Function Spaces

a and an uncountable set c Ker

e

e

C l :=

U rEQn[O,l]

re + [0, eo] and C2 :=

U

(X + r)e + (0, ex].

x+rE[O,l] xEA\{O}, rEQ

If C is endowed with the metric induced by the norm of F, then, by a proof similar to the one given in Theorem 1.2, C is a connected metric space that is not separably connected.

42

RICHARD M. ARON AND MANUEL MAESTRE

ACKNOWLEDGEMENT. This note evolved during visits the first author made to the Departamento de Amilisis Matematico of the Universidad de Valencia, and while the second author was a Visiting Professor in the Department of Mathematical Sciences at Kent State University for the 1998-99 academic year. The authors express their thanks to the Departments concerned for their hospitality. In addition, the authors are very grateful to the Referee, whose careful reading of the original version of this paper led to the present considerably simplified proof of Theorem 1.2.

References [A] Balbas, A., Estevez, M., Herves, C., and Verdejo, A. Espacios separablemente conexos, Rev. R.Acas. Cien. Exact. Fis. Nat. (Spain), 92, 1, (1998), 35-40. [B] J. Candeal, C. Herves and E. Indurain, Some results on representation and extension of preferences, Journal of Mathematical Economics 29 (1998), 75-81. [C] R. Pol, Two examples of non-separable metrizable spaces, ColI. Math. 33, (1975),209-211. DEPARTMENT OF MATHEl\IATICS, KENT STATE UNIVERSITY, KENT. OH 44242 USA E-mail address: aronClmcs. kent. edu DEPARTAMENTO DE ANALISIS MATEMATlCO, FACULTAD DE MATEMATICAS, UNIVERSIDAD DE VALENCIA, 46100 BURJASOT (VALENCIA). SPAIN E-mail address: manuel.maestreCluv. es

Contemporary Mathematics

Volume 328, 2003

Weighted Chebyshev Centres and Intersection Properties of Balls in Banach Spaces Pradipta Bandyopadhyay and S Dutta ABSTRACT. Vesely has studied Banach spaces that admit weighted Chebyshev centres for finite sets. Subsequently, Bandyopadhyay and Rae had shown, inter alia, that Lt-preduals have this property. In this work, we investigate why and to what extent are these results true and thereby explore when a more general family of sets admit weighted Chebyshev centres. We extend and improve upon some earlier results in this general set-up and relate them with a modified notion of minimal points. Special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of Lt-preduals. We also consider some stability results.

1. Introduction

Let X be a Banach space. We will denote by Bx[x,r] the closed ball ofradius r > 0 around x EX. We will identify any element x E X with its canonical image in X**. Our notations are otherwise standard. Any unexplained terminology can be found in either [6] or [10). In this paper we continue the study of Banach spaces that admit weighted Chebyshev centres that began with [3). DEFINITION 1.1. Let Y be a subspace of a Banach space X. For A ---? ~+, define

~

Y and

p:A

¢A,p(X) = sup{p(a)llx -

all : a E A}

A point Xo E X is called a weighted Chebyshev centre of A in X for the weight p if ¢A,p attains its minimum at Xo. When A is finite, Vesely [18) has shown that if X is a dual space, A admits weighted Chebyshev centres in X for any weight p, that the infimum of ¢ A,p over X and X** are the same, and THEOREM 1.2. [18, Theorem 2.7) For a Banach space X and aI, a2, ... ,an E X, the following are equivalent : 2000 Mathematics Subject Classification. Primary 4IA65, 46B20; Secondary 41A28, 46B25, 46E15, 46E30. Key words and phrases. Weighted Chebyshev centres, minimal points, central subspaces, I-complemented subspace, I PI,oo, Ll_preduals. © 43

2003 American Mathematical Society

44

BANDYOPADHYAY AND DUTTA (a) Ifrl, r2, ... , rn > 0 and ni=IBx " [ai, riJ =10, then ni=IBx[ai, riJ =10. (b) {al,a2, ... ,an } admits weighted Chebyshev centres for all weights rl,r2, ... ,rn > O. (c) {aI, a2,' .. , an} admits f-centres for every continuous monotone coercive f : IR+. -4 IR (see [18J for the definitions).

In this work, we investigate why and to what extent are these results true and thereby explore when a more general family of sets admit weighted Chebyshev centres. Extending the notion of central subspaces introduced in [3], we define an A-C-subspace Y of a Banach space X with the centres of the balls coming from a given family A of subsets of Y, the typical examples being those of finite, compact, bounded or arbitrary sets. The first gives us the central subspace a la [3J and the last one is related to the Finite Infinite Intersection Property (I Pj,oo) [8J. We extend and improve upon some results of [3, 18J in this general set-up and relate them with a modified notion of minimal points. We also improve upon one of the main results of [4J on the structure of the set of minimal points of a compact set. As in [3], special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of Ll-preduals. We also consider some stability results.

2. General Results We first extend VeselY's result in [18J on dual