closed in E. Proof.

Nis a decomposable section 0150 by Mm E* then

If we show, for some bounded convex neighbourhood V of 0 in E,

for all r>0

(*)

181

DECOMPOSABILITY AND EXACT INTERPOLATION

then the Gauge Lemma (1.3.2) applies with

to yield d(= d(A, B)) = d on D = (S + M) . But x eD implies, by (*), that Hence d(x) <00 so that x E S + rV M (some r) CS +M d(x) To show (*) it suffices to demonstrate

(S + rV nM)° C Now,

(S°

N, (S + Er)°).

50 is decomposable by N then we can choose a bounded convex

neighbourhood V of 0 in E such that V° C W, where IIlr2xII dw(x) c'ii Then the Minkowski functional p((S + rV M)°) is the support functional Hence for x E S° and

1 p((S + rV C'i M)°)x = p(S°)x +p((rV M)°)x

p(S°)irix +p(S°)ir2x +rdw(x) + (p(S°)+ rh' hI)lr2x = p(S°)'rr1x +p((S +E,)°)ir2x.

Hence, if A =p(S°)rrix,

p(S+Er)01T2X and a

and

x= where 0/0 is taken to be 1. Thus (*)°, and therefore (*), holds and the proof is

complete.

If E is a closed subset of the compact Hausdorif space X then the space subspace of M(X) and the range of the projection M(E) is a subsets of M(X) are split under irk, including the Various = positive cone (in the real case), the generalized positive cones of Section 3, the unit ball and more generally the sets U° of Theorem 5.1. We discuss next when this is inherited, in the form of decomposability, under the quotient map q is

We assume in the following that E is an interpolation set, so that q(M(E)) in A*.

Let 0€ 5, a

convex set split under

and let W be any

convex norm-neighbourhood of 0 such that

ir2(W)c

14'.

by 4 E A* and assume that the image A*. Since F is an interpolation set, N (for N = M(E)) is Denote images

is

-closed in We note

COMPLEX

182

SPACES

that

=p(S+A 6.2. PRoPosITIoN.

E S with p(S)p =

then

(i =1, 2). Proof.

p(SA)(lri,.6)A=p(S+A

=p(S+A1)/h —p(S+A ')1T2/.i.

p(S)p, —p(S)1r2/h 6.3. PRoPosITIoN.

The Minkowski functional

p(W+N+Aj=p(W+i'r2A 1)0 1T2. Proof. = r'rr2 v ÷

E

Conversely, if 1r2/h = rv +

r'rr2 W +

then

6.4. PROPOSITION. If Wand N are

p(W+N+A

sets and

E aAM(X) then

=p(W÷IT2aAA1)o 1T2/L.

then

Proof.

,2=rv'+n'+e'; Since /L

I c r w + ir2A I

aM(X),

p'E9W,

(EA1.

v'EaN,

so that

But it follows, as in Proposition 6.3, that 1r2.

If for each x S satisfying for some k >0 and Was above (1) 4 =x and p(S)ji (ii) p(W)ir2p s kp(W +

6.5. LEMMA (DECOMPOSABILITY LEMMA).

there exists a

E

then = (S

If Wand N are

N)A

and

is a decomposable section of can be chosen in

satisfying (i) and

DECOMPOSABILITY AND EXAcI INTERPOlATION

183

(ii)' then the same conclusion holds.

1. Choose

Proof. Assume first define

according to the hypothesis and

E(SriN) and IT2X

=x—

—

(7r2/L)".

If

=0 and = 1 then again choose and define as above. to be a positive homogeneous Since p(S)x = p(S)p we can extend function on S. Then p(S)x and, by Proposition 6.3 =

If x E x

(S

=

= 0 so that = 0. Hence 0 = then and N) . This shows S =(S riN) is a decomposable section

of S. The proof using (ii)' is the same with Proposition 6.4 in place of Proposition 6.3.

As an application of Lemma 6.5 we develop another, measure theoretic, peak set criterion based on the of Sections 3 and 4. Let A be a closed separating subspace of C(X) containing constants and let E be a closed interpolating (with respect to A) subset of X. We take, as in Theorem 3.1 with generators a, a for and u = (re a)'. As in Section 4, we take

Z=co(aSuãS)cM(X) ZE co(aSEUäSE)cM(E) ZE

=co(aFuäF).

6.6. THEOREM. Let E be a closed interpolation set for A. If there is an s, 0s<1, such that for each m jm(E)I sIIlr2mII then ZE is a positively decomposable section of ZA and k(E) is a generalized peak set.

COMPLEX

184

of P* C such that lal = 1 and s =

Choose the generators a,

Proof.

rea =1/u.

SPACES

then IlI.Lllu

= u re

= re

=

and generally, 11u

We apply Lemma 6.5 with S the cone spanned by Z and U the in M(X). Let x E ZA = Z and E Z with 4 = x.

unit ball

Then, given in EA, sllir2p, + 7r2m

— sIIir2,z

1114

sljir2mll —

Jm(E)I—s re

re ir1m(u)—s re ir2j.t(u)

= —s re ir2m(u)—s re 1T2/L(U) = (1 —s) re

re ir2m(u)+re

(1

—IIslr2m +1T2#LIIu.

Hence

((1 +s)/(l —s))lIs'7r2m + Since

m A was arbitrary this establishes (ii) of Lemma 6.5 and therefore

ZEZA is

a decomposable section of ZA. We show next that ZE is a positively decomposable section. It suffices to

show the positive cone in A*/N is a-additive for some a >0 under the quotient norm induced by

in M(X) (I = 1,.. . , n), we have

Thus, given

+

=

+

=

+

((1—s)/(1 ((1 —

Finally, we note that

Thus h(E)

+s)) s)) E

(MT (E))A =

E, so that

k(E) is a generalized peak set, by Theorem 4.1.

COROLLARY. Let E be a closed interpolation set for A such that N +M(E) is A-stable. Then if there is an s, 0 s <1, such that for each 6.7.

DECOMPOSABILITY AND EXACT INTERPOLATION

185

m E 3A

m(E)j

(*)

sIIlr2mII

then ZE is a positively decomposable section of ZA and k(E) is a generalized

peak set. ball is A-stable and that the measure Theorem 3. 1(iv) shows the such that 4 = x can be chosen to be a boundary measure. Thus the same computation as in Theorem 6.6 shows, with Proposition 6.4, that

Proof.

17r2/LIJu

((1 +s)/(1

+

Then Lemma 6.5(i) and (ii)' yield the same conclusion. The positive decomposability follows in exactly the same fashion.

We turn now to some general measure-theoretic conditions sufficient for exact p-interpolation. We shall apply Theorem 5.6, employing the closedness condition of Theorem 6.1. 6.8. COROLLARY. Letp be a bounded dominator orletE be an interpolation set with p any dominator. Then, if

is a decomposable section of Ci in A*, E is an exact p-interpolation set for A. Proof.

In either case N is

in A* so that Theorem 6.1 yields

Theorem 5.6(iii). Let E be an approximate p-interpolation set for A with p a bounded dominator such that for some x, 0 s <1, 6.9. THEOREM.

forall m EA1.

lirim

(*)

Then E is an exact p-interpolation set for A. If, in addition, U° and N are A-stable and (*) holds with m E aA only, then E is an exact p-interpolation set. If p 1 andp 1 on E then (*) also implies k(E) is a generalized peak set.

so, by Corollary 6.8, it Theorem 5.3 shows Ci = (U° remains only to show the decomposability of U in A*. We apply Lemma 6.5 with S = U° = W. If lixiL = 1 in A* then we can choose /L with 4 = x and U°. Since U° is split, = 1. If U° is A-stable then we can choose J) + A%, (1 = 1, 2). Given m E A 1(aA = Proposition 6.2 shows choose n E with Proof.

1I—irim

Since

IL

and

II are

sIIlr2mIIP.

equivalent there is a k such that for i' E M(X),

COMPLEX FUNCtiON SPACES

186

Thus sillr2m —1r2/Liip

5117r2#.Lllp

m

+

n — m)

—

(7r2/L —

—

ifl'2/LIip —111T21L

Hence iilr2I.L liP

(1

+ ks)/(1

which shows Lemma 6.5(u), or 6.5(u1) in case U° and N are A-stable. If p 1 and p 1 on E then = im(.E)l

iiirim and ii— 7T2m lip

ii—lr2mil

=

that Theorem 6.6 or Corollary 6.7 applies to show k(E) is a generalized peak set. so

6.10. COROLLARY. LetE be a closed set and letp be a bounded dominator such that for some s, 0s< 1,

foralim

iiirim +A

(*)

Then E is an exact p-interpolation set for A. lip is A-superharmonic and E satisfies either

(i) EcaAX,Or (ii) E = F Xfor F a closed face of SA, and if (*) holds for m E

only, then E is an exact interpolation set for A.

Proof. If (*) holds for m E A ', (aA1), then Theorem 5.3 or Corollary 5.4 shows E is an approximate p-interpolation set for A. Since un m


the rest follows from Theorem 6.9.

We note that if s =0 in (*) then E is an M-sct and (*) is satisfied for any p (superharmonic p in the I9A1 case). This is just the conclusion of Corollary 5.8 in this case. We consider now some examples illustrating the various dual interpolation criteria of the preceding sections. First we present an approximate interpolation set which is not exact.

DECOMPOSABILITY AND EXACT INTERPOLATION

187

Let X be the unit ball in the real sequence space 11 = =1. Then take A = c0, the pre-dual of 11, so that sup Let E be the singleton {x°},

and let = hail =

n=l,2 Let A, E, X, p be as above. Then E is an approxim ate, but not exact, p-interpolation set for A. 6.11. EXAMPLE.

= 1, then Proof. If a E c0 and (a, x°) = = 1 implies each a,1 = 1. But a E c0 means that -*0 so exact p-interpolation is impossible. It is clear, however, that we can find a E c0 with (a, x°) =1 and hail 1+ s for any given 6>0. Thus £ is an approximate p-interpolation set for A.

The next few examples (as well as some in Section 7) are based on the disc algebra whose basic properties we now sketch.

6.12. The Disc Algebra Let

be the unit disc in the complex plane C and let

={fE

f is holomorphic on mt

denote integration on T C using normalized Lebesgue measure Let and define for p M(T) the Fourier coefficients

n=0,±1,±2 If fe C(T) then by 1(n) we mean (fdu)A(n) so that 1(n) =

If g E C(T) and

frf(z) du(z).

E M(T) then define the convolution

(g * p)(w) =Jg(wi)

w e T.

(1) The polynomials in z are dense in Proof.

Take fr(Z) =f(rz) (fE

and 0 r <1) and note that fr jS

holomorphic on a neighbourhood of hence uniformly approximable by a polynomial, and f, -*f uniformly on & (2) The circle T is a boundary for

COMPLEX FUNCTION SPACES

188

That the map! -*flr is an isometry is immediate from the Maximum Modulus Principle (Rudin [187]). Proof.

We denote

(3) Let E M(T). Then =0 if and only if 4(n) =0 for all integers n. If p is real then ,a. =0 if and only if 4(n) =0 for n 0, 1,2 If g E C(T) then and (g * Proof.

The Stone—Weierstrass Theorem shows that polynomials in z, are e C(T) = {0}. If is real

dense in C(T). Hence 4 =0 for all n implies

L

then

4(—n) = 4(n) so the second statement follows. The continuity of g * the convolution formula follows from the calculation (g *

=

j

*

=

du(w)

is easy to check and

dcr(w)

=

du(uz) d/.L(z)) =

=

where we have used the change of variable w = uz, Fubini's Theorem and the translation invariance of do on T.

(4) A(T)={f€C(T):f(n)=0;n=—1,—2,...}. From (1) and the fact that do(z) = 0 for n = 1,2,... we see that A B, where B is the set on the right. We show next A C B. Proof.

Let ii E A1 and define

=

dv(z);

fe C(T).

Then

4(n)=Ji"diz(z)=Jz"dv(z)O;

n0,1,2

If gEB then

(g * ,4)(n) = j(n)4(n) =0 for all n

DECOMPOSABILITY AND EXACT INTERPOLATION

so that g *

189

=0. But

0=g *iz(1)=j' g(2)dp =j'gdv so that J'EB. (5) (Wermer Maximality Theorem). Let A(T) C B c C(T) where B is a closed sub-algebra of C(T). Then either B = A(T) or B = C(T). E B then B = C(T) by the Stone—Weierstrass Theorem. If If i = B then z generates a proper maximal ideal in B so that the Gelfand— Naimark Theorem (see, for example Stout [204]) shows there is an algebra

Proof.

homomorphism d E SB such that represents b with respect to B then

=0. Thus, if

(X) and

E

n=1,2,... =0; n = 0, 1,2,... and

so that (p

= cr. But then if g E B and

have

=

du(z) =

=

=0.

Then (4) shows g E A(T).

6.13. EXAMPLE. Let A = A(T) c C(T) and let E be a non-empty proper closed subset of T. Then

(1) AlE is dense in C(E). (2) If E contains an arc then Efails to be an interpolation set for A.

Let J be a closed sub-arc of T and let f E A with f 0 on .1. Then 0 on T, as can be seen by taking a finite number of rotations

Proof.

f,(z)

such that the product g =

a, E T

f,, is identically zero on T. But then the

holomorphic extension of g is identically zero on A implying that one of the factors, and hence f, has uncountably many zeros on the interior of A. This on T. means 0 in A(A), and thus, Now let B = (AlE) C C(E), M = {fE C(T): flE = 0} and let A1 = (A +M) in C(T) so that A1 is a closed sub-algebra containing A with

Since E is proper in T there is a closed sub-arc J c T\E and on E and on J, fE C(T) with A lIE = B.

f=l+(f—1)EA+M.

COMPLEX FUNC11ON SPACES

190

But the above shows

A so that

= A 111! = C(E). Hence 6.12(5) gives = C(T) so that Similarly, if E contains an arc then there is a C(E) such that f is identically zero on a sub-arc of E. But again, such ant is not the restriction of an element of A. We will discuss A' (A the disc algebra) at the end of Section 7 and use this to

relate the exactness criteria of Corollary 5.8 to the original Rudin and Carleson theorems. For now we return to the more general subspace exactness criteria.

If E is a closed subset of X with N = M(E) A-stable and p is Asuperharmonic then we have the following measure theoretic conditions: (i) 6.10(*) (ii) 6.9(*) (iii) 6.7(*)

sII—lr2mIIP

Ilirim

for alt m

Ilirirn +A'110

for all m ('IA

m(E)I sIIir2mII

It is clear that (i) implies (ii) and, for p

for all m

1,

Ilirim so that (ii) implies (iii). Thus any of these conditions for p 1 imply, by Corollary 6.7, that E is a face of SA. Thus the condition (*) of Corollary 6.10, while consonant with the

traditional formulations, is geometrically rather restrictive. The next example shows that it may be more productive to combine Lemma 6.5 and Corollary 6.8 to obtain an exactness condition based directly on decomposability. Let X be a square in R2 with vertices denoted {1, 2, 3, 4}. Let E = {1, 2} with 1 and 2 diagonally opposite vertices. Take A = A(X) (real affine functions) and = = 1. One can verify with elementary calculations that E is an exact p-interpolation set for A, but it would be reassuring to have this confirmed by a theoretical result from this section. The basis is of course the decomposability of X by the diagonal through E. 6.14. EXAMPLE. In the above set-up, Corollary 5.4 shows E is an approximate p-interpolation set. The condition (*) of Theorem 6.9 fails. However Lemma 6.5(i) and (ii)' hold (for S = U°) so that the conclusion of Corollary 6.8 holds, showing that the diagonal

F=coE is a decomposable section of X and E is an exact p-interpolation set for A.

DECOMPOSABILITY AND EXACT

191

Proof. We note that aAM(X) is the four-dimensional space spanned by the at the vertices. Then point-masses the one-dimensional space generated by

m =rj+s2—s3--e4. If = A8, then liz U° = M1(X) with

=

A1

Since

=

E C t9AX, N is stable and so is

au° =co Clearly

'n N = {O} since F is a (one-dimensional) simplex. Thus rr1m+A £ mN

=

and so Corollary 5.4(iii) shows E is an approximate p-interpolation set but

Theorem 6.9(*) fails. Let S = M1(X). To show Lemma 6.5 applies it suffices to demonstrate (ii)' since the stability requirements and (i) are immediate. then Let = If A3 and A4 are opposite in sign then .

.1.

ir2/z—ir2)

If, say, 0 A3 A4, consider v = Ilvit

—A31

jz + A3m. Then

=

+ A2'

.1.

and —

IA3',

=

and

Hence in either case Lemma 6.5(u)' holds.

Let X be the square and now E = {1, 3} consists of adjacent vertices. Again, corollary 5.4 shows E is an approximate p-

6.15. EXAMPLE.

interpolation set forA(X) but Theorem 6.9(*) fails. However Corollary 6.7(*) holds showing that the edge

F = co E is a positively decomposable face of X and hence an exact p-interpolation set. Proof.

The first part is the same as in 6.14 and to check Corollary 6.7(*) we

have lm(E)i =

—

0.

Gamelin [114] and Briem [50] present interpolation results in terms of an extension constant e(A, E). We conclude by illustrating the relationship

COMPLEX FUNCTION SPACES

192

between this and Corollary 5.4 and Corollary 6.10. We shall deal with the

boundary measure case only, noting that similar but somewhat easier arguments apply in the earlier version. Let X be a compact metric space, 1 E A a closed separating subspace of C(X) and E a compact subset of äAA. Let e(A, E) denote the smallest number k such that for aiim

IIi'im +A

Let '& be the collection of all compact subsets G aAX\E and let

for (x,t)EExT for (x,t)€GxT

1

p(G,k)(x,t)= k 1vk

otherwise.

Then p is l.s.c. and hence a gauge on X. 6.16. THEOREM.

(1) The extension constant e(A, E) k if and only if E is an approximate p(G, k)-interpolation set for all G E (2) If e(A, E) <1 then E is an exact p(G, k) interpolation set for all k such

that e(A, E)< k. Proof.

Let p

p(G, k) and note that, by Theorem 5.1(iii)

Ili.'IL = IlL

for all

+ kIIlL Gil + (k v 1)il,L vu

E M(X). The stability of N and U° are immediate. Now we have

+A

llirim

If

(Y = X\(E u G))

E ÔAM(X) then, using the metrizability of X, = Il.41(8AX\E) = sup {l,L1(G): G E

Thus

p

(3)

p(G, k)and GE

Now, if e(A, E) k then

Ilirjm +A

(4)

kllir2mll

for all m E 8A-

so that (5) and

Ilirim +A' E is an approximate p(G, k) interpolation set by Corollary 5.4.

M-HULLS AND FUNCTION ALGEBRAS

193

Conversely, the inequalities (5) imply, by (3), that (4) holds. If k0 =

e(A,E)
m +'

= Iiirim + A'

Nil < ko(Iim all + urn

(ko/k)(kIimIaJI+(k v

I

=

so that Corollary 6.10 gives the exactness of solutions to p(G, k). Notes

The usual procedure in the literature for showing E is an exact interpolation set is to show E is a generalized peak set and then use the 2 " iteration

of Bishop [39] and Glicksberg [118]. Thus Corollary 6.10(*) with s =0

is shown in these papers to imply E is a (generalized) peak exact pinterpolation set (for continuous T-invariant p). Gamelin [113] shows that Corollary 6. i0(*) with p 1 implies that E is a (generalized) peak exact interpolation set. He also relates the smallest value of s to the extension constant of Theorem 6.16 and proves the pre-boundary measure version of Theorem 6.16. The version given in Theorem 6.16 is due to Briem [50]. Andô [19] gives a general Banach space formulation of Corollary 6. 10(*) and introduces the polar geometry which is instrumental in the development herein (cf. Asimow [28,29]). The peak set conditions and the notion of positive decomposability in Theorem 6.6 and Corollary 6.7 are given in Asimow [29]. The cases =0 is proved in Alfsen and Hirsberg[9] and the fact implies k(E) is a peak set (X that Corollary 6.10(*) (p 1, rn metrizable) is given by Briem [50]. He also gives the condition in Corollary 6.7 for the case E a singleton. The application of decomposability to exact interpolation criteria is given in [28, 29].

7. M-.HULLS AND FUNCTION ALGEBRAS We say the closed subset E of X is an M-hull for A (1 E A, a separating subspace of C(X)) if (i) M(E) is A-stable, (ii) rn E aAA implies m = rnIE E A' (iii) E = k(E) (k(E) = E) X c SA c A*) As in Section 4, if F satisfies (i) and (ii) E is termed an M-set and Corollary 4.5 shows that if E is an M-set then k(E) is an M-hull. It is easy to see that an

M-set F may be a proper subset of k(E)—for example, if X is the unit interval and A A(X) then E = {0, 1} is an M-set whose hull is X. For A a function algebra the natural objects to study are the M-hulls and we shall see• first that the characterizations of Theorem 4.4 can be substantially sharpened

COMPLEX FUNCTION SPACES

194

Throughout this section we take the complex state space ZA to be determined by a =1 and f3 = —i so that ZA = Co (SA U —ISA).

We refer to convex subsets of ZA as symmetric if they are of the form

co(F u-iF). In

E c SA) is symmetric.

= Co (F u —iF), (F =

particular the set

7.1. THEOREM. LetA be a function algebra in C(X) andEa closed subset of X. The following are equivalent: (i) E is a generalized peak set; (ii) m E A' implies m e A (iii) E is an M-hull; (iv) E = k(E) and ZE is a split face of ZA; (v) E = k(E) and ZE is a face of ZA.

LetE be a generalized peak set and let m EA1', gEA and s >0 be given. Choose K, a compact subset of X, such that Proof.

K ri E = Choose 1€

A such

that

0

IlfIl

and

ImI(K) + 8.

ImI(X\E)

=1 and

IfI<1

fEl onE.

onK,

for n sufficiently large, that

Since A is an algebra we can assume, by

IfI<e onK. Now ir,m = mIE

so that

irim(g)I =

fdmJ

JJg dm1 =

=

L1X\E1

dImI+e) Hence ir1m E A

and (1)

implies

(ii)

is shown.

(ii) holds and x k(E) then there is a probability measure on E such that represents x. Thus If

supported

EA so

that /

Since

= 1 it follows that x E E. Thus E

k(E) and,

if

is a maximal

M-HULLS AND FUNCTION ALGEBRAS

195

measure representing x, then

EA1 so that /h(E) 1. This shows E is and therefore E is an M-hull. Now Theorem 4.4 and Corollary 4.5 show (iii) implies (iv) and (iv) implies (1). Thus (i)—(iv) are equivalent and trivially imply (v).

If (v) holds then we shall show (ii). Thus let m At, g E A and E >0 be given.

Choose K compact in X such that K E =0 and

Now K is a compact subset of ZA disjoint from the face 2E.

Thus

Proposition 2.10.6 shows there exists Of E A(ZA) such that

0Of onZA OfS

OflZE

kOf onK where 8 and k are chosen as follows: Let h = e -f E A and note that fl i128 on E so that 6 is chosen sufficiently small that

1—hi8 onE. Since hi exp (—re f) we have hi ihie onK. Hence =

1 On X and can choose k so large that

IL (gh +g(1 — h)) dmj

IL gh dm1 + j'

—

h)l dimi

gill —hldm

iIgll[e(imi(K)+ 1)+rImKE)]. We see that the proof of the last implication is a variation on the argument

used to prove the Bishop Peak Point Theorem (4.3) where we show co (x, —ix) is a face of ZA. The equivalence (ii) shows that once E is known to be an M-hulI the key

property (ii) of the definition holds for all m E not just the boundary measures. 6.10 (s = 0), the M-hulls are exact p We note as well that, from

interpolation sets for any dominator p and that the extension constant e(A, E) is either zero or greater than or equal to one. We discuss an example of the later phenomena in Example 7.20.

FUNCTiON SPACES

196

It follows from Theorem 7.l(iv) that the collection of M-hulls for A is closed under intersections and finite unions. We shall use the notation dAX = ext (54)

and

Since F = co E c 54 is a face, aAE = ext F, and, if E1 (/=1, 2) are M-hulls then E1 c E2 if and only if aAE1 C 8AE2. Statements (iv) and (v) serve to characterize the symmetric split faces of ZA. We now investigate the properties of A pertaining to general split faces

of ZA, interpreting some standard function algebra concepts from this geometric perspective. This will be in the same spirit as the result in Theorem 0.2.

A useful tool in relating facial structure in ZA to A is the Silov decomposition. Let A be a closed subalgebra of functions in C(X), but do not assume now that A contains constants or separates points. Let A0 = {u E A: u is realvalued}. We say E is anrisymmetric for A if fE A and real-valued implies is constant. If X is antisymmetric for A we say A is an anti-symmetric algebra. e ext (A1). Then supp /L is LetA be as above and anti-symmetric. If 1 E A then supp is not a singleton. 7.2. PROPOSITiON.

real. Choose a, b real (a 0) Proof. Let E supp and let f E A with so that if g = af+ b then 0< g < ion E. Then, g need not be in A, but since A is an algebra

(1—g)/.LEA.

and

Now 1= I,.'ll Hence

jL

g

is

j'

=J

g

+

a convex combination of g

(1— g)

= Hgpit+ 11(1 — g),.'I1.

g

so is is clear that no element of A' can be a non-zero point mass.

Decomposition of X with respect to A to be the collection S9 of the maximal sets of constancy for functions in A0. Thus, for

We now define the

each XEX,if = {y: u(x) = u(y) for all u A0}

M.HULLS AND FUNCTION ALGEBRAS

then = subsets.

197

The elements of .9' form a partition of X into compact

7.3. THEOREM. Let

be the closed algebra of functions A.

decomposition of X with respect to the

(i) If 1 EA then each EE.9'is a generalized peak set. (ii) If E0 is antisymmetric for A then E0 c E for some E E .9g. (iii) A ={fE C(X): fIne AJE, VE E .9'}. Proof.

(i) If E there is a u A() such that UIn 0, u(x) #0 and IIuII< 1. If v = 1 — u2 then IJvII = 1, v(x) < 1 and E c v'(l). This shows E is an intersection of peak sets. Statement (ii) is by definition. (iii) Let f C(X) and lIE AlE for all E E .9'. If A1 {0} then choose

€extA. Then F0 = supp /L C E Thus, if fIE =

El'.

g e A,

=f fd,L—f gdp.=fgd,L =0 so that

If A is a seif-adjoint function algebra (separating and containing constants) then A0 separates points so that .9'= {x}XEX. Then (iii) shows A = C(X), which is precisely the complex Stone—Weierstrass Theorem. If K is closed in X we denote by 'K the closed ideal of all functions in C(X) vanishing on K. Clearly ('K

).L

= {/L

C M(X): supp

c K).

If K = 0 then 'K = C(X) and if K = X, 'K = {0}. We observe next that all closed ideals in C(X) are of the form 7.4. COROLLARY. LetA be a closed ideal in C(X) and let

K —{x:f(x)=OVfeA}. Then A = 'K. Proof.

Clearly A c

Let .9' be the If

If x K then clearly K C f(y)=1. Let

decomposition with respect to A.

K then there exists f cA such that

g=1fJ2=ffeAo.

COMPLEX FUNCTION SPACES

198

Then g(y) hE

1. 11 z E X\{y} use the Tietze Extension Theorem to choose with

h(y)=l

hEIK,

h(z)=O.

and

Then u = hg A0 with u(y) = I and u(z) = 0. Thus = {y} and it follows that consists precisely of K (if K 0) and the singletons {y} for K (if

K X). Thus f€

implies

VE€.'/ so

that 1KCA.

We now take A to be a function algebra so that 1 E A and A separates points.

The largest closed ideal of C(X) which is contained in A is called the essential ideal of A. This is well-&fined since if and '2 are ideals in A then + 12) . Then Corollary 7.4 shows the essential ideal is 'K for a closed subset K of X which we call the essential set. if K = X (equivalently 'K = {0}) then we say A is essential.

so is

7.5. THEOREM. Let A be a function algebra and let K be the set for A. (i) K is a generalized peak set (equivalently an M-hu!l).

(ii) K cE (E closed) if and only if supp cK (iii) If x e K then (iv)

c Efor all

E

={x}.

(v) A ={f€ C(X):fIK EA1K}. (vi) is essential in C(K). (i) follows from the Tietze Theorem and the fact that 1 E A. For (ii), c (JE) . Parts (iii) and (iv) are straightforward applications of the Tietze Theorem. in (v), if for = Proof.

K E if and only if 'E c 'K if and only if

g A then f— g 'K c A so that f€ A. For (vi) let K0 c K and suppose IK0IK AIR. Jig E IKOC C(X) then Thus 'K0 c A, implying K = K0.

1K4IK c AIK so that (v) gives g E A.

We now return to the representation OA = A(ZA) and identify the split faces of ZA. 7.6. THEORF.M.

LetF,(j =

1,

2) be closed and convex in SA such that

F=co(F1 u—IF2) is split in ZA. Then E, = F, Proof.

Xis an M-hull.

Let G be the complementary face of F in Z4. Then

G=co(G1 u—iG2)

M-HULLS AND FUNCTION ALGEBRAS

199

where and

G2=i(Gn(—iSA)).

Let h be the affine function on ZA such that

hJF=OlIp and h

is a well-defined u.s.c. affine function such that

h

1 and

<1

hi

on SA\FI. Now Theorem 1.6.1 (ix) shows there is a

pointwisc to h. Then O((1 + Ofa function h0 on ZA such that

in A such that

converges pointwise to an affine

h0(x)= h(x)--h(—ix) for XE SA ho(—ix) = h(x)+ h(—ix)

for —ix

E

Hence

Osh01 OflZA h0

1

on

co (F1

—iF1)

and

hI) <1 on ZA\co (F1 u —iFi).

Therefore co (F1 u —iF1) is a face of ZA and the conclusion for F1 follows from Theorem 7.1(v). The argument for E2 is the same. If E1 and F2 are closed sets in X we say F1 U —iE2 is an A-interpolation pair if F1 u —iE2 is an interpolation set for A(ZA) as a real suhspace of as

in Theorem 3.1. As usual this just amounts to the real linear span, N, of F1 u —iF2 (F, = cö E, c SA) being in A*. Here so that

is indeed the quotient image of N.

7.7.

only if N +A'is

The pair E1 u —iF2 is an A-interpolation pair if and in M(X).

Proof. We have E1 u —iF2 an A-interpolation pair if and only if iQ in A* if and only if N +A is in M(X).

is

We let

I is a real subspace in C(X) and jO = N. Thus, if B = I n A then B° = (I°+A°) = (N

in M(X)).

COMPLEX FUNCTION SPACES

200

7.8. THEOREM. LetE, (/ 1,2) be M-hulls for the function algebra A such thatX = E1 U E2. If E1 U --iE2 is an A-interpolation pairthen the essential set Kc E2 so that

c A. Let

Proof.

(/= 1,2)

on

and let I he as above. We note that

c (B2)0 (I

(the real-valued functions in B2)

(*)

C i(B1)0.

We show that the consists of the set E1 clearly E1 E2 c

decomposition of B1 as a subspace of C(E1) E2 and singletons in E1\(E1 E2). If x E E1 n E2 then E E,, let

={y EE,: f(y) =f(x) Vf€ I C Then (*) shows is a singleton if and it suffices to show E1 is a generalized peak set for E2. We observe first that since z E constant and f(z) implies there is an fE I A with 1ff =1 then (*) shows

= iu2

and if f=2 then fIE2 = u1

(iii real, 1

1, 2).

Thus for e >0 sufficiently small

g =1 ± e(f—f(x))2 (+ for j = I and — for j=2) satisfies

and

g(z)<1.

is a generalized peak set for AlE,, hence an M-hull for AlE,, and Thus is an M-hull for A. therefore (since E, an M-hull for A), c E1 E2 if and only if ÔAKX C E1 E2. Thus one can assume Hence xE and y x. Then, since x E aAX, CorolE2). Suppose y E lary 4.3 shows there is a pcak function fin A such thatf(x) = 1 and fl <1 on we have (E1 E2) U {y}. Since x, y E Cx

which is

=p.

by Proposition 7.7. Thus —iv2);

p. €A

p1

real and supp ,.',CE1

(1=1,2).

M-IIULLS AND FUNCTION ALGEBRAS

If Jo is the point-wise limit of

201

is a characteristic function

as n —>00 then

gives the contradiction

and evaluating at

1 = iv2(fo).

= {x} if x e E2) (j = 1,2). This shows is as desired. Since Et E2 is a generalized peak set we see that

Hence y must equal x and {O} for x

so that Theorem 7.3(iii) gives

E1\(E1

in C(E,).

B, = If

then

EM(E,): supp Thus A' implies supp E2 and

cE1 mE2} (j = 1,2).

c E, E2 so that Theorem 7.5 shows K E1 c A.

The case E1 = X and E2 =0 amounts to a theorem of Hoffman and Wermer [130]. 7.9. COROLLARY.

If re A is uniformly closed in CR(X) then A = C(X).

Proof. We have A(ZA)1s4 = {uIsA; u = ref, f€ A} so the hypothesis says X u — i 0 is an interpolation pair, giving

C(X)cA. Let E1

E2 = (E1\E2) u (E2\E1).

7.10. COROLLARY. If E1 (f = 1,2) are M-hulls with E, u —iE2 an Ainterpolation pair then p. E A1 implies IL I(E, Proof.

E2)

= 0.

If E = E1 u E2 then E is an M-hull and so p. E A' implies ILIE€

By applying Theorem 7.8 to E, we see supp p.IE c E1 uE2 so that If we combine Theorem 7.6 and Corollary 7.10 we have a characterization of the split faces of ZA. 7.11. COROLLARY. Let A be a function algebra with complex state space ZA. The following are equivalent:

COMPLEX FUNCTION SPACES

202

(i) E,(j

1,

2) is an M-hull and E1 u —iE2 is api A-interpolation pair,

(ii) E1 is an M-hull and A1 implies A E2) 0; (iii) E1 = F1 m X and F = Co (F1 u —iF2) is a split face of ZA.

Corollary 7.10 gives (i) implies (ii). If (ii) holds and = — with on X —iX as in Theorem 3.1 then 1.Z A(ZA) implies so that

Proof.

=

ref

f

imfd,h2=ref

Theorem 7.6 gives (iii) implies (1).

If we apply Corollary 7.11 to the case E1 refined version of Corollary 7.9. 7.12. COROLLARY.

E and E2 =

The following are equivalent:

(1)

E is an M-huil and (re A)IE is closed in cK(E);

(ii)

E is an M-hull and AlE =

(iii) E

0 then we obtain a

where Fc SA is a split face of ZA.

7.13. COROLLARY.

The following are equivalent:

(i) co (F u —iSA) is split in ZA; (ii) E implies supp c E F n X; (iii) the essential set K E. Proof. of (1)

The case F = F1 and SA = F2 in Corollary 7.11 gives the

equivalence

and (ii) since (ii) implies E is an M-hull. The equivalence of (ii) and (iii)

is Theorem 7.5(u).

7.14. COROLLARY. If A is a function algebra in C(X) then A is reflexive if and only if X is finite and A = C(X). Proof.

Assume A is

reflexive. If

x E ext S4 then {x} is an M-hull with

C({x}) so that Corollary 7.12 shows {x} is split in Z4. But then the complementary face is compact. The intersection of all the complementary Af{x} =

of extreme points of Z is a compact face disjoint from ext Z4 and hence empty. But if x0 is an accumulation point of ext ZA then x0 must faces

belong to each complementary face. It follows that ext ZA is finite and hence a finite dimensional simplex. Thus A is seif-adjoint and equals C(X). Since

f2 A whenever fE A it must be the case that X = ext SA. A theorem of Sidney and Stout [194] shows that the equivalence of (i) and (ii) in Corollary 7.12 holds for arbitrary closed sets E, not just M-hulls.

M-HULLS AND FUNCTION ALGEBRAS

7.15. THEOREM. AlE = C(E).

203

If A is a function algebra and (re A)JE

is closed then

Let B = (AIE). Thus if g B then there is a sequence in A such that -> g uniformly on E. Hence rc E -'re g uniformly so that there is an f€ A with re lIE = re g. This shows re B = (re A)IE is closed in C(E) and hence, by Corollary 7.9 B = C(E). Thus re AlE = and AlE ={O} and it suffices to is dense in C(E). Therefore (AIE) = A1 show there is a c such that, for = /J.1E and 1T2p, = Proof.

lkimlIcIIlT2mlI Given

for all meA'.

m At and e >0 there is a Borel function ifs

[0, 2ir) and

on

E with range in

dIml = c" dm.

By Lusin's Theorem there is u E C'R(E) such that lull

and

dlml<E. Since

the map f-+ reflE is a real bounded linear operator from A onto

(re A)IE there is a constant K (not dependent on u) and an fE A such that re flE = u and IIfII
0= m(e'1)= Thus

On the other hand, with =

= u + iv,

dm 'B

LIE

—e") dmj

uIeiIb_e:idImI (1117' mu

Thus IIirimll

The preceding results indicate the importance of identifying the measures in A. If we consider the disc algebra A(T) (Example 6.12) we are impressed by the special role of the (normalized) Lebesgue measure o' on T. It is the unique representing measure on T for the multiplicative state L defined by

L(f) = f(0);

I the holomorphic extension off.

204

COMPLEX FUNCTION SPACES

More generally, if SA is a simplex then each multiplicative state has a unique (boundary) representing measure. This is, for example, the case whenever A 'where in fact SA is a Bauer is a Dirichiet algebra (re A dense in simplex. The abstract F. and M. Riesz Theorem (Theorem 7.17 below) says that if is a unique representing measure on X for a multiplicative state in with respect to SA then each m E A' has Lebesgue decomposition ma + such that ma,

Let A be a function algebra and let be a multi.. plicative linear functional in S,1. Then b has a unique representing measure on X if and only if for all u E 7.16. PROPOSITION.

&T=sup{J refdu:JEA and refu}. Consider u on X SA and let ü be the lower envelope of u on SA. Then by Theorem 1.6.1 Proof.

à(b)=sup{Of(cb): Ofu on But Of = ref on X c Ix u dcr. Conversely,

5,1

XcZSA}=inf{J' udv:

so that uniqueness of r implies the formula for

Ju for all U E

shows the uniqueness of a-. Let A be a function algebra with a- a unique representing

7.J 7.

measure for the multiplicative state

5,1. Then m E A' if and only if

E A where m = ma +

m with

ma, respect to cr.

singular to a- and dma p da- where p is a Let m with a--integrable Borel function on X. Let h e A and e >0 be given. Choose k so large that if Proof.

B={xEX:lpj(x)k} then 211h11

(1) JB

Let E be a Borel set such that (r(E) = 0 =

205

M-HULLS AND FUNCTION ALGEBRAS

Then choose K compact, n positive, S >0 and U open satisfying

KcEcU (2)

e"

(3)

<6

(4)

fl(r(U)< 8. Let u E

with uIK

1, 0 u 1 and supp u E U. Then

Ju so that

f_nu do —,wr(U)>---8. Choose, by Proposition 7.16, fe A with (by adding an imaginary constant)

'r(imf)=O ref —flu o(ref)>—8. Let g=e1. Then =e

i

and

onK.

(5)

Thus

But since o

is

multiplicative, o(g) =

=

= re cr(g) = o'(re g).

Hence

(f 1

do')

2(1

(6)

Now

0=j'gh dm =Jh dma _f(1 —g)hp do'-I-j'

gh

(7)

COMPLEX FUNCTiON SPACES

206

with

fgh

e

by (5), (2) and (3).

Likewise

i L (1—

g)hp doj

211h11

f

P1 d(T +

L 1(1 —

du

by (1), (6) and (4).

Therefore (7) gives

Since e

is

arbitrary, ma E A and thus

m

so is

—

m4.

7.18. COROLLARY. Let A be the disc algebra on T and a' normalized Lebesgue measure on T. If =0 for allf E A such that o(f) =0 then m is absolutely continuous with respect to a'.

M

Proof. Let ={fEA: a'(f) =0}. Then, since cr(zk)=0 for all k = 1,2,... we have A spanned by M and the constants. Thus if m €M and c = m (1), =0 for E by the theorem. Thus m — CuE A-so that (m — = If dm5. We have all k =0 for all k n then consider di' =

0.

for j

1

LzuI so

that v E

M'.

But the

above then yields

o=f dv =f(i)"

We conclude ifzç(n) =0

= ii A1. Hence

" dm.ç(z)= rhç(n +1).

for all n so that

=0.

7.19. CoRol LARY

THEOREM). LetEbe a subset of T with u(E) =0. Then E is an M-hullforthe disc algebra A on T. In particularE is a peak set and a full exact p-interpolation set for any dominator p. Proof.

Let m e A1. Then dm = h du for a u-integrable h on T, by the

preceding corollary. Thus mE =0. Then Theorem 7.1 shows E is an M-hull and the rest follows from Corollary 2.3 and Corollary 6.10.

207

M-HULLS AND FUNCTION ALGEBRAS

We can apply Corollary 7.18 to obtain another example of interpolation. Let B= so that B is the subalgebra of C(T) given by and

B=A' Let E={—1, 1}.

7.20. EXAMPLE. With B and E as above, E is an approximate, but not exact, full interpolation set forB in the uniform norm. Proof.

then

mEA1 and hence dm = h dcr with

h d(r =0. Thus

=hodoH-ae1; if N = M(E) then clearly B

N = {0} and = al.

Since Ct and h0 dcr are mutually singular

= llh0 doll j h0 doj = Ial = lkri,LIl. This shows E is a full approximate interpolation set. 1ff E C(E) is ZIE then

there is no extension g (even in C(T)) satisfying

1

=

dt7.

As noted after Theorem 7.1 the extension constant e(B, E) must be 0 or 1. Here it is one and exact interpolation fails. Notes

General reference books for function algebras are Browder [57], Gamelin [114], Leibowitz [150] and Stout [204].

The fact that, for function algebras, generalized peak sets are polation sets was shown by Bear [36] and by Hoffman and Singer [129]. in Theorem 7.1 the equivalence of (i) and (ii) is given by Glicksberg [1181 using

COMPLEX FUNCTION SPACES

208

arguments of Bishop [38],

as

indicated in the notes to Sections 2 and 4; the

equivalence of (iv) and (v) is due to Ellis [102]. For Dirichiet algebras Glicksberg [118] has shown that interpolation sets are generalized peak sets.

Proposition 7.2 is due to De Branges [76]. The antisymmetric decomposition for function algebras was introduced by Bishop [38], where he shows that maximal antisymmetric sets are generalized peak sets. The

notion of essential set was introduced and developed by Bear [35]. Theorem 7.6 and Corollaries 7.11—7.14 are due to Briem [52] and Ellis [102, 103]. Corollary 7.9 is a theorem of Hoffman and Wermer [130] (see also Browder [56]); the generalization of the Hoff man—Wermer Theorem given

in Theorem 7.15 is due to Sidney and Stout [194]. Theorem 7.17 and Corollary 7.18 are versions of the F. and M. Riesz Theoreni, and for the history of these results we refer to the book by Stout [204]. Example 7.20 is due to Glicksberg [118] and Gamelin [113]. 8.

FACIAL TOPOLOGIES AND DECOMPOSITIONS

We recall from Chapter 3 that the facial topology on ext K (K compact convex) has as its closed sets all subsets of ext K of the form (ext K) F, F a closed split face of K. This topology is Hausdorif if and only if K is a Bauer X0 = ext K. We shall interpret this in simplex, in which case A(K)

the context of A(ZA) where A is a (closed, separating with constants) subspace of

and ZA

Co (SA

—ZSA) is the complex state space.

We denote aZA ext ZA (=aAX —IaAX) with

= ext SA. We define the closed sets of the symmetric facial topology on aZA as those of the form aZA F, F a symmetric split face of ZA. Since the collection of symmetric split faces is closed under intersections

Now F is a and finite convex hulls this is indeed a topology on symmetric split face if and only if F X c SA is an M-hull for A. Thus a set E0

is (relatively) closed in aAX if and only if

EO=aAE for an M-hull E of A. For this reason we call the restriction of the symmetric facial topology to f3AX the M-topology. This is of course weaker than the restriction of the facial topology of 9ZA to 8.1. THEOREM. aAX is compact in the M-topology.

The M-topology is weaker than the facial topology on äAX in SA which is compact by Theorem 3.1.1. Proof.

We next develop the notion of central functions and the result of Theorem 3.1.4 to the present context. Let (IA = A(ZA) and let f be a real function on

FACIAL TOPOLOGIES AND DECOMPOSITIONS

209

aZA. We say f is symmetric if f(x) =f(—ix) for all x I9AX. If u is a real function on aAX then we denote the symmetric extension to aZA by 17. We see that f is facially continuous if and only if there is an h E A such that for X

äAX re

h(x)=f(x)

im h(x)=f(—ix)

and for each a E A there is a b E A such that

(re h)(re a)=re b

(imh)(ima)=imb pointwise Ofl aAX.

It is easy to see that if f is facially continuous and symmetric then f is continuous in the symmetric facial topology. Likewise, if u is continuous on aAX in the M-topology then 17 is symmetric facially continuous on As in Section 7 we denote A11

{u E A: u real-valued}.

Finally, observe that 1ff is real-valued on

then f is real-valued on X.

Let u0 be a real-valued function on aAX. The following are equivalent: (i) u0 is continuous in the M-topology, (ii) 170 = OhIÔZA; h = U + iu, u E A0 and Oh central in A(ZA), (iii) = ulax for u A11 such that for each a E A there is a b E A with ua b pointwise on 8.2. THEOREM.

If is symmetric facially is M-topology continuous then continuous, and hence facially continuous, on OZA. Thus there is an h E A with = and the symmetry of shows h = u + iv with u = v on aAX. But then h — ih e A and is real-valued on aAX, hence on X. This shows U = v E A0 and thus, (i) implies (ii). If (ii) holds the above formulas show

Proof.

u(re a) = re b

u(ima)=imb on so that ua = b on aX. This in turn shows that O(u + lu) is central in A(ZA) so that 170 is symmetric facially continuous and hence u0 is Mtopology continuous.

The following are equivalent: (1) The M-topology on aAX is Hausdorff; (ii) aAX is compact and A is isometric to C(aAX).

8.3. COROLLARY.

COMPLEX F(JNC1JON SPACES

210

Proof. If (i) holds w*-.topology on

then the M-topology, the facial topology and the

8.4. COROLLARY.

Let A be a function algebra.

coincide. Thus SA is a Bauer simplex with A(SA) isometric to Nowf E A implies re flax is continuous and = hence continuous in the M.-topology. Then Theorem 8.2 shows re f€ A so that A is self-adjoint, so that in particular re AIax is closed in But re is always dense in A(SA) = CR(aX) so that re AIax = and (ii) follows. The converse is clear.

(i) A real function u0 on ax is M-topology continuous if and only if u0 =

for some Li

A0.

(ii) The M-topology is Hausdorff if and only if A = C(X).

Theorem 8.2 shows (i). For (ii) we note that Corollary 8.3 gives A isometric to C(aX) and in particular, A is self-adjoint. Then the Stone.-. Weierstrass Theorem gives A C(X). Proof.

8.5. THEOREM. LetA be a function algebra. The following are equivalent: (i) Otis central in A(ZA); (ii) f=u+iv; U, v EA0 on the essential set K.

If (1) holds we can apply the multiplicative formulas for central functions with a = 1 to obtain b1 A such that on ax Proof.

u = re b1 v =im b1. Thus b1 is real and so b1 A0. If a = o

—i

we obtain b2 e A with

re b2

= im b2

on ox. Then b2 is imaginary so that ib2 A0 and equals v on ax. Therefore b1 — b2 =fon x and it follows that b1 = u on x, ib2 = v on x, giving u, v A0. Let w = u — v. We show Ow is central in A(ZA). Given a E A choose h A such that

u(re a)=re b

v(ima)=imb on ax. Then b — va = w(re a) on OX. But b —va A and hence b — va E A0. Thus (Ow)(Oa) = O(b — va)

onax on —iax.

FACIAL TOPOLOGIES AND DECOMPOSITIONS

211

Thus Ow is central. Now E aZA:

for a closed split face F of Z. But —iaX c F implies

F=co (Flu—iSA): F1

X. But then WIK.

If (ii) holds and a E A is given let b = u(re a) + iv(im a)

on X.

Then b C(X) and b ua on K. But ua A and so Theorem 7.5(v) gives b A. It follows that Of is central. The function algebra A is antisymmetric if and only if the only centralfunctions in A(ZA) are constants. 8.6. COROLLARY.

In general we say the centre of A(K) is trivial if the only central functions are constants. We next consider abstract versions in A(K) of the Silov decomposition for function algebras. We first define the Bishop decomposition for function algebras and then give A(K) analogues of both decompositions.

Thus, let A be a function algebra and let

denote the collection of

maximal (with respect to inclusion) antisymmetric subsets of X. Since each {x} is antisymmetric and the closure of the union of all antisym.metric sets containing x is again antisymmetric, we see that is a partition of X into closed subsets. is called the Bishop decomposition of A. 8.7. THEOREM. (1)

(ii) each E E

is a generalized peak set;

(iii) a ={f€ C(X): fIEEAIE for allE E (i) is the conclusion of Theorem 7.3(u). For (ii), let E1 denote the intersection of all peak sets containing E. Then E1 is a generalized peak set and hence an M-hull. If E is a proper subset of E1 then the maximality of E shows there is an f€ A, fjE1 real and not constant. But fIB is constant, say fIB = c, so that for e >0 small, Proof.

g= 1—B(f—c)2 is

a real peak function on E1 with

Ec{xeEj:

COMPLEX FUNCTION SPACES

212

But then E2 is an M-hull for AlE1 and hence an M-hull for A. It follows that E2 is a generalized peak set such that

E E2 c E1. Since this contradicts the minimality of E1, it must be the case that E = E1.

The proof of (iii), using Proposition 7.2, is identical to that of Theorem 7.3(iii).

We see that in both .9' and

the elements are M-hulls and therefore

correspond to (symmetric) split faces of ZA. If K is a compact convex set we let be the collection of non-empty closed split faces of K. A sub-collection of is called a decomposition of K if it consists of pairwise disjoint split faces covering 3K (= ext K). Let be the sub-collection of sets FE 3' such that fIF is constant wheneverf is a centralfunction in A(K). We denote by the collection of maximal (with respect to inclusion) elements in 8.8. THEOREM. The collection isa decomposition of K with restriction to aK consisting of the maximal sets of constancy in 9K of central functions in

A(K). 0 and 1€ A(K) is constant on F1 If F, (I = 1, 2) E and F1 and F2 then f is constant on F1 u F2, hence Ofl Co (F1 u F2) which is also a are pairwise disjoint. If x E split face. It follows that the elements of let E = {y E aK: f(x) = f(y) for all f central in A(K)}. Proof.

Since each such f is facially continuous, E is facially closed in Thus E aK F, Fe Since F = aF (= ext F), if f is central then flE constant implies fIF is constant so that F e 3". If F c F1 €3" then the definition of E c Hence F is maximal and so, 3's covers aK. shows that The collection is termed the abstract S1ilov decomposition of K If A is a function algebra then the abstract Silov decomposition of ZA determines the Silov decomposition of A in the following sense. 8.9. THEOREM. Let A be a function algebra with complex state spa-e ZA. Then E e .5°, the Silov decomposition of A if and only if

E=Fr'mXCSA

for Fe Proof.

the

abstract

decomposition for ZA.

Given Fe .9' then F is an M-hull so that, with 8E F

c SA,

213

FACIAL TOPOLOGIES AND DECOMPOSITIONS

Let x E 1E c aZA and let x E FE

If Of is central in A(ZA) then Theorem

8.5 shows ref = u E A0. Hence u is constant on E so that Of aF = UIaE is cF X and hence E c constant. It follows from Theorem 8.8 that Xc Conversely, u E A0 implies (by Theorem 8.5) that O(u ÷ iu) is central and hence constant on F. Thus u is constant on F X so that This shows E

We shall see that the abstract

decomposition determines A(K) in a

fashion analogous to Theorem 7.3(iii) for function algebras. We develop this pcoperty in conjunction with the abstract Bishop decomposition which is described as follows. We let, for the closed non-empty split faces of K,

= {F

the centre of A (F) is trivial}.

The elements of 2?' are called abstractly antisymmetric. Let maximal elements of 2?'. 8.10. THEOREM.

The collection

is

denote the

a decomposition of K such that for

with F1 F2 0. Let F = Co (F1 u First, let (1= 1, 2) E E Now, central functions in A(F) restrict to central functions in A(F,) and since F is as well. This shows elements of are pairwise disjoint. If x E t9K let be the smallest element of containing x. Then must be antisymmetric, for if f is central in there such that is a split face G of Proof.

G = {y Hence x E 0 =

aFt: f(x) = f(y)}.

so that f is constant and

€2?'.

if

={G: XEGE2?'} then

U

is non-empty. Thus, let F be the smallest element in containing We show F is antisymmetric. Let f be central in A(F) and let

E = {y äF: f(y) =f(x)}.

If G all G

then G c E. But E = 9F n G0 for

f is constant on G and

a split face of F which contains IIG.for Thus G0 = F and so f is constant on F. Clearly F is maximal in 2?'

and FcF0€ As in the decomposition for function algebras, we can associate the abstract Bishop decomposition with the Bishop decomposition of A.

COMPLEX FUNCTION SPACES

214

8.11. THEOREM. LetA be a function algebra with Bishop decomposition Then E 6 if and only if E = F X SA for some F€ the abstract Bishop decomposition for ZA. Proof. Let E E (F0 u 6

so that E is an

with F0 =

(E) and G =

the split faces of ZA. Now G is the complex state space of

AlE so that if h is central in A(G) then h = Oflo, fE A, and Theorem 8.5 shows f = u + iv with tilE, VIE 6 (AIE)() and u v on the essential set of AlE. But E antisymmetric for A implies that uJn, VIE are constant functions and that AlE is essential. Thus u = v on E and Of is constant on G. Thus G E so that

with F1 c SA. Let E1 = F1 F Co (F1 u—iF2) Theorem 7.6 shows E1 is an M-hull and hence so is E = E1 u E2. Let 0 be the symmetric split face of ZA such that G X = E. We show F is antisymmetric for A. Let satisfies Theorem 8.5(u) so that fEA such that fJE e(AIE)o. Then

O(f+ if)Io is central in A(G). Thus its restriction to F is constant, say, c. Then O(f+ if)IE, = re tIE, = a = re fIE2 = O(f+ so

that fJE = re fIE = a. Hence we have, since E is antisymmetric,

We now take up the question of to what degree these abstract positions determine A(K). Let X = (ext K) and consider A = A(K)Ix as a subspace of CR(X) with state space K. As usual, for subsets C of MR(X), aAC or äC denotes the intersection of C with the boundary measures in for (ext K) E for subsets E c K. MR(X). We continue to use We first establish the abstract analogue of Theorem 7.6. 8.12. THEOREM. Proof.

then supp1a

If IJ

Let F0 be the smallest element in

containing supp p. We show F0

is antisymmetric. Let f be centraFin A(F0) and assume Of 1. Then for each a€A(F0) there is a b€A(F0) with af—b on and hence on supp

This gives

j so that f f dp + (1—f)

af

=0

and (1—f) EAt. Since is extreme in At and we have f = A A a constant. Then f = A a.e.

FACIAL TOPOLOGIES AND DECOMPOSITIONS

A on supp

SO that

213

If G is the split face of F0 such that aF()

G = {x aF0: f(x)

then fA on G so that suppjicG, implying

On F,,. Thus F0 is antisymmetric.

f

Similarly

We need to show next that if f€ C(X) is annihilated by a(ext At) then f is To accomplish this it is necessary to introduce a annihilated by all variant of the Choquet ordering on a set. If W is a compact convex set then we say a subset T is hereditary-up if partially ordered

T and

ii

>

implies

e

We say T is extremal if

x=Ay+(1—A)zET; y,ZEW, impliesy,zeT. For example, T may be a union of faces. A function f on W is isotonic if

v> '1 implies f(v) 8.13. LEMMA. Let (W, >-.) be as above with the continuous convex isotonic functions separating the points of W. If T is a closed extremal hereditary-up subset of W then there is an element of T which is maximal with respect to and belongs to ext W. Proof.

Let

= {E c T: 0 E is closed, extremal and hereditary-up}.

Since the properties of elements of are preserved under intersections every sub-collection with the finite intersection property has non-empty intersection. Thus there is a minimal (with respect to inclusion) element 1ff is continuous, convex and isotonic and if E00 is the subset of E0 E() E is non-empty, extremal and, where f attains its maximum on E0, then since E0 is hereditary-up, E00 is as well. Thus E00 E Therefore E00 = E0 and must be a singleton by the point separation hypothesis. It follows that and extremal, hence extreme in W. this point must be maximal

For f€ C(X) let / and J be the upper and lower envelopes of f on K as in Chapter 1, Section 6.

8.14. PRoPosrrroN. Let X = (ext K) and f€ C(X). If Proof.

Define >. on Af by p >..

if it(p1)+

=0 for all

COMPLEX FUNCI1ON SPACES

216

for all p1' P2 continuous convex functions on K. If p is continuous, convex and non-negative then the maps (p)

are continuous and convex on A By considering the pairs p, 0 and 0, p they are seen to he isotonic as well. Since —P2 :p1 continuous, convex on K} is

dense in CR(K) we see that the elements of At are separated by the continuous convex isotonic functions. Now, givenfE C(X), let E with 0. We show v(f)> 0 for some ii E a(ext A Since is a boundary measure, are maximal so that /z. and

=

(--I) =

P2 continuous and convex on K with

Therefore there exist

and —

(f) +

/L (Pi) + /L (p2)

(—f)) =

I

0.

Let

a = sup {p '(pi)+ ir(p2):

ii E

At}

and

T is closed and hereditary-up in At. Furthermore, T is extremal since a >0 implies lvii = 1 for all v T. But then if ii E T and

vEAj

(i1,2)

then Ikil = 1. Since these measures annihilate the function 1, liv

I

1 I

and

ii,

12 I

1.—

If B1, B2 is the Jordan decomposition of X for ii then v

1

(B1) 1

so that p

+i4)

and

+i.',).

Thus

and it follows that E T. T and 11 maximal Therefore Lemma 8.13 yields i' (ext are maximal measures so that latter clearly implies that p and

vca(ext Ar).

The

FACIAL

T gives ii

If X = (ext K), let A = A(K)Jx

and denote

1=1=1

L ={f€

(p2)=a >0.

on Xc

K}

where / and fare the upper and lower envelopes off relative to K. 8.15. PROPOSITION.

A ={fEL:

Ea(extAt)}.

Clearly A is contained in the set on the right. Suppose then, f€ L Ai). Then Proposition 8.14 shows = Ofor all E =0 for all p e A , implying that We prove that fEA A. Thus, given p A with p Owe have p = p — p and we can 1. Now choose v1 and maximal on assume Ip 'jJ = p(l) = p. (1) = K with Proof.

and f is annihilated by all

and v.>p. If a EA(K) then

'(a)—(a)=p.(a)=() so that 7.' =

Also

— P2 E

p.

=

= p.

(f) =

<

vi(f)

SO that v1(f) = p. (f) and similarly P2(f) = p. (f). Consequently

8.16. THEOREM. LetX = (ext K) andA

1ff E L and

for all

E

(the abstract Bishop decomposition of K) then fE A. Proof.

Let p. e a(ext A) so that by Theorem 8.12

supp p. c X F for some F

f

If g E A(F) and g = on X F then choose h E A(K) with

/L(f)=j'

Xr'F

fdp.=f

Then Proposition 8.15 shows 1€ A.

gdp.=p.(h)=0.

=g

218

COMPLEX FUNCTION SPACLS

8.17. COROLLARY.

If X

c

then f€ A if and only

U

and

iffE

E A(F)IX(,F for all FE Proof.

Theorem 1.6.1 shows

E Mt (X) and

1(x) = supp

But x E F implies supp /L C F whenever

represents x}.

represents x. It follows that

f(x)=a(x)=f(x) for aEA(F)

with

Similarly 1(x) =f(x) for all x X so that fe L and Theorem 8.16 applies. 8.18. COROLLARY.

1ff E C(X) then tEA if and only if

forallF€ Proof.

Since the abstract Bishop decomposition is liner (Theorem 8.10)

than, if f€ L and satisfies E

A by Theorem 8.16. To show fc L it suffices to show

as in Corollary 8.17. Thus, let x X and G be the smallest closed split face of K containing x. If h is central but not constant on G then choose a strictly between the minimum and maximum values of h on G. Then there are split faces G1 (i= 1,2) of K such that

={y€aK: Now K = co (G1 u G2) and K = (aG1 aG2) c (aG1) (aG2) so that X E G1 or x G2. But G G is a proper split face of K which contradicts the minimality of G. Hence h is constant on G so that x

E

G C F€

8.19. COROLLARY. K is a simplex if and only if each

of the

abstract Bishop (Silov) decomposition for K is a simplex.

if K is a simplex then so is each closed face. Assume then that the -. = {O}. If not, then Proposition are simplexes. We show members of which, by Theorem 8.12, has supp 8.14 shows there is a E a(ext FE But then E aA(F) so that =0. Therefore K is a simplex. Proof.

LINDENSIRAUSS SPACES

219

Finally, if each element of is a simplex then the same holds for Theorem 8.10, and again, K is a simplex.

by

We conclude with an example showing that in general {(0, t) E 11 x R; tI 1} and G = {(x, 0) c 1' x x 0 and lix co(FuG)in l1xR. 8.20. EXAMPLE.

Let F Let K

The set K above is a metrizable simplex for which .5 LI

Proof.

1}.

0—

It is easy to see that

ext

K = {(0,

±1),

and

(ext K) = (ext K) u {(0, 0)}. Also, each clement of K has a unique representation as a convex combination of an element of G and (0, ±1). hence K is a metrizahie simplex. We have

A(K)=c0xR2;

f-(a,s,t);

and elements of the form (a, 0, 0) are central. Hence each

O)} E

1ff is

central in A(K) then there is an h E A(K) such that =f2 on {(0, 0), (0, ±i)}. h Thus f is constant on F and = {F,

On the other hand, every clement of A(F) is central (since F is a Bauer simplex) so that (0,

1) E

and

is

finer than

Notes

Theorems 8.1, 8.2, 8.4 and Corollaries 8.3, 8.5, 8.6 are given in Ellis [102] and Hirsberg [127]. The study of facial decompositions for compact convex sets was introduced by Ellis [1041, where the results 8.7 to 8.11 are proved.

Theorems 8.12, 8.16 and their corollaries are due to Briem [53], and Example 8.20 is due to Asimow.

9. LINDENSTRAUSS SPACES A real or complex Banach space A is called a Lindenstrauss space if A* is We isometric to a concrete L'(/L) space for some measure space (X, .5°,

220

COMPLEX FUNCTION SPACES

shall restrict our attention here to the case where A is a (close separating, with constants) suhspace of C(X). If A is a real subspace of CR(X) then A* is a simplex, as per the is an L' space if and only if the state space

discussion in Chapter 2, Section 6. We now have the tools at hand to characterize the complex Lindenstrauss function spaces using a general characterization due to Effros. The set-up here is the same as in Section 1 where U denotes the unit hail of A* We are interested in the subspace of C(U) consisting of the T-hornogeneous functions, that is, functions f satisfying f(tx) = tf(x) for all t T. We define the projection horn of C(U) onto the T-honiogeneous functions by

hom (f)(x)

=

dt

where dt denotes Haar measure on T. It is easy to check that horn is indeed a norm-decreasing projection. We also have use for the projection in

M(U) given by

fEC(U). We define U to be a T-simplex if whenever A,

are maximal measures on U

representing the same L 6 U then horn A = horn Then the theorem of Ethos referred to above says A is a Lindenstrauss space

if and only if U = Ar is a T-simplex. For the purposes of our discussion here we shall consider this the defining property of Lindenstrauss spaces. 9.1. TIIE0REM. Let 1 6 A be a separating subspace of C(X). The following are equivalent: (1) A is a Lindenstrauss space; ={O};

(ii)

is a simplex; (iv) A is seif-adjoint and SA is a simplex; (v) each E c X which is a peak set for re A is an M-hull; (vi) A is self-adjoint and each L U is represented by a unique maximal probability measure on U.

(iii)

Proof.

(I)

lvI °

implies (ii): Suppose v 6 t9AA.L and full = 1. Then let j as in Section 1. Thus = h(x)x where =h and hi =

on X and

Jf(tx) d1(x, t) = J

f(h(x)x)h(x) dv(x).

I

LINDF.NSTRAUSS SPACES

221

Hence, if f is T-invariant then

i

is

A' and

dv = f,1f di for T-homogeneousf, we have 1 represents 0. Thus horn i =0. Given f€ C(X), we then extend f to TX by k = g then f(tx) = tf(x) and then to g E C(U) by the Tietze Theorem, if horn maximal. Since ii

k(tx) = tk(x) = tf(x) =f(tx) and

0= (horn i)(k) =

d1(x, t) = = JTX tf(x)

Jf(x) dv(x) = v(f).

Thus v0. Now (ii) and (iii) are equivalent since Theorem 3.1 shows aAA' ={0} if and only if 3A(ZA) = {0} which is equivalent by Theorem 2.7.3 to

being a

simplex. The equivalence of (iii) and (iv) is immediate from Theorem 0.2. The equivalence of (ii) and (vi) is essentially Theorem 1.5(u). be maximal on U representing the same L U. Let (ii) implies (i). Let ii ii by HjZ, Thus so that v are boundary measures with = v so that if f is T-homogeneous on U

Jf(tx) di(x, t)

= J'i dv

dji(x, t) so that hom ji = horn

E c SA. If E is a peak set for re A then F is an (iii) implies (v). Let F (exposed) face of Sn, hence a face of ZA. But then F is split in ZA so that Co (F u — iF) is split, showing E is an M-huIl. (v) implies (iv). If F0 is an exposed face of SA then E0 = F0 X is a peak set for re A. Thus (v) gives that E1, is an M-hull and hence, co

u—iFo) is split is a simplex. To show A is self -adjoint we need to prove (Theorem 0.2) that SA is

in ZA. In particular F0 is split in

so that Theorem 2.7.2 shows

split in ZA. We use the set-up of Section 3, identifying = — '-iX) andf with OfEA(ZA), where with

c M(X)

Ofd1L2. Let 10111, 11,111 denote

If

aA(ZA)L then

the norms in A(ZA), A(ZA)*. a boundary measure on X c Sn. We show =

—'0. We consider first any support pair (a, x) in that is, (A(ZA), A(Zn)*) with loll 1 and a(x) = lix Ii. Let he a boundary measure on X u —iX c ZA representing x with lxii

where

,

=

= I1C11±llfl

is the Jordan decomposition of lieil=

(a)

on Z4. Then ii

COMPLEX FUNCTION SPACES

222

so that (since hail = 1)

and

=

r(—a) = fl Then

a(z)= supp

c{z EZ4; a(z)=

where F4, F are disjoint exposed faces of ZA. Then

F =co(F F :co(Fi u--iF2) where F; (j = 1,2) are exposed faces of SA. The hypothesis then shows that

(j=1,2) is split in Thus if f is the function on ZA which is 0(i) on and 0 elsewhere then h is affine on ZA, 0 h t 1 and satisfies (since is boundary)

=((F11). Also, using Theorem 1.6. 1(ix) and the fact that h t Similarly there is an affine l.s.c.

and, since h

0 on —iX,

with

0 on X,

= 0. Proceeding as above with

(j = 1,2) we have affinc Ls.c.

with

(hi) = (—IF2)

Now if X() we can assume (replacing with a suitable boundary measure on that = lixoll. Given r >0 we can, by the Bishop—Phelps Theorem (Chapter 1, Section 7), choose a support pair (a, x) with flail = and fix — xoji < s. Then using the above iixii

=

= ((Fl) — =).

Thus iixoll

lix — xoli + llxil < 5e.

)+

)+

= 411x —xofl<4s.

LINDENSTRAUSS SPACES

223

We can combine Theorem 4.4 with the above to characterize the symmetric split simplicial faces of ZA.

9.2. COROLLARY. LetA be as above and letE = k(E), F = iF) C Zn. The following are equivalent: ZF Co (1) E is an M-hull and AlE is a Lindenstraus.s space,

E C Sn and

=0, (ii) /L E implies (iii) ZE is a simplex and a split face of ZA. 9.3. COROLLARY. If A C C(X) is a Lindenstrauss space and &4X = E is closed in X then AlE = C(E). then If E so that P4E = = 0. Thus C(E); but E is a boundary so that AlE is closed.

Proof.

9.4. COROLI.ARY.

=

If A is a function algebra in C(X) and a Lindenstrauss

space then A C(X). Proof. A is self -adjoint. If A is a function algebra in C(X) such that A and re A have the same peak sets then A = CCX). 9.5. COROLLARY.

Proof.

Combine Theorem 9.1(v) and the preceding corollary.

Notes The underpinning for this section is the characterization of T-simplexes due to Effros [96]. The various equivalences of Theorem 9.1 (except for (v))

are due to Hirsberg and Lazar [128] and Fuhr and Phelps [112]. The equivalence of (v) to the others and the peak set characterization of C(X) is due to Briem [55].

CHAPTER 5

Convexity Theory for C *..Algebras

1. THE STRUCTURE TOPOLOGY AND PRIMITIVE IDEALS IN A(K) Much of the motivation for the development of infinite-dimensional convexity theory during recent years has come from the study of C*_algebras and W*..algebras. In this section we will develop the notion of structure topology for the primitive ideal space of A(K) in direct analogy to the In a later section we will tie Jacobson structure topology for the abstract and concrete cases together, after we have studied the facial structure of the state space of a unital C*..algcbra. we write Let K be a compact convex set. For each x E for the smallest closed split face of K which contains x. The face is called a = {fE A(K) :f(y) = Vy is primitive face of K and its annihilator called a primitive ideal in A(K). If F is a closed split face of K then the set F is called an M-ideal (or near lattice ideal). Since F is evidently the closed convex hull of all the primitive faces of K which are contained in F, dually F.. is the intersection of all primitive ideals containing If J is an M-ideal in A(K) we will write 1(J) for the set of all primitive ideals of A(K) which contain J; I (J) is called the hull of .1. If E = {Ia} is a set

of primitive ideals in A(K) then we write 1(E) for the largest M-ideal where F is the smallest closed split Ia. Dually, 1(E) = face of K which contains Ua (fa)L, where = {x E K :f(x) =0, VfE Ia}. contained in

The M-ideal 1(E) is called the kernel of E. lt is clear that for all such) and £ we have

1(1(J))

1(1(E))

E.

Denote by Prim A(K) the set of all primitive ideals in A(K) and call a subset

E of Prim A(K) structurally closed if E =1(J) for some M-ideal J. If E is 224

THE STRUCTURE TOPOLOGY AND PRIMATIVE IDEALS IN 4(K)

225

structurally closed then

Ec

ñ(J) = E,

=

so that E = ñ(1(E)). Moreover, E = {L.} is structurally closed then F contains every primitive ideal which contains 1(E), and consequently if

1(E) = Let E1, E2 be structurally closed with F1 = ñ(J,), j 1, 2. Then, if J = n J2, I is an M-ideal and 2 F1 u F2. l-Iowever if E 1(1) then = co (It U 12), and since x c we must have either thus ñ(J) c E1 u E2. Therefore E1 u E2 is structurally c I or c closed. If is a family of structurally closed sets with Ea = for each a, then if I = (fla I )_ we see that I is an M-ideal and fla F,, = .1(J), using the facts that F = (F1)', 1= (1 ') for every closed split face F of K and every J1

ft

M-ideal I in A(K). Hence we have shown that the structurally closed sets induce a topology on Prim A(K) called the structure topology. This topology will only be Hausdorif when K is a Bauer simplex (see Corollary 1.2), but it is always T0. In fact if I, I are primitive ideals with I I then either I or if then is a structurally closed set containing I but not I. 1.1. THEOREM. LetK be a compact convex set and letf: Prim

R be

a bounded structurally continuous function. Then there exists a central function g in A(K) such that XEaK. Proof.

1ff is constantly equal to 1, we can put g = 1. Therefore, without loss

of generality, we assume that 0 f 1. Let ii be U,

=

r'(

1+1))

a natural number and define

for j=o

-

Then, if X denotes Prim A(K) with the structure topology, X\U1 is closed

and soJ1 =fl{I : IEX\U,}is anM-ideal. Nowwe havel0+11+'' .+J,, = f there would exist a primi{fl7=0 I) = A(K), because if F = fl'...0 I tive face G c F and hence a primitive ideal G.L (X\U1) = 0. If h E A(K) satisfies 0 h 1, and if e >0, then we will prove by induction on n that there exist h, .1, j=0, 1,. . . , ii, such that h = h, and —e

0. if +1,, then is an M-ideal and . . +1,, = A(K), so that there exist

1+8, that

)'- = F,,_1

h,,-1 €A(co

€J,_1 for some 6E(0,e). Let F,, F,,=J: so = 0 on F,,_1 F,,. Clearly we can define nF,, and

LF,,)) so that h,,. = 1

on F,, and h,,1 = 0 on F,,_1; in

226

CONVEXITY THEORY FOR

u F,,). Since co (F,, F,,) is particular we have 0 h i øt CO a closed split face of K we can, by Theorem 2.10.5, extend h,,_1 to K so that A(K) and 0 h71_1 h, — 8). Now if we put h,, = — J,,, 0 h,,_1 < 1 + e, —e
2(1-re) ——. n

A(K) for each n, it follows that is a Cauchy sequence for all XE (3K. In converging to some u A(K) such that u(x) A(K)such that g(x) =f((F) 3 the particular case when h 1 we obtain for all x E (3K. Therefore, in general, we have u(x) = g(x)h(x) for all x E (3K, and consequently we deduce that g belongs to the centre of A(K). Since

In the special case when every x E (3K is a split face (for example when K is a

simplex or the state space of a function algebra) the primitive faces of K

coincide with the extreme points of K. It is evident in this case that Prim A(K) with the structure topology is homeomorphic to (3K with the facial topology, and so Theorem 1.1 coincides with Theorem 3.1.4 in this situation. Notes The description of the structure topology for Prim A(K) given above follows the paper of Alfsen and Effros [8], and differs somewhat from that given in Alfsen's book [5] where Størmer's axiom for K is assumed. Theorem 1.1 is

due to Alfsen et a!. [7, 8] and is an abstract version of a special case of a theorem of Dauns and Hofmann [71]. The proof given above is due to Elliott and Olesen [99]. A much more general version of 1.1, which applies in any real Banach space, appears in [8]. For the connections between 1.1 and spectral theory for A(K) we refer to the paper of Edwards [86].

2. THE IDEAL STRUCTURE OF A UNITAL AND THE FACIAL STRUCTURE OF ITS STATE SPACE Let A be a unital let K={4' E A*: tb(1) = 1}, with the topology, be the state space of A and let Ah be the Hermitian elements of A.

Then Ah is a partially ordered real Banach space, for the ordering of

THE IDEAL STRUCTURE OF A UNITAL

227

positivity of operators, or equivalently for the positive cone {xx*: x A}. Moreover, I is an order unit defining the norm in A,,, and so 2.2.3 shows that

A,, is isometrically and order isomorphic to A(K). For the basic theory of C*_algebras we will refer to Dixmier [81], or Sakai [188]. If F c K and if M c A we will write M = {x K: a (x) 0, Va c M},

Fo={xEA:çb(x)=0,VçbEF},

we

note that LF F0 by the Cauchy—Schwarz inequality. Note that (FJ' is a face of A(K1. If N is a closed face of A(K)' and if L = {a cA : a*a E N}

then L is a closed left ideal in A. In fact a E L implies that Aa E L for

A >0, and a, bEL implies that 0(a +h)*(a so that a+bEL. Finally if hEA and

a EL we have (ba)*ba E and hence ha cL. It follows that is a closed left ideal in A. We now prove that there is a one-to-one correspondence between the closed left ideals of A and the closed faces F of K. 2.1. THEOREM. Fis a closed face of K if and only if F = L in A in this case we may take L = LF. Proof.

Let L he a closed left ideal in A, and let F =

L. Let 0

let a L. Then a*a L so that 0(a*a) =0, whence

E

(Li1 and () by the

Cauchy—Schwarz inequality for positive linear functionals. Therefore = (L)', and hence F is a face of K. Conversely, let F be a closed face of K. Then is a closed left ideal, by

EA*: 0(a**)= the previous discussion. Let M some b**EA**, some 'T'EF}. M is a linear subspaccof the normal dual space A* of the A**, and AF c A**F c A in A**. We will prove (i) MnK=F, and (ii) M is w*.closcd. From this it will follow that F=(L0) , and that M is the of AF so that L0 = {a E A: f(ha) =0, Vb E A, E F} = L,.. (i) For any subset E of K the carrier of E is defined to be the smallest (Hermitian) projection e in A** such that Oe = for all e E, where for all a E A, and 0(e) 0. When 0 K let = denote the smallest norm-closed face of K containing 0. If e is the carrier of {th} then the set G4, = 0 K} is and convex (using the fact that with K, we have 01(e) = I and it follows using the CauchySchwarz inequality that for / = 1, 2. Hence we have c; = obtain the reverse inclusion, since each '1' c has carriei' it is sufficient to show that every 'I' K belongs to Fe,, whenever carrier N'} e. Since the ((Oe)e =

and is a face of K containing 0 since if Aq5, i- (1 — A )02 E

0
only elements of A** that we need to belong to = K. e that Corresponding to let p denote the usual GNS

ot A as a weakly-closed algebra of operators on a Hubert space 11, with cyclic vector

CONVEXITY THEORY FOR

228

z. Then, since carrier = 1, p(A**)sz is strongly dense in H, where p(A**)1 is the commutant of p(A**). Let s Ep(A**)S with IIsII 1, and let t (1 If 0 is defined for all b A** by 0(b) = (p(b)sz, sz) we clearly have 0 0, and further th(b) — 0(b) = (p(b)z, z) — (p(b)sz, sz) (p(b)t2 z, z) 0 for all h >0, so that 0 Therefore contains a multiple of 0. However, since the mapping )x, x) is strongly continuous, and since is strongly dense in

.H, and also since every positive normal functional on A** has the form 'I'(a) = (p(a)x, x) for some x F!, it follows that F,, = K. We conclude that =

Let e' be the carrier of F, and let be a maximal orthogonal family of projections in A** that are carriers of elements of P Put e" = SUPU and I = e' — e" 0. If f 0 then ç6(f) >0 for some F, since otherwise we would have the = = for all E F, which contradicts the definition of C'. —e")] >Owe can assume that the carrier g of th is

Since

disjoint from e". This contradicts the inaximality of {e1j, and therefore = C,,. Suppose

that we have 0(e') =

0 is a normal functional on A** 0 0 6(e,j, and so 0(e') = 6(e1) for a countable subfamily {ej} of the {ea}. Because 0(e' -- e,) 0 and the carrier h of 0 is we see that h c1. If th1 F are such that e1 is the carrier of then 2 'th1 belongs to F with carrier 2 'c1 h. The first part of the proof of (i) now gives 0 E F. Consequently we obtain F {Oe: 0 E K, 0(e') 0}, the reverse inclusion following by the corresponding argument for given above. The

preceding result gives M={a**e'Oe?:OEK, a**EA**}. Now if '1' a**eIOe'EK we have and hence M n K F, and the proof of (i) is complete. (ii) Let denote the closed unit ball of We need to show that

Al r is

is a Suppose that net in M with limit /. The polar decomposition (cf. Sakai [188, 1.14.4], Dixmier [81, with w,, E A**, Ija 0, 12.2.4]) enables us to write = =

=

kind

In

particular we have ii,, M (r)

co (F u {0}). We can

therefore, by choosing a subnet if neccssary, assume that is convergent to some ij e co (Pu {0}). We have, for each a E A, 1&(a)i2 so that Ø(a)12r,(a*a) and consequently for all b in A**. Now let 4, be the polar decomfor all

position of th. Then

subnet such that we see that If a E A,, we obtain, for all = IthII ku = a

= urn

= urn

A

A2

and this implies that is

+A)j2

ij((a +A)2)

Consequently 4, =

r,(a2)

M

r

THE IDEAL STRUCTURE OF A UNITAL C*.AI.GEP3RA

229

2.2. CoRolLARY. Every closed face of the state space K of the C A is semi-exposed.

We prove directly that F is a semi-exposed face of K if and only if and the result then follows from 2.1. Let F be a semi-exposed face of K, so that F = (F1)' = ((F1)1)- by 2.8. If aE then a2 Ilalla, so that qS(a2) =0 for all q5 in F, and hence a E Proof.

F=

Therefore we have (F ) c

F'0, and it follows immediately that

F=(LF)'. and let c/, E K\F. There exists an a E L11 with Conversely, let F = 0 by the Cauchy—Schwarz inequality. Since 0, and hence and hence F is semi-exposed. Lj; is a left ideal a*a belongs to Our next investigation in this section is the connection between closed two-sided ideals in A and closed split faces of K. A subset E of K is invariant belongs if for each th E F and each a E A with gó(a*a) >0 the functional = for each b E A. to F, where

2.3. THEOREM. Let F be a closed face of the state space K of a unital A. Then the following statements are equivalent.

(i) Fis invariant; (ii) F0 is a two-sided ideal in A; (iii) F,. is positively generated; (iv) Fis a split face of K. (iv) implies (iii). This follows from (2.10.5). (ii) implies (iv). We have F = (Fr) by Corollary 2.2. Let ç& 6 K and let ir denote the representation of A induced by 1, with cyclic vector in the sense of the Gelfand—Neumark--Segal construction. If H denotes the HilProof.

bert space associated with IT let L be the closed linear subspace of H spanned by {rr(a)x: a EF0, XE H}, and let M be the orthogonal complement of L. Since F0 is a two-sided ideal L is invariant under ir(A), and hence also is M; write IT1, 1T2 for the restrictions of IT to L and M respectively, and write for the projections of onto L and M respectively. Forj = 1, 2 where a 6 A. Then and are clearly positive define th1(a) = linear functionals on A and, using orthogonality, we have for a 6 A

+ b2(a) = =

=

+

+

1(a).

Similarly we obtain 4.'2(a) =0 whenever a 6 F0, and also that lkb1Il

Suppose

now that b =

112

=

+ '/4 where

= 0 on F0 and

=

Foil.

Then if {u,j is an approximate identity for the ideal F0, and if IT' denotes the

CONVEXITY THEORY FOR

230

representation associated with

and with cyclic vector con= converges to (x) for each x A, and similarly converges to = vanish on F0, and F0 for each a, it follows that b1(x). Since i52 and and Finally the set N' K: II'YII = II'YIF0II} is convex because I'YIF0I! = urn (cf. Dixmier [80, 2.1.5]), and it is therefore evident that F is a split face of K. (iii) implies (ii). As in the proof of 2.2 we have c Fo. Since is self-ad joint it follows that F0 = line; (Fj' c Lp, so that F0 = LF. Hence F0 is a seif-adjoint left ideal and therefore a two-sided ideal. (i) implies (ii). We know that F = Let u E A be unitary, so that EF it follows that u*au whenever b E F. Hence if a and therefore au = u(u*au) e is a left ideal. Since A is the complex linear span since of its unitary elements we deduce that LF is a right ideal also. In particular

verges to

Consequently

is a strongly Archimedean ideal because Ah/(A,, n L1.) is isometrically order-isomorphic to (A/LF),,. Therefore Ah = F1 (cf. Chapter 2 Section 8), and also LF = F0. (ii) implies (i.). Let F = I', where 1 is a closed two-sided ideal in A,.and let F and a E A with *a) > (1. Then if h E I we have a */,a E I, so that

&(h) =0. Hence & F and F is invariant.

If A is a unital

with state space K then is an algebra homomorphism. Moreover, the following statements are equivalent: (i) A,, is a vector lattice; (ii) K is a simplex; (iii) A is commutative. 2.4. COROLLARY.

is a split face of K if and only if

Proof.

If tb is an algebra homomorphism then

'(0) is a closed two-sided

'(0)) is a split face of K by Theorem 2.3. ideal, and hence b = 1(0) is a two-sided ideal, Conversely, if is a split face of K then by Theorem 2.3. It is elementary now to verify that q5 is an algebra

homomorphism. (i) implies (ii) is trivial. If A is commutative then the Gelfand—Neumark Theorem shows that A = C(X), and hence that A,, = cR(X), for some compact Hausdorif space X; consequently we obtain (iii) implies (i). Finally,

if K is a simplex then each b aK is a split face of K, and hence b is an algebra homomorphism. Therefore = for all a, h E = A and &K, so that ab = ba by the Krein--Milman Theorem; consequently (ii) implies (iii).

State spaces of unital concerning split faces.

provide some interesting examples

TILE IDEAl. STRUCTURE OF A UNITAL C*.ALGEBRA

231

2.5. EXAMPLE. Let A denote the (the bounded linear and let K be the state space of A and F = operators on where L is the

two-sided ideal of compact operators jn A. Then F is a closed split face of K

and there existf, g A(K)4, h

with h f, g on F, and such that no extension h'€A(K) of h exists satisfying gon K. Proof.

for n

1

e21,

be the coordinate basis By 2.3, F is a closed split face of K. Let define and for n =0, = —1 -2 -'1/2 e2,,, --1 —n (n — n )e2,, 1—n

1 +(n

1

1

C2n.

It is easily checked that f and g =f+k are projections withL kwhile gonF. Puth = L.Wehaveof, g onK f, that there exists an h' A(K)' extending h with h' g. If q denotes the range projection for h' we have q f, g, so that the intersection A'! of the ranges of f and g contains the range of q. However, inspection shows that A'! = {0}, giving a contradiction.

The above example should be compared with 2.10.5. In Chapter 2 we saw that every closed split face is semi-exposed; the following example answers a natural question in this connection. Let Hhe a non-separable Hilbert space, let A = B(H) and let K be the state space of A. Then if L denotes the compact operators on H, F = L1 is a closed split face of K which is not an intersection of exposed split faces of K. 2.6. EXAMPLE.

Proof. Since L is a minimal closed two-sided ideal in A, F is a maximal proper closed split face of K. It will therefore be sufficient to show that F is f(x)> 0 for not exposed. Suppose that there exists an f€ A(K) with fE all x K\F. Then f identifies with a compact Hermitian operator T on H, H with and since H is non-separable there exists a 1 and =0. But then if j5 is defined on A by K and = (at, we have for H, then f(qS) = (TE, =0. However, if a E L satisfies a-ri = çb(a) = I so that j5 K\F, giving a contradiction.

We now give a generalization of Corollary 4.9.5. 2.7. ThEoREM. If a complex unital Banach algebra A is a complex Lindenstrauss space then A = C(X) for some compact Hausdorff space X. Proof. Let e denote the algebraic identity of A and let S denote the state space of A. The second Banach dual A** of A is also a complex Banach algebra with respect to the Arens product and has unit ê, the canonical embedding of e in A**. Let S** denote the state space of A** relative to ê.

CONVEXITY THEORY FOR

232

Since A is a complex Lindenstrauss space A** is linearly isometric to the space C( Y) for some compact Hausdorif space Y. Let T: A** -÷ C( Y) The Bohnenblust—Karlin Theorem implement this isometry, and let u = [43] shows that ê is an extreme point of (A**)i, and so U is an extreme point of (C(Y))1. Hence ui = 1 on Y. Let S' denote the state space of C(Y) and define p: (C( Y))* by r(d4(f) = q5 (fü) for q5

(C(Y))* and f€ C(Y). Now r is a linear isometry onto (C(Y))* and

= 1 = b(1) so that hence p is a linear isometry onto A***. If b ES' then = = = çb(uu) = 1, giving Jp(b)Ii = I, and also =I= we have = Conversely, if E and (p p(b) E = (T* = S'. = ili(ê) 1, so that We have shown that is an affine isomorphism between the simplex S' and S**, and therefore S** is a simplex. Similarly pco (S' u —iS') is an affine isomorphism between the simplex co (S' —iS') and co u _j5**) so that S** is a split face of Z**. Therefore A** is a (see 4.0.2), and Corollary 2.4 shows that A** is commutative. The subalgebra A of A** is consequently a function algebra, and so the result follows from Corollary 4.9.5.

Notes

Theorem 2.1 is due, independently, to Ethos [92, 5.1] and Prosser [176, 5.11]; our treatment follows the proof of Prosser. Corollary 2.2 was noted by Chu [66], and the particular case for extreme points follows from a result of Glimm [120, Theorem 9]. The connection between closed two-sided ideals in a and the split faces of the state space as presented in 2.3 was

developed in the papers of Effros [92], StØrmer [206] and Alfsen and Andersen [6]. The implication (ii)

(iv) in Theorem 2.3 can also be proved

by a method of Smith and Ward [197, 5.3] by which they show that the M-ideals and the closed two-sided ideals coincide for any The equivalence of (i) and (iii) in Corollary 2.4 was first proved by Sherman [192]. Example 2.5 is due to Stefánson [200], Example 2.6 is due to Andersen and Theorem 2.7 to Ellis [105].

3. COMMUTATIVITY AND ORDER IN C*..ALGEBRAS

The relationship between commutativity and order properties in Cd'algebras has been considered by many authors; in this section we discuss some of their results. subspace of the bounded seif-adjoint operators Let V be a on a Hubert space H. Then if a E V we denote by a j the positive square root

COMMUTATIVITY AND ORDER IN

233

of a2. Even if a belongs to V it is not in general the lattice modulus of a in V unless the operators in V commute. 3.1. THEOREM.

Let V be a linear space of bounded linear seif-adjoint

operators on a Hubert space H. Then if a E V whenever a E V, and if V has the Riesz decomposition property, V is commutative. Proof.

Let L be the closure of V in B(I-f) for the strong (or weak) operator

topology. Then al e L whenever a EL (cf. Kadison [137]). If P {a c L: 0 a, flail 1} then its strong closure 0 is compact for the weak operator topology. If

Q

the Krein—Milman

Theorem, and hence so is V because V is positively generated since I

aj a, —a for each a E V.

is commutative we will prove that V is a vector Before showing that lattice with a equal to the lattice modulus of a, for each a E V. First suppose and that ab =0. If x E V and ..v a, b then, by the Riesz that a, h E

interpolation property, there exists y E V with 0, x y a, b. If z"2 denotes the positive square root of any z E V we have 0 a' "2ya"2 = 0 since a'12 is the limit of certain polynomials in a, and ab = 0. /2)? I/2)* so that a112y 1/2 = 0 and Therefore 0 = a '12ya 1/2 = (a "2y hence ay 0. Moreover, we have 0 y2 y"2ay112 =0 so that y2 = 0 = y. Hence every lower bound x for a and b belongs to — V , and so inf (a, h) 0. Now for any a V we have Ia I + a, a — a 0, and (lal a )(a I — a) =0, + a, a j — a) =0. Thereand therefore the preceding argument gives inf I

= laI, and V is a vector lattice with the required property. The strong continuity of the modulus operation now shows that L is an M-space with al the lattice modulus of a for each a EL. Since 0 is compact for the weak operator topology, and is also a directed set, Q contains a greatest element SO that L is a GR(X) space, for some compact Hausdorif space X. The extreme points of Q correspond to characteristic functions of open and closed sets in X. Hence if e,, e2 E &Q then e1 A e2, e1—e, A e2 and and are disjoint. Suppose that a, h E 0 and e2 — e, A e2 all belong to a A b =0. Then we have fore sup (a, —a)

Ia I — inf (Ia I + a, Ia I —a)

(a+h)2=(avb+aAb)2=(avb—aAh)2=Ia--b12=(a—b)2, so that ah ± ba 0. But then a2b2 —ahab = h2a2 and so the positive square root a of a2 also commutes with b2. Applying this argument again gives ab = ha, so that ab 0= ha. It follows that e, A e2, e, — e1 A e2 and A e2 commute, and hence e, and e2 commute. C2 — The above result enables us to extend Corollary 2.4 to the non-unital case immediately.

CONVEXITY THEORY FOR

234

Let A be a only if A,, is a simplex space. 3.2. COROLLARY.

Then A is commutative if and

Then A is commutative if and Let A be a only if it has the property that 0 a b in A implies that 2 b2. 3.3. COROLLARY.

Of course a belongs to A,, whenever a E A,1. The result will follow if we can show that A,, is a vector lattice; for this it is sufficient, as in the proof of 3.1, to show that a, h E with ab =0 implies that inf (a, h) =0. Suppose that x E A,, with x a, b. Then we have 0 b — x <2a + b — x and 0 a — x a + — x. Hence we obtain (b — (2a + b — x)2, so that 2a2 —ax — xa 0, and consequently x2 (2a — x)2. It now follows that xj 2a — x. [In fact if 5, T arc positive operators on a Hubert space with 2j + 2A T + — S2) + T2 then, for A >0, (Al + S + T)(AI + T — S) = (A i(iTS — iST) = X + iY, where X and Y arc Hermitian. Moreover X is Proof.

S I

positive and invertible and since X "2yx""2 is self-adjoint the spectrum of I —iX'12YX 1/2 lies in {z C :rez = 1} and consequently (X + iY) =

X'12(1+iX

1X1"2 exists. Therefore it follows that the spectrum of (T — S) lies in the positive half-line, giving S T.] Similarly we + x) a, b. obtain xI 2b — x, and the two inequalities together give 0 I

As in the proof of Theorem 3.1 the fact that ab =0 now implies that x 0. Consequently we obtain 0 = inf (a, b) as

required. Notes

An extensive investigation of the relationship between order properties and commutativity in was carried out by Topping [208], where, in particular, he proved Theorem 3.1. Corollary 3.3 was originally proved by Ogasawara [166].

4. THE CENTRE OF A

WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS

In Section 3.1 we introduced the abstract notion of centre for A(K) spaces. We now show that this notion coincides for A,, with the algebraic centre, that is those elements in A,, which commute with all elements of the algebra. In version of carrying out this process we investigate the concrete the Dauns—Hofmann Theorem that we studied in Section 5.1. If A is a unital with state space K, and if bc&K, then the usual GNS construction gives an irreducible of A, and the kernel of is called a primitive ideal in A. The space of all primitive ideals in A will be denoted by Prim A, and will be endowed with the Jacobson (or hull-kernel) topology (see Dixmier [81]).

WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS

235

4.1. THEOREM. If K is the slate space of a unital then the following statements hold. (i) us a primitive ideal in A if and oniy if!' = {x E K; a(x) =0, Va E I} is a primitive face of K.

(ii) K satisfies Strmer's axiom. (iii) 1/f: Prim A R is continuous for the Jacobson topology then there exists an a E A,,, with a in the algebraic centre of A, such that for all çb €aK. f(ker (iv) The algebraic centre of A,, coincides with the centre of A(K).

so that I is a primitive ideal in A, and let I = ker *xy) e iiF, 'I' is the w*..lirnit of states of the form (see Dixmier [81, Corollary 3.4.3]), and since the primitive face containing j5 is invariant we see that Therefore we have F = E,,, a primitive then I = ker face of K. Conversely, if E is a primitive ideal, and again we obtain Fqs (ii) Let {Fa} be a family of closed split faces of K and let the family of closed two.sided ideals in A with F,. = (see 2.3). Then if I = fla and if E = aFa we clearly have Proof.

Let

E

F = I'. Then, If

a

a

a

4,e.E

Therefore 1 contains the set

the

=fl

(F,,,)o.

U, Fa. Now, if

of states of the form 3.4.2]). Therefore we obtain 'V E U, It follows that U. Fa face of K.

(ker ira)'), 'I' is )E, E) where / E (see Dixmier [81, (ker and consequently 'V and hence is a closed split

(iii) By the result of (i) the Jacobson topologies on Prim A and on Prim A(K) are equivalent. Therefore, by Theorem 1.1, there exists an aE belonging to the centre of A(K), such that f(ker = 4(a) for all We need to show that a belongs to the algebraic centre of A. Let I be a primitive ideal in A and let p E sf1. Then ker is a primitive ideal containing I, so that ker irs, belongs to the closure of I in Prim A(K) be

and thus f(I) =f(ker

In p(a —f(ker ira)) =0 for all p E

particular we see that p(a —f(I)) = so that a —f(I) belongs to I by the

Krein—Milman Theorem. Now for any b E A we have ab —f(I)h E I and ba —f(I)h E 1, so that ab — ha E I for any primitive ideal 1. Because A is semi-simple it now follows that ab = ha, that is a belongs to the algebraic centre of A. (iv) Let a belong to the centre of A(K). Then, by Theorem 1.1, the function f: Prim A R defined by f(ker = 4(a) is continuous. Hence the proof of (iii) shows that a belongs to the algebraic centre of A.

CONVEXITY THEORY FOR C*.ALOEBRAS

236

Conversely, let a belong to the algebraic centre of A, a Ah. We may assume that If 4) E .3K then 8 = — 4)(a) E (0, 1), and we can define A. If b 0 then we have 0 'P(b) =8 b — a)) for all b Therefore we see that 'V E K and also (1—8) '(4) — 8'I') that 0 'I' S K. Since 4) is extreme in K the equation 4) = 8'P+(4) —6'!') implies that 4) = '1', and hence 6q5(b) 4)(bd— a)) = (i— 4)(a))di(b). Consequently we obtain 4)(ba) = 4)(ab) = 4)(a)4)(b), and therefore a belongs to the centre of A(K). and Bishop decompositions for a compact In Chapter 4 we defined the convex set K, and we investigated their relationship with function algebras. A then these decompositions If K is the state space of a unital

give rise to natural decompositions for A. Using some of the results denotes the developed in this chapter it is not difficult to see that if is the family bf closed two-sided ideals in decomposition for K then A which arc generated by the maximal ideals of the centre of A. Similarly, if {F0} denotes the Bishop decomposition for K then {(F0)0} forms the family of closed two-sided ideals I which are minimal subject to the property that

A/I has trivial centre. If K is the state space of a

then the Bishop decomposition situation, and it is not known in general need not cover 0Km the if the Bishop decomposition determines A(K) amongst Banach subspaces of However, if some subspace L of has the form L = B,, for some unital B containing A as a *..subalgebra, and if L contains {fE

I

A(K)I(F0

then L = A,, by Glimm's Stone—Weierstrass Theorem [119]. In general the best result we have is the following, which shows that the Bishop decomposition always determines the centre of A (K). 4.2. THEOREM. Let K be a compact convex set satisfying Størmer's axiom and let {F0} he the Bishop decomposition for K. Then we have (centre A (K))

{f€

:fI(F0

is constant, V0}.

Proof. 1ff belongs to the centre of A(K) then certainlyf is constant on each F,3 because of antisymmetry. Conversely, letf E C14(.3K) be constant on each 3K. Since each primitive face of K is contained in some F0 we can i1 by g((FX).I) =f(x) for all x E 3K. If we define a function g: Prim can show that g is structurally continuous it will follow that f€ (centre A(K))IOK by Theorem 1.1.

F0

It will be sufficient to show that g is upper semi-continuous for the :f(x) a} is non-empty. structure topology. Let a R such that E =

WEAKLY CENTRAL ALGEBRAS, PRIME ALGEBRAS

237

Since K satisfies StØrmer's axiom the set F = {UFX :f(x) a } is a closed Certainly we have (Fr).. J for each (Fr) E E. split face of K; put J =

J then is contained in F. Now the set On the other hand, if f(y) a} is compact and contains all with f(u) a, and G = {y E consequently G 2 by Milman's Theorem. Hence we havef(x) a so that belongs to E. We have therefore shown that E = h(J), and thus E is structurally closed and g is upper semi-continuous as required.

A is said to he weakly central if whenever I, J are maximal two-sided ideals in A, and Z is the centre of A, then I Z = .1 Z implies that I = J. A compact convex set K is weakly central if whenever F and G are minimal closed split faces of K, and is the centre of A(K), implies that F = G. We show next that the Bishop then F1 = A

decomposition always determines A(K) when K is weakly central. 4.3. THEoREM. If K is weakly central then the Bishop and &ilov decompositions for K coincide, consequently if A is a -algebra, or any weakly central unital the Bishop decomposition for the state space of A determines A.

Let F be a face in the decomposition for K, and let be the centre of A(K). Then if 0 and H are distinct minimal closed split faces of F we have = since each is constant on F. Therefore Fl = G because K is weakly central and this contradiction shows F has no non-trivial, mutually disjoint, closed split faces. Hence A(F) must have trivial centre, and consequently the Bishop and decompositions for K coincide. The preceding general theory now shows that the Bishop decomposition determines A(K) and hence A, whenever K is the state space of a weakly central unital Since every is weakly central (cf. now follows. Misonou [160]) the result for Proof.

4.4. A compact convex set K is weakly central if and only for every closed split face Fof K the centre of A(F) equals the restriction toFof the centre of A(K). consequently a unital A is weakly central if and only if for every closed two-sided ideal Tin A the centre of A/f is the quotient of the centre of A.

Suppose first that centre A(F) = (centre A(K))IF for every closed split face F of K. Let 0 and H be minimal closed split faces of K, so that, in particular G and I-f are disjoint. Let F = co (0 u I-I) and let f€ A(F) such that f(x) =0 for all x G and f(y) = I for all y I-I. Clearly f is central in

Proof.

238

CONVEXITY

FOR

A(F) and so there exists a g central in A(K)with g belongs which shows that K is weakly central. to G but not to Conversely, suppose that K is weakly central. Let F be a closed split face of K, and let q A(K) A (F) be the restriction (quotient) map. If Max denote the map centre A(K) and = centre A(F) let c7 : Max where N belongs to the maximal ideal space of q(N) f(x) = O} for some x aF so that belongs to Max because N = : f(xi)=O}he distinct from N, with E oF. Then there exist disjoint closed split faces G and G1 of F containing x and x1 respectively. But then x and x1 cannot belong to the same member of the Silov decomposition for K, arguing as in Theorem 4.3. Hence there exists a g Ej with g(x) g(x1), and consequently 4(N) 4(N1), which shows that 4 is an injection. It is easily seen that c7 is continuous. In fact let Na = {f E

:

f(Xa) = O} be a

f(x) = O}, that is for all net in Max j' converging to N = {f€ -* g(x), which means that so that f€ But if g we have 4(N). This argument shows that, under 4, Max is homeomorphic to a closed subset of Max 4_I to Max so Let f and let g be a continuous extension of fo we have g(x) = g({h h(x) = O}) = that g Max1. If x g(4({h€j': h(x)=O}))=f(x), so that g is an extension of f.

counterpart of Theorem 5.1. A We now consider the algebra A is called a prime algebra if whenever 1, J are closed two-sided ideals in A then IJ = {xy: x E I, y J} = {O} implies that either I = {O} or J={O}. 4.5. THEOREM.

Let A he a unital and oniy if Ah is an antilattice.

Then A isa prime algebra if

By Theorem 3.5.1, Ah will be an antilattice if and only if the state space K of A is a prime compact convex set. Let A be a prime algebra and let F, 0 be semi-exposed faces of K with Co (F G) = K. Then, by Theorem 2.1, there exists a closed left ideal L and and G = R . Now we have a closed right ideal R in A with F = RL R L (co (R u L' = {O}, and therefore AR and LA arc closed two-sided ideals with ARLA = {O}. Hence we obtain either AR = {O} or LA = {O}, that is either R = {O} or L = {O}. Therefore either F = K or G = K so that K is prime, and A,, is an antilattice. Conversely, suppose that K is prime and that I, J are closed two-sided = {O}. In fact I nJ is a ideals in A with JJ = {O}. We must also have I closed two-sided ideal containing If and, since (I J) A,, is positively Proof.

generated, we can deduce that I n I = If if we prove that every b €

UNiT TRACES FOR

239

belongs to If. Since b"2 is a uniform limit of polynomials in b, with no constant terms, we have b112 €1 n J, and hence h b 112b"2 E If as required. But now it follows that co (I = K, and hence = (I either I = K or f = K. Therefore either I = {O} or J = {O}, so that A is a prime algebra.

AZ

4.6. COROLLARY.

If A is a

with state

space K then the

following statements are equivalent.

(i) A isa factor; (ii) K is a prime compact convex set; (iii) A1, is an antilattice.

Prool The equivalence of (ii) and (iii) has already been established. If K is prime then A(K) is an antilattice, and so clearly the centre of A(K) must be the constants. Consequently A is a factor, by Theorem 4. 1(iv). Conversely, suppose that A is a factor. If A is of type H or type III then aK= K (see 011mm [119, p. 239]) and hence K is prime by Corollary 3.5.2. If A is of type I then there exists a f E uK associated with a faithful irreducible representation of A. If F is the smallest closed split face of K aK is the of states of A of the form containing then every (see Dixmier [81, Corollary 3.4.3]), and the invariance of F now gives F = K. If I, J are closed two-sided ideals in A with If = {O} then we K = Co (J1 u f'), as in the proof of 4.5, and since either €1' or conclude that either P = K or J = K. Therefore either I {O} or f {O} so that A is a prime algebra. Consequently A,, is an antilattice by Theorem 4.5. Notes The results contained in Theorem 4.1 arc due to Størmer [206] and to Alfsen and Andersen [6, 7, 16]. Theorems 4.2 and 4.3 were proved by Ellis [106]. was studied by Vesterstrøm in The theory of weakly central

[211], and Theorem 4.4 is an extension by Chu [64, 67] of some of Vesterstrøm's results to compact convex sets. Theorem 4.5 is due to Chu [66]; it was also proved by Archbold [20] using quite different techniques. Corollary 4.6 was first proved by Kadison [138]. Sonic further results of this kind are proved by Dixmier [82].

5. UNIT TRACES FOR We saw in Section 2 that the state space K of a unital

A is never

a simplex unless A is commutative. Nevertheless K does possess an algebraically important subset which is a simplex. Let T(A) = Va, b E A) denote the unit traces on A. We will e K: =

240

CONVEXITY THEORY FOR

require the fact, proved in the next section, that every Hermitian E A * has with 'V', a unique decomposition 'P = 'it — positive and II'Vll = II"Pil÷II'i'lI; this result is due to Grothendieck [123]. We will prove that link T(A) is an L-space with = sup ('iF, 0), 'P = sup (—'I', 0) for each 'PE "AR T(A).

5.1. THEOREM. The u/lit traces T(A) on a unital

A form a

compact simplex. Proof. Let V = T(A) and let 4) V. If u E A is unitary define and i41 belonging toA* by 'I'(a) =4) *au), 'P(a) =4) (u *aii)for each a E A, where 4) + and 4)- are as mentioned immediately prior to the theorem. We have, foreach a EA, and also ll'Pi1 + Il'Vil. The uniqueness of the 1I'i'÷II + I1'r + 114) 11=114)11

= decomposition 4) = and 'V = 4), that is — 4) now implies that = 4)(u*au) and 4)(a) = V(u*au) for all a E A and all unitaries u €A. Therefore cj'(ua)= cb+(u*uau)= a a linear combination of unitaries it follows that 4) and the theorem will follow if we V show that inf (p, q) exists in V for each pair p. q in V, or equivalently that [0, p1 is lattice-ordered for all p in Firstly, let p £ (A*) and let be the Hilbert space associated with p in

the usual Gelfand—Neumark-Segal construction, and let denote the representation of A on H,, induced by the left regular representation of A. the completion of A/Ne for the norm induced by the inner product = {x E A: p(x*x) = O} and I denotes the image of (a, b) = p(h*a), where x A in A/Na under the quotient map.] if q A* with 0 q then the Cauchy—Schwarz inequality for q gives q(bi*ai) = q(ba) whenever h1 =b and = ci, and hence there exists a Dq such that q(b*a) = (a, Dqb) for all a, b A. Since 0 q p it follows easily that 0 Dq where I is the = q(b*a*c) = identity operator on Moreover DqLj,(a)b) = (a*c,Dqb)=(Lp(a)*ê,Dqb)=(ê,Lp(a)Dqb), and hence Dq belongs to the commutant A}. Conversely, if D = of and we can define q(a)=(a,Di), and it is easy to check that 0 q p and that D = Dq. From this reasoning it follows that {q £ 0 D I}. A*: 0< q p} is order isomorphic to {D Now take p V with 0 q p. In this case the right regular and the representation of A induces a *..representation R,, of A on such that .Jâ = involution in A induces an isometry J on for a e A and (x, y) = (Jy, Jx) for x, y in 1-I,,. For all, a, b, c in A we have (a, = (ab*, Dq5) = = q(b*c*a) = (a, Dqcb) = (a, so that Dq where q' E A* and 0 q' p, Conversely if then

UNIT TRACES FOR C*.ALGEBRAS

241

q'(ab), that is q' V. Therefore we have shown that {q E V: 0 q p} is 0< D I}. order isomorphic to {D E

r If we can show that is a commutative then and T€ the result will follow. Let be 1-lermitian. Then in Ah with -* Si and 37, JS1 = Si, JT1 = Ti and so there exist Ti. = urn For each a E A we have Sâ = and Ta = similarly. Since each pair urn and commute it follows YPZ

that (STd, b) = (Td, Sb) = urn (TSd,

lim

=

h)so that ST= TS.

The first part of the above proof shows that, for p K, is an irreducible *representation of A if and only if p aK, whereas the second part of the proof shows that, for p T(A), L,, is a factor representation (i.e. is trivial) if and only if p aT(A). If p 9K T(A) then is a two-sided ideal and also, byTheorem 2.l,{p}= Nt so thatp is a split face of T(A) coincides with the characters of A. In K; consequently the set T(A) is not equal to 3T(A); in fact if A is the general of complex 2 x2 matrices then (cf. Størmer {201}) the state space of A is affinely isomorphic to a ball in R3, and the unique trace maps into the centre of the ball. In the case when A is separable a result of Dixmier [80] shows that is a primitive ideal whenever p E in this case is a primitive face of K containing p. Another subset of K which is a simplex, and which is of algebraic interest,

is the commutator face Cm(A) of K. Here we define Cm(A) = (comm where the commutator ideal comm 4 of A is the closed two-sided ideal generated by the commutators ab — ba, a, b A. Since T(A) = {ab —ba: a, b €A} it is clear that T(A)Cm(A). In the case when A = B(C"), T(A) is a singleton whereas Cm(A) is empty, because A is a simple algebra. 5.2. THEOREM. The commutator face Cm(A) of a unital A is the largest closed split face of the state space K which is a simplex. Moreover,

Crn(A) is the closed convex hull of the characters of A. Proof.

By Theorem 2.3, Cm(A) is a closed split face of K; also Cm(A) can

be identified with the state space of the commutative A/comm A, and hence Cm(A) is a Bauer simplex. It is evident that Cm(A) contains all the characters of A. If x A\comm A then 0 I A/comm A,

and since A/comm A is commutative there exists a character 'I' of = 0; hence defines a character of A with q5(x) 0. It follows from this reasoning that comm A is precisely the intersection of the kernels of all the characters of A. Since each character of

A/comm A such that 'I'(I)

A is a split face of K, and since K satisfies Størmer's axiom, the closed

CONVEXITY THEORY FOR C*.ALOEBRAS

242

convex hull of the characters is also a closed split face F of K and hence F= (F1) L= (comm A) l. = Cm(A).

Suppose that 6 is a closed split face of K which is a simplex. Then, if aG, q5 is a split face of G and hence b is a split face of K. But then, by Corollary 2.4, is a character of A so that 4) Cm(A). Consequently Cm(A) contains G. q5

Notes

Theorem 5.1 is a result of Thoma [207], but the proof given here is due to Effros and Hahn [97]. A consequence of 5.1 is that each 4) T(A) is the barycentre of a unique maximal measure on T(A). This measure is the central measure corresponding to 4, (see Sakai [188, Theorem 3.1.18]). For a complete discussion of central measures for compact convex sets and for we refer to Alfsen [5, § 2.8], Sakai [188, Chapter 3] and Wils [214, 215,216]. The use of Choquet boundary theory in the decomposition of lower semi-continuous semi-infinite traces on separable is discussed by Davies [74]. is given by Alfsen Another approach to the trace simplex for and Shultz [10, § 12], and some further results concerning simplexes of states have been proved by Batty [33]. The theory of commutator ideals in C*_algebras is discussed by Arveson [21, § 3.3].

6. THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS AND OF We firstly consider some recent work of E. M. Alfsen, F. W. Shultz and E. Størmer concerning characterizations and representations of certain Jordan algebras.

Let A be a real Banach space which is a Jordan algebra over R with identity I and such that (i) Ia . bil Ilaillibli, (ii) 11a211 = Ia and (iii) 11a211 112

a, b in A; A is called a J13 -algebra. The connection between A(K) spaces and JB-algebras is demonstrated in the following result. fla2

1)211 for all

6.1. THEOREM.

If A is a JB-algehra then A can be identified with an A(K)

space such that A(K) = A2 = {a2: a A}, 1(x) = a for all x E K, and 0 a2 1 whenever —1 a 1. Conversely, if A(K) is a Jordan algebra over R

with identity 1, and if 0 a2 1 whenever haIl

1, then A(K) is a JB-

algebra.

Let A be a JB-algebra. If a A then (cf. Jacobson [133, p. 36]) the polynomials in a form an associative subalgebra, and hence their closure Proof.

THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS

243

C(a) is a commutative Banach algebra. Let a, b €A2\{O} and put a =llall, = llbll. Since la/a 11= 1 we can find d C(a) such that 1—a/a = d2, by the Binomial Theorem. Hence if a = c2 and f= 'lad we have a — a = ad2 so that la —all 111211 11c2+f211 = a; similarly we obtain —bli
Ia + (3 —(a + b)iI a ÷ /3, or Ill — (a + b)/(a + /3)11 1. As above we can find a and hence a ÷ h A2. This shows that A2 is a g e A with g2 = (a + b)/(a + convex cone. If a2=—b2 then 1a112=11a21111a2+b211=O, and thus A2 is a proper cone. Moreover, the preceding discussion implies that A2 = (a A: hi lila 1— a Il} from which it follows that A2 is norm-closed.

Suppose that —1

a

1

and write 1—a2 = (1—

a) = c2 d2, for

some elements c, d in C(a). Since C(a) is associative we have a2 + 1. Further we obtain 1, so that 0 a2

llalI2Ila2ll
d)2 =

1.

On the other hand 1 + a A2 and 1— a A2 whenever hall 1. Therefore we andthefirsthalfofthetheorem have{a €A: IIahI 1}={a €A: —1

is proved. Conversely, let A(K) he a Jordan algebra over R with identity 1 such that

()a2 1 wheneverflall 1. Nowif IlaIl 1 andlIbIl 1 + b)]2 1 and 0 — b)]2 and 1 so that 0 — b)ll --b)]2 <1 and consequently Ila —1 + b)]2 — =

+b)ll

.

—b)]211 1. It follows that ha In particular we see that 11a211

0a2S 1 sothata

+ b)]2 —

Ilbil for all a, b in A(K).

.

Ia 112

1

1. Therefore

for all a A. However, if la 2Ils 1 then

1) 1;similarly—a land

1— (a —

1. Therefore we obtain 11a211 = ha for all a in A. Since all squares so Ia we clearly have 11a211 la2 + bij for all a, b in A. Hence belong to A(K) is a JB-algebra. An immediate consequence of the above result is that every JB-algebra has the property that a =0 implies that a1 =0 for j = 1, . . . , n. The basic ideas in the theory developed by Alfsen and Shultz [10] are those of P-projections and projective faces. 112

Let V=lin K bea base-norm space and letA =Ah(K) = LetP: v-÷ : A A be the dual (positive) proV be a positive projection and let and similarly for jection. P is said to be smooth if ((ker = (((ker Since (ker P*)0 = PV, which is positively-generated, we have (ker p")0 =

((ker

and hence P" is smooth if and only if (ker p*)0 K is a norm

semi-exposed face of K. A positive projection Q: V -+ V is said to be quasi-complementary to P if (ker = (QV) and (ker = (PVY, and It is not difficult to verify (cf. 10, similar definitions apply to P"' and Theorem 1.81) that P and Q are smooth and quasi-complementary if and and Q* are smooth and quasi-complementary. only if

CONVEXITY THEORY FOR C*.ALGEBRAS

244

If P is a smooth positive projection with IIP1I 1, and if there exists a smooth quasi-complementary projection Q with JIQII 1, then P and are

called P-projections. When this situation attains we have (PV) K = (ker P*)1)

K, which is a (norm) semi-exposed face of K; such faces are called projective faces of K. It is easy to see that every split face F of K is projective with associated P-projection satisfying P(Ax + (1—A )y) = Ax for Let P be a P-projection with quasi-complement P' (necessarily unique) and let = PV K and and are called quasi= P' V K; then complementary projective faces. The connection between projective faces and split faces is demonstrated more clearly in the following result. 6.2. THEOREM.

Let F, G be semi-exposed faces of K and let M =

un (F u G). Then Fand G are quasi-complementary projective faces of K if

and only if the following two conditions hold:

(i) F and G are split faces of co (F u G) and the associated projections M lin F and ir' : M -> un G are continuous; (ii) there is a continuous positive projection

Vonlo M with (15(K) c K

and such that ker c15 ç F G, where F =

and 0 =

First let F and G be quasi-complementary projective faces with and G = FQ. The restriction of P to M maps F onto itself and G into 0, and is continuous, so that statement (i) follows. Now let 'f = P + Q: V -> M. Then, since PQ = QP =0, is a continuous positive projection onto M. = We have 1 0 and P*(1 _P*1) = 0 so that Q*(1 — j Q*1, since =0. Therefore, if x E K we have = (P* + Q*)(1)(x) = 1, so that is contained in K. Finally, if qS(v) =0 then (Fo) then f€ (ker Q)° since Q is smooth, Qv = Q2v = Q(—Pv) =0; if and hence f(v) =0 so that v F. Similarly we have ker 15 c G. Conversely suppose that the semi-exposed faces F and 0 satisfy conditions (i) and (ii). Let P = so that P and Q are and Q = ir'° continuous and positive. We clearly have 11(1)11 1 and, since IT, IT' 0 and ('ir + = '1' for '11€ we have IIir('1')II Il'P11 and IIir'('V)II Ii'i'V so that 1 and 11i4 1. Therefore we obtain IIP1I I and 1Q11 1. We evidently Proof.

F=

have PV K = F and QV K =0. Let x E (ker Q) K and write x = it' is the identity operator on M = x — 15(x)

ker 15

F we have x

and therefore

E

Since

F = (PV), because F is semi-

exposed. Suppose that Pu =0. Then u — Qu (ker F) (ker Q) c ker (15 and hence u =Qu + (u — Qu)E linO-4- ker 15 c 0. The preceding paragraph gave

F(ker Q), and G = (ker P

similarly, so we obtain ker P c (ker that is smooth. The relationship

THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS

245

between P and 0 which was found in the preceding paragraph now implies that P and Q are quasi-complementary P-projections, and that F and G are quasi-complementary projective faces of K.

We now consider the relevance of the above concepts to

and

6.3. THEOREM. Let A =B,,, where B is a

and let K be the : A A is a P-projection if and only if normal state space of B. Then p*a = pap, a E A, for some projection p E Bh. Every norm -closed face FofKis projective with corresponding projection PF E B,, such that F = {x E K: pF(x) = 1}. Moreover, the map is a bijection of the set of norm-closed faces of K onto the projections in Bh, which map the split faces of K onto the central projections in Bh.

We first note that A can be identified with A"(K), the dual space of the base normed space V = un K. (V is the Hermitian part of the normal dual space of B, see Sakai [188]). Let p B,, be a projection and define, for a E A, P*a = pap and Q*a = (1 —p)a(1 —p). Then F'> and Q* are w'>-continuous, and hence are the duals of projections F, 0: V V. Clearly we have IIPII, 1Q11 1. Suppose that Then, for p E K, the Cauchy—Schwarz inequality p*a =0 for some a = 0. Hence pa =0 and also ap = (pa)* =0, so gives p(pa)12 c (Q*A)+. Similarly we that Q*a = a — ap —pa +pap = a, that is (ker (p*A) obtain (ker Q*)+ Now if a E (p*Ar we have pap = a, so that Q*a = (1 —p)a(1 —p) = (1 —p)pap(1 —p) =0. Hence (P*A)+ = (ker and similarly (Q*A)+ = (ker p*)f• In order to show that F'> is smooth we need to shiw that when p 0 and p*)0, then p E (ker p*)0 Since PV is closed we need to show that pE we see that b p PV. Now if b = 1 so that b(p) = 0, and hence = 1 without loss of = p(l) = p(P*l) = IPpil, and we can assume generality. Note that p(l — p) = p(l — 1) =0, and so for each a E A we have, using the Cauchy—Schwarz inequality, p(a) = p(pap)+p(pa(1 —p)) ± p((l —p)ap) +p((l —p)a(1 —p)) = p(pap) = p(P'>a)). Therefore we obtain p = Pp and thus P'>, and similarly Q*, is smooth. Consequently is a p-projection. Conversely, let F'> : A-> A be a P-projection with quasi-complement Q* and let P'> 1 = p. Let 0< A <1 and p = Aa + (1—A )b with a, b [0, 1]. Then so that a, b (P*A)' = 0Aa, (1 —A)b Q'>, and also a = P'>a b =P*b But then Aa =p —(1 —A)b Ap, and we obtain a =p = b. This shows that p is an extreme point of [0, 1] and is therefore a projection in B (see Prosser [176,2.2]). Now if R* : A -> A is defined by R *a = pap we know that R * is also a P-projection with the same range as F'>, namely the order Proof.

CONVEXITY ThEORY FOR

246

ideal generated by p. The quasi-complement S* for R* will be given by S*a = (1 —p)a(1 —p) and since Q*1 = 1 —p, we can deduce that S*A = Q*A It follows that ker R* = ker and the first equivalence of the theorem is established. Let : A -+ A be any P-projection and let x e K PV, SO that (P*1)(x) = 1(Px) = 1. Conversely, let x EK such that (P*1)(x) = 1. Then, if Q* is

the quasi-complementary P-projection for P*, we have (Q* 1 )(x) = 1)(x) =0 so that x (ker = (PV). Hence we have shown that the projective face F,,, associated with P, has the form F,, = {x K: (P*1)(x) = 1}. (In particular, F,, is norm-exposed.) Now if F is any norm-closed face of K there exists a projection pp E B,, (the carrier of F) such that F = {x K: pp(x) = 1} (see proof of Theorem 2.1). The first part of the proof, together with the preceeding remarks, now shows that F is the projective face associated with the P-projection Ppa = (1

a€A. Suppose that F,, = for some P-projection S: V V. Then P and S have the same range and so arguing as we did above for the equality of is a bijection as required. P = S. Therefore the map F

If F is a split face of K then it is easily seen that the associated = A, so that PF = is central in A, and in B,, by Theorem 4.1(iv) (see also Section 3.8). Conversely, suppose that is central in A. Then if Q* is the quasi-complement to

P-projection P* is order-bounded in

I. It follows that K = co (F,, u F0) and hence, by Theorem 3.17, that F,, and FQ are complementary split faces of K. A is precisely the normal state Since the state space K of a unital A** the preceding result shows that every normspace of the closed face of K is projective. The following result proves, and generalizes, the result concerning unique decompositions which was used in the proof of Theorem 5.1. 6.4. THEOREM. Let K he a compact convex set such that every normexposed face is projective. Then each p E A(K)* has a unique decomposition p = o — r, where o', 0 and = HI + HI. Proof. Let p = — r where cr, r 0 and = flcrll + lrll = 1,0W, r 0. Choose f€A"(K) such that I =II/1I=f(p), and let g +f), h = 1 —g so that it 0g, h 1. Since 1 = (g—h)(p) =g(o)—g(r)—h(cr)±h(r) =

that g(o-) = licrIl, h(-r) IIrII and g(r) = hfrr) 0. Put F = K: g(x) = 0} so that F is a norm-exposed face of K containing r/IIrII, and let P: V - V (V = A(K)*) be the P-projection such that F = F,,. Let k A"(K) be defined by k = (PI)*l where P' is the quasi-complement of P. Now follows {x

247

TIlE STATE SPACES OF JORDAN OPERATOR ALGEBRAS

g =0 on F,, = = (ker gE(kerPs)O=(P?)*Ab(K). Therefore, since (P?)*1 = k

P' is smooth we have we have

1.

It is evident that Pr = r and hence P'r 0. Moreover, since g(u) 114 we have = ho-Il so that (P*1)(cr) = (1 ._(pl)* 1)(o-) 0. Hence o- belongs to (ker = (im P')', and therefore P'p = Since P' did not depend on the

choice of decomposition p =

—r

it follows that this decomposition is

unique. Continuing with our notation in which V = un K is a base normed space with dual space A = A"(K), we define (V, A) to be in spectral duality if for each a E A and A R there exists a P-projection P, with quasi-complement 0, such that a (x) A for x E F,, and a (x) > A for x E F0, and such that P satisfies the following conditions: (1) P*a ± Q*a = a; (ii) whenever S and T are

quasi-complementary P-projections on V with S*A ± T*a =

a,

then P

commutes with S and T. In the particular case where A and V satisfy the hypotheses of Theorem 6.3, (V, A) is in spectral duality. In fact let a €A, A €R and let eA be the

spectral projection for a corresponding to the value A. Then eA A and commutes with a, so that eAaeA + (1 — eA)a(l

—

eA) = a,

and thus condition (i)

holds for the P-projection P corresponding to eA. Moreover if p E A is a projection such that pap + (1— p)a(1 —p) = a then, multiplying on the left by p and also on the right by p, we see that p commutes with a, and therefore p commutes with eA. Evidently now (ii) is also satisfied. Finally, if x F1 then 4(x) = I so that a(x) = J A, while if x EFQ we have (x) =0 so

that a(x) =

> A.

In the general case where (V, A) are in spectral duality Alfsen and Shultz

([10, Theorem 6.8]) have proved that for each a A there exists a unique family A E R, of P-projections in V the spectral resolution for a, such = that the family satisfies for A

(i)

(ii)

= inf

= 1, and such that a = JA Riemann—Stieltjcs Integral.

(iii) inf

in the sense of a norm-convergent

When K is a compact convex set such that (V, A b(K)) are in spectral duality, K is said to be spectral. If, in addition, for each a A(K) the family consists of upper semi-continuous functions in A"(K) then K is said to be strongly spectral. 6.5. THEOREM. (1) Every compact simplex is spectral.

(ii) A compact simplex is strongly spectral if and only if it is a Bauer simplex.

(iii) The state space of a anita!

is strongly spectral.

CONVEXITY THEORY FOR Ca-ALGEBRAS

248

(i) Since every norm-closed face of a compact simplex is split and hence projective, the P-projections in V clearly commute. Hence, given a Ab(K) and A E R we need to find complementary faces F and F' such Proof.

that a (x) A on F and a (x)> A on F'. Without loss of generality we can take A =0. Let a = a — a - be the lattice decomposition of a and let F = {x E K: a 4(x) = O} and G = {x E K: a(x) = 0}. Then F and G are split faces of K with a (x) 0 for x E F. If we can show that K = co (F u G) it will follow that

forx€F'. F'cG, and hence that Fix an x in K, and note that a (x) = sup {a (y): 0 y x}. Thus we can choose {y,,} with 0 y,, x, such that a — a(y,,) < 2". We have, for each

ii,

a(y,,)—a(y,, A y,,ii)=a(yn Vy,, <2 "+ a a(y,, A

obtain a iflfk {y,,

a(y,,

<

1

2"'.

By induction we

for each

k. If A Yn-'-k} then {;v,,} converges in the norm of V to y = Sup,, —.

A

A

and 0 y x. Moreover, we have a y) a (y) = that a4(y)= Writing z =x —y we see that a'(z)= 0, x = y + z, and a — a(y) =0. It follows that K = co (F 0). =a (ii) Let K be a strongly spectral simplex. For x K let denote the maximal measure on K with resultant x; and for each a EA(K) define a regular Borel measure on R by 4) for each = JK (qS ° a) and hence a (y) = a so

bounded Borel function 'b on R. The formula j((—oO, A]), x €K, now gives the spectral resolution for a. (For the precise details see Alfsen— Schultz [10].). For each x E K define so that For = we have 3K then for some x = a(x)2. On the other hand if

a E A(K) the support of /L that

(f A

{x

K: a

will consist of more than one point, so

= a(x)2. Evidently then we have 9K = = a (x)2}, and it will follow that 9K is closed if we can

A(K). show that For any open interval (a, /3) we have

f

a)

=J =1

(Xa.m0 a) —

J

(—a))

which is l.s.c. in x. Since every open set U in is a disjoint union of open intervals we see that J (xu o a) is also I.s.c. in x. Finally, the function 4)(t) = t2, t R, is the limit of an increasing sequence of non-negative simple is functions of the form akXuk, with Uk open in R, and therefore

l.s.c. on K. Carrying out the same procedure with the function 'P(t) = l.s.c. and hence at21 is continuous. The converse implication of (ii) is just the commutative version of (iii). B, so that K is the normal (iii) Let K be the state space of a state space of A = B**, in the usual way. We have already noted that, if 11a112—t2 gives

THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS

249

V = A*, then (V, A) are in spectral duality, and hence K is spectral. Since B,, = A(K) is closed under composition with continuous real-valued

functions we can choose a decreasing sequence converging ° a} converges to Hence is u.s.c. in x, and consequently K is so that strongly spectral.

I

There exist further geometric characterizations of spectral duality and strongly spectral compact convex sets. We will state these characterizations, without proof, for compact sets K. Two elementsf, g are said to be

orthogonal, written fig if there exists a projective face F =0 on F;, while g =0 on the quasi-complementary projective face Fr.. ThEOREM. Let K he a compact convex set. Then: (i) Kis spectral if and only if every norm-exposed face of K isprojective and every f€ A"(K) admits a unique decomposition f = g — h, where g, h

and gih. (ii) K is strongly spectral if K is spectral and, in addition, every f A(K) admits a (necessarily unique) decomposition f g — h with g, h E and g I h. Alfsen eta!. [14] have developed a much more detailed theory concerning P-projections, JB-algebras and There is insufficient space in

this book to give a detailed description of their work, so we will just summarize some of their main results. If K is a compact convex set and if E c K, then we will denote by cr co E 1, where 0 E for each the set of all x K such that x = n and the sum is norm-convergent. We will say that K has the Hilbert ball property if, for every pair, x, y the smallest face, face (x, y) of K containing x and y is a norm-exposed face of K which is affinely isomorphic to the closed unit ball of some finite or infinite-dimensional Hubert space FL (If x y we take dim H = 0.) 6.7. THEOREM.

(a) Let A be a JB-algebra. Then there is a unique Jordan idealf in A such that A/Jhas a faithful isometric Jordan representation as a Jordan algebra of self-adjoint operators on a Hilbert space. Moreover, every factor representation of A which does not annihilate J maps A onto the exceptional Jordan algebra of all Hermitian (3 x 3) matrices over the C'ayley numbers. (b) A compact convex set K is the space of a JB-algebra if and only if K has the following properties: (i) K has the Hilbert ball property; (ii) 0 co 9K is a split face of K; (iii) every norm-exposed face of K is projective; (iv) every f€ A(K) admits a decomposition f= g — h, where g, h E A(K)t and g I h.

250

CONVEXITY THEORY FOR C*.ALGEBRAS

(c) A compact convex set K is the state space of a Jordan algebra of seif-adjoint operators on some Hubert space (with Jordan product S ° T = hST + TS)) if and only if K satisfies the con'ditions of (b) and such that face (x, y) is a split face of K whenever x, y E 9K and dim (face (x, y)) =9.

The methods that we have touched on in this section have culminated in the solution of Alfsen and Shultz [12]of one of the main problems in the subject, namely the characterization of the state spaces of The extra

ingredient that is required to distinguish the state spaces of amongst the state spaces of JB-algebras is the concept of orientability. Kis said to have the 3-bali property if it has the Hubert ball property with

the dimension of each face (x, y) being either three or one. Suppose that B = face (x, y) has dimension three. Using the affine isomorphism between we can define two opposite vector products in B; an B and the unit ball in orientation of B will mean a choice of one of these vector products for B.

Let A be a JB-algebra with state space K and let a E A, p E K. Then if p(a2) Owe can define Pa e K by pa(b) = p(aba)/p(a 2) b A. Let 9' denote the collection (face (p, where face Pa) has dimension three. Then K is said to be orientable if it is possible to choose orientations for the balls in 9' such that, for each a E A, the map p wherever it is X Pa is defined. 6.8. THEOREM. A JB-algebra is isomorphic to the self-adjoint part of a if and only if its state space K has the 3-ball properly and is orientable.

If A is a then the Jordan product in A,, is determined by the geometry of K. There may however exist different in A, and the choice of such products is connected with the orientabiity of K. For a detailed discussion of these topics, and for the proof of 6.8 we refer to the fundamental paper of Alfsen and Shultz [12]. Notes

This section is intended as an introduction to, and a brief survey of, a large body of fundamental work due to Alfsen et al. (see [10, 11, 12, 13, 14]). Theorem 6.1 is due to Alfsen, Shultz and Størmer [14,2.1]. Theorems 6.2 and 6.3 are due to Alfsen and Shultz [10, 3.5 and 11.5]. Theorems 6.4 and 6.6 appear in Alfsen and Shultz [13, 1.3, 2.2 and 2.4]. Theorem 6.5 was proved in Alfsen and Shultz [10, 10.4, 10.9 and 11.6]. Theorem 6.7 gives the main results of Alfsen and Shultz [11] and Theorem 6.8 summarizes the main results of Alfsen and Shultz [12].

THE STATE SPACES OF JORDAN OPERATOR ALGEBRAS

251

The geometric characterization of the state space of a given by Alfsen and Shultz in 6.8 represents a formidable achievement. For many years this problem had been one of the most important in the subject, and had occupied the efforts of many mathematicians. In particular the earlier work of Topping [209] on Jordan algebras of self -adjoint operators deserves mention. No such geometric characterization of the state spaces of function algebras is known. An up-dated version of [12] has recently been produced: E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz, "State spaces of

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121—124.

Index of Definitions

224 104

232

A (K), 43, 113, 143, 245

H(A), 147 H(A*), 147 h(E), 157

A,,, 226

A(K), 13

A(K), 117 A°, polar, 2 A0, 196

224 horn 1' 220 horn 220

146

A0(Pr), 75 Ar{x E A: IIxIkr}, A(T), 188 A(s), 187

1k,

197

invf, 151

3

224 75

k(E),

B, 156

157

224

bal, 2 co, 2

LF, 227

co, 2 'tf(E), 141 Cm(A), 241

M', 227

ma, rn,, 204

MR(X), 16

M(X), 16

Mt(X), 15 C(X), 11

d(A,B),9 d(A,B),9

e(A,E) 192

C(E), 142 PA, p(A), Minkowski functional, 2 Prim A, 234 Prim A(K), 224

(E, P) 31 (E*, p*), 32 epi f, epigraph off, 18

qx, 154 S", 196 64

e(X), 63

SM, state space

£ upper envelope off, 22, 26

137

f, lower envelope off, 19

T(A), 239

F0, 227

V, 172 wcak*, 3

fig, 249

F1, 224, 227 224

Z, 146 ZA, 194 261

of Al, 12, 149

INDEX OF DEFINITIONS

262

125

aM(K),

'&,, 229

149

p(A**), 228

118

p(G,k), 192 p(X), px, support function, 2 co E, 249

aM(X), 149

150

aMX, 12

ax, ext X, extreme points, 3

r,

17

125 227

0,

146

<,23 >, 215

Subject Index

Approximately directed, 39 Approximately generated, 33 Archimedean order, 37 Archimedean order ideal, 78, 93 Archimedean order unit, 37

A A-convex hull, 157 A-hull, 157 A-interpolation pair, 199 A-stable, 156, 193 A-superharmonic, 169 Absolutely dominated, 99 Absolutely order convex, 99 Abstract Bishop decomposition, 213 Abstract L-space, 58. Abstract M-space, 57 Abstract decomposition, 212 Abstractly antisymmetric, 213 Additive ordered space, 39 Affine function, 113 Affinc function space, A(K), 13 Alaoglu theorem, 3 Alfsen—Hirsberg theorem, 178 Algebra (of sets), 15 Almost Archimedean, 76 a-additive, 39 a-directed, 39 a-generated, 31 a-normal, 32 (a, n)-additive, 39 (a, n)-directed, 39 Andô's theorem (stability of split sets), 84 Antilattice, 124, 238 Antisymmetric algebra, 196 Antisymmetric set, 196 Approximate solution, 160

B

Baire measure, 15 Baire sets, 15 Banach—Grothendieck theorem, 6 Banach lattice, 49 Base, for a cone, 36 Base norm space, 36 Bauer simplex, 72, 103, 121, 247 Bishop decomposition, 211, 236 Bishop peak point theorem, 166 Bishop—Phelps theorem, 27 Boolcan algebra, 63 Borel measure, iS Borel sets, 15 Boundary for a function space, 12 Boundary measure, 96, 149 Boundary with respect to M, 12 Bounded weak* topology, 4 C

characterization of the state space, 250 Carrier, 227 Central measure, 242 Centralizer, 143 Centre of A(K), 104, 143, 210, 235

Approximate p-interpolation set, 160, 175, 187 263

264

SUBJECT INDEX

Characteristic function, 64 Characteristic of a face, 110 Choquet—l3ishop--de Leeuw theorem, 25, 149 Choquet boundary, 12 Choquet—Meyer theorem, 71 Choquet ordering, 23

Choquet simplex, 69, 106, 110, 113, 218, 230, 240, 247 Choquet's theorem, 24 Clopen, 63 Closed convex hull, 2 Closed range theorem, 10 Commutator face, 241 Commutator ideal in a sub-cone, 52, 87 Complete Banach lattice, 52 Complex state space, 147, 194 Cone, 2, 30 locally compact, 38 Convex balanced hull, 2 Convex function, 18 Convex hull, 2 Convolution, 187 Cunningham algebra, 141

241

Extension constant, 192 Extremal set, 215 Extreme point, 3

F Face, 3, 52 Face (x, y), 249 Facially closed sets, 102 Facially continuous functions, 105 Fourier coefficients, 187 Freudenthal unit, 65 Full interpolation set, 156 Function algebra, 166, 193, 223 Functional, sub-additive, 45 Functional, super-additive, 45

G Gauge functional, 7, 159 Gauge lemma, 9 Generalized peak set, 164, 194 Graph, 18 Gurarij space, 138

H

D

(DP), 50 D-regular, 10 Dauns—Hofmann theorem, 225, 235 Decomposability lemma, 182 Decomposable section, 80,93, 164, 179 Decomposition, 212 Decomposition property, 44, 50 Dilated set, 113 Directed ordered space, 39 Dirichiet algebra, 148, 204 Disc algebra, 159, 187, 206 Dominator, 159 Dual Cone, 30

E Edwards' separation theorem, 72 Epigraph, 18 Essential ideal, 198 Essential set, 198 Exact solution, 160, 177, 193 Exposed, 78

Harmonic functions, 124 Hereditary-up set, 215 Hermitian element, 147 Hubert ball property, 249 Hoffman—Wermer theorem, 201 Hull, 224 Hustad's theorem, 152 Hypostrict ideal, 91 I

Ideal in A(K), 117 Interpolation set, 154 Invariant, 229 Invariant function, 151 Isotonic function, 215 J

lB-algebra, 242

characterization of the state space, 249

265

SUBJECT INDEX

p

K

Kakutani's theorem (L-spaces), 67 Kernel, 224 Krein—Milman theorem, 3 Krein—Milman property, 144

theorem, 4, 7, 48 L

L-idcal, 140, 167, 169 L-projection, 140 Lattice cone, 44 Lattice homomorphism, 60 Lattice isometry, 62 Lattice (of sets), 15 Lattice ordered space, 48 Lattice, regular, 57 Lindenstrauss space, 219, 231 Linearly compact simplex, 138, 153 Lower envelope, 19, 26 Lower-semicontinuous (I.s.c.), 18 M

M-hull, 193 M-ideal, 224, 232 M-set, 167, 169 M-topology, 208 Maximal measure, 23, 24, 70 Minkowski functional, 2

N

Near lattice ideal, 91, 224 Norm, equivalence of, 32 Normal, partial ordering, 32 Normed lattice, 49

0 Open mapping theorem, 10

Order ideal, 76 Order isometry, 35 Order unit, 35 Order unit space, 35 Order-bounded operator, 141 Ordered Banach space, 30 Orientahie state space, 250 Orientation of a Hilbert ball, 250 Orthogonal affinc functions, 249

PEP, 78

P-projection, 244 Parallel face, 98, 1 13, 159 Partition, 64 Partition of unity, 134 Peak set, 164 Peaked partition of unity, 134 Polar calculus, 2 Polar set, 2 Polish space, 126 Positive cone, 30 Positive extension property, 78 Positive generation, 31, 77 Positively decomposable section, 81, 93 Positively generated ordered space, 47 Poulsen's simplex, 130 Pre-gauge, 7 Pre-gauge set, regular, 7, 33 Prime 238 Prime compact convex set, 124, 239 Primitive face, 224 Primitive ideal in a C'-algebra, 234 Primitive ideal in A(K), 224 Projective face, 244 Q

0(K) continuous convex functions, 18 Q(K) Ls.c. convex functions, 18 Quasi-complement of a face, 117 Quasi-complementary projection, 243 Quasi-complementary projective faces, 244

R (RIP), 50 R.N.P. 144 Radon—Nikodym property, 144 Representing matrix, 138 Representing measure, 16 Riesz Interpolation Property (RIP), 50 Riesz norm, 99 Riesz Representation theorem, 15 F. and M. Riesz theorem, 204 Rudin—Carleson theorem, 206 p-interpolation problem, 160, 169 S

Section (of a cone by a subspace), 76

Self-adjoint, 12, 147

266

SUBJEC1 INDEX

Self-determined sub cone, 76 Semi-exposed face, 76 Separation theorem, 1 Sidney—Stout theorem, 203 decomposition, 196, 236 Simplex see Choquet simplex, Linearly compact simplex Simplex spaces, 69, 106 Simplexoid, 154 Smooth projection, 243 Spectral compact convex set, 247 Spectral duality, 247

Split face, 70, 94, 110, 147, 202, 229, 241, 245 boundary measures, 97 Split sets, 83 Standard compact convex set, 110, 129 State space, 11, 12, 125, 149 State space, order unit space, 35 Stone's theorem, 64

Stone—Weierstrass theorem, 61, 122, 123

Størmer's axiom, 116, 236 Strong hull, 165 Strongly Archimedean order ideal, 79, 93

Strongly spectral compact convex set, 247 Structurally closed, 224 Structure topology, 225 Support cone, 27 Support function, 2 Support pair, 27

Symmetric facial topology, 208 Symmetric function, 209 Symmetric set, 194 T T-simplcx, 220

Three-ball property, 250 Topologizing family, 4 Totally disconnected, 63

U Uniform algebra, 166 Unique decompositions, 240, 246 Uniqueness property, 153 Unit traces on a 239 Upper envelope, 22

V Vidav—Palmer theorem, 147

w 53 Weakly central compact convex set, 237

Weak hull, 165 Weakly central 237 Weak order unit, 65 Weak* topology, 3 Wermer's maximality theorem, 189

BOOK REVIEWS

268

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Number 2, March 1983

©I983American Mathematical Society 0273-0979/82/0000-0977/sO 1.75

Convexity theory and its applications in functional analysis, by L. Asimow and

A. J. Ellis, Academic Press, London, v + 266 pp., $56.00. ISBN 0-12065340-0

This book focuses on the role of compact convex sets in functional analysis.

We will begin this review by trying to indicate why this role has been an important one and by giving a brief description of the historical evolution of research in this area. We will then turn to commenting directly on the contents and contribution of the book under review. One reason for the central role of compact convex sets in functional analysis is their ubiquity—as evidenced by the Banach-Alaoglu theorem that the unit Compact convex ball of the dual space of a Banach space is

sets play a key role, for example, in the fields of function algebras, group representations, and If A is a function algebra, the state space S(A) is defined to be the set of linear functionals in the dual unit ball with value 1 at the identity; this compact convex set contains all multiplicative linear functionals and thus contains a homeomorphic copy of the spectrum of

A. If G is a locally compact group, the set of continuous positive definite functions with value at most 1 at the identity is compact and convex. There is a construction which for each such function associates a representation of G; all strongly continuous unitary representations of G arise in this way. For another example, if A is a unital the states of A (i.e. the positive functionals with value 1 at the identity) form a compact convex set, and again states of A yield representations of A. For a final example, let X be a compact Hausdorff space, C( X) the Banach space of continuous real-valued functions on X, and M a closed subspace of C( X), separating points of X and containing the constant functions. Let S(M) be the set of functionals in the dual unit ball of M which take the value 1 at 1.

(S(M) is called the state space of M.) Then S(M) is convex and compact in the weak* topology and completely determines M. In fact, the evaluation map is an isometry (and an order isomorphism) from M onto the space A(S(M)) of continuous affine functions on S( M) (where we give A( S( M)) the supremum norm and pointwise ordering). This example is universal in the following sense. If K is any compact convex set in a locally convex topological vector space, then the evaluation map is an affine homeomorphism of K onto the state space of A(K) (the ordered Banach space of continuous affine functions on K). This

duality of K and A(K) was discovered by Kadison and has come to be the standard context for working with compact convex sets. In general if K is any compact convex set, we will refer to the elements of K as states.

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The preceding examples illustrate the kinds of ways in which compact convex sets arise in functional analysis. For any convex set K, a convex combination x = ty + (1 — t )z can be thought of as expressing x as a "mixture" of y and z. Points that cannot be expressed as such a mixture (for y, z distinct and t + 0, 1) are the extreme points (" pure states") of K. As one might expect, they play a key role in analyzing the structure of K and in applications.

The starting point here is, of course, the Krein-Milman theorem: every compact convex set (in a locally convex space) is the closure of the convex hull of its extreme points. In the case of locally compact groups, the extreme points of the compact convex set of positive definite functions described above are precisely those which give rise to irreducible representations. This was used by Gelfand and Raikov to establish the existence of a complete set of irreducible

representations. In the context of again pure states give rise to irreducible representations; Segal used this to establish the existence of a complete set of irreducible representations for (This construction has become known as the G-N-S construction.) Finally, in the function algebra

context, each pure state is multiplicative, but not all multiplicative linear functionals are pure states. The closure of the pure states can be identified with the Shilov boundary of the algebra—the minimal closed subset of the maximal ideal space on which each function in the algebra achieves its maximum. Given the existence of many pure states, it is natural to pursue the question

of decomposition of arbitrary states into mixtures of pure states. In finite dimensions, Minkowski's theorem asserts that every state is a convex combination of pure states. The Krein-Milman theorem gives one generalization: every state is a limit of convex combinations of pure states. It seems reasonable to

hope each state could be expressed as an integral of pure states. One way to make this precise is to recall that convergence in K is equivalent to pointwise convergence on A(K). Thus if is a measure on K and x is any state in K, then we write x

ifa(x)

Then the for all a in A(K). (If this holds, x is said to be the barycenter of question becomes whether the measure can be taken to live on the set 0eK of = 1. Choquet showed that this can always be pure states of K, i.e. done if K is metrizable. (If K is not metrizable, so that 8eK may not even be measurable, then the Choquet-Bishop-de Leeuw theorem states that can be chosen to be a boundary measure; this reduces to the previous result if K is metrizable.) As one might expect, these decompositions for states can be used to establish decompositions of representations into irreducible representations

for locally compact groups and

269

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270

What about the uniqueness of this decomposition of states into pure states? In finite dimensions, each point is a unique convex combination of pure states iff the convex set is a simplex. Choquet gave the definition for an arbitrary

compact convex set being a simplex. (The intersection of homothets of K should again be a homothet of K; equivalently A(K)* should be a vector lattice for the positive cone generated by K.) He then showed that these compact convex sets were precisely those with the desired uniqueness property. This general theory has come to be known as Choquet theory; it dates from

the late fifties. The theory and some of its applications are dealt with in R. Phelps' book Lectures on Choquet 's theorem and in E. Alfsen's Compact convex sets and boundary integrals. Choquet theory is the main topic of Phelps' book and the subject of almost half of Alfsen's book. Given the excellent presentation of Choquet theory in these two books, the authors of the book

under review wisely decided that another presentation in depth was not needed. Instead, they very quickly derive the basic results of Choquet theory,

and use this as a tool throughout the book. The main focus of the book is instead the developments in the field in recent years. The direction of most work regarding compact convex sets in the last twenty

years (and the primary subject of this book) has been the use of K as a dual object in exploring the structure of A(K). As indicated above, K determines the order and norm on A(K), and so it is not surprising that related properties for A( K) are reflected in corresponding properties of K. For example, A( K) will be a vector lattice iff K is a simplex whose set of pure states is closed (a "Bauer simplex"). However, it often turns out that if K arises as the state space of an algebra, then this algebra structure on A(K) is reflected in the geometry of K. For example, if A = C( X), then isomorphisms of A are in 1-1 correspon-

dence with affine homeomorphisms of S(A). (In fact, the multiplication on A = A(S(C(X))) can be recovered by noting that fg = h iff f(k)g(k) = g(k) for all pure states k.) More generally, Kadison showed that affine homeomorphisms of the state space of a unital induce Jordan isomorphisms

of the

(A Jordan isomorphism is a linear map preserving the symmetrized product ab + ba.) (In this case, however, recovering the Jordan product from the state space is much more subtle than for C( X).) Thus for such algebraic notions as the center, one-and two-sided ideals, the primitive ideal space, etc. have been found to have characterizations in terms of the state space. For function algebras, the state space has been used fruitfully in investigating the notions of peak sets and interpolation sets. A related direction has been the study of certain classes of compact convex sets and the related A(K) spaces. For example, much work has been devoted to the study of simplex spaces (spaces A(K) for K a simplex) and the question of characterizing the state spaces of The book under review focuses on these relationships between the structure of K and that of A(K). We will now summarize the chapters of the book. (It should be said that research in recent years has gone in a variety of directions, quite a few of which are discussed here, and we will not attempt to present a thorough summary.)

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The first chapter ("Preliminaries") gives a speedy development of prerequisites (with the assumption that the reader has taken at least a basic course in functional analysis). The approach here focuses on the tools of the polar calculus and gauges (Minkowski functionals of convex sets containing zero), and leads up to the Choquet-Bishop-de Leeuw theorem and the Bishop-Phelps theorem.

The second chapter ("Duality in Ordered Banach Spaces") includes a discussion of Choquet simplexes, of abstract L and M spaces, and of the Riesz decomposition property, as well as the basic facts on the K-A( K) framework. The authors often have chosen to state results in a more general context than Alfsen and Phelps (who stick primarily to spaces of the form A(K)).

The third chapter deals with simplex spaces. There are many beautiful results here. Several characterizations of simplexes among compact convex sets are given. (For example, a compact convex set is a simplex iff each closed face

is split iff each closed face is parallel.) The Poulsen simplex is discussed in is dense in K, and it detail. (This amazing simplex K has the property that is uniquely determined by this property among metrizable simplexes. Furthermore, if K is any compact metrizable simplex, then K is affinely homeomophic to a closed face of the Poulsen simplex.) The authors also give Haydon's proof of the result of Choquet that a topological space Xis homeomorphic to 8eK for

K a compact metrizable simplex iff X is a Polish space (i.e. a separable complete metrizable space). In this chapter, as throughout the book, the authors give very nice constructions to serve as counterexamples to natural conjectures and to illuminate the spirit of the theorems.

The fourth chapter (" Complex Function Spaces") is the primary chapter that works in the context of complex scalars. Included here are the complex

versions of results on uniqueness of representing measures and on Lindenstrauss spaces (Banach spaces whose dual is isometric to a space However, the main thrust of the chapter is the study of function algebras. In general a function algebra A may contain almost no real valued functions, and so will not contain A(K) where K is the state space of A. Nevertheless, S(A) proves to be useful in analyzing facts about A. The primary tool of the authors is the use of the complex state space, the convex hull of aS(A) and /3S(A) for suitable complex scalars a and /3. The authors use this framework to derive

many of the classical results of peark and interpolation sets for function algebras. (The Hoffman-Wermer theorem, the Bishop peak point theorem, and the Rudin-Carleson theorem appear as corollaries to the developments here.) The final chapter discusses convexity theory for The center and primitive ideal space of a are given geometric characterizations, and the Dauns-Hoffman theorem (that the center consists of those elements of A which induce a continuous map on the primitive ideal space) is proved. The results of Effros and Prosser giving a duality of ideals of A and faces of S(A)

are derived. The book ends with a discussion of the results of Alfsen, Hanche-Olsen, Størmer, and Shultz characterizing those compact convex sets which are affinely homeomorphic to the state spaces of Jordan and bras. The authors do an excellent job of giving a self-contained summary of these characterizations, including proofs of several key results.

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The authors have done a nice job of presenting a large amount of diverse material in a cohesive and self-contained fashion. The material is presented clearly and succinctly, with well-chosen examples. The choice of topics is excellent— a variety of the most appealing results of the last twenty years or so

in this area. (This is also an area in which the authors themselves have been quite active.)

The book would perhaps be a little difficult to use for quick reference. Results are sometimes stated in fairly complicated notation, with the notation (and some assumptions) explained in the preceding text. Thus a reader who is only browsing may have some difficulty following the spirit of the lemmas. (This should not cause any problems for the serious reader; in any case, this is

only a localized problem.) In general the book makes for very enjoyable reading.

The material here has little overlap with that available elsewhere in books (e.g. the books of Alfsen and Phelps). I highly recommend the book for the functional analyst interested in a self-contained presentation of many of the most interesting results in this field in recent years. FRED SHULTZ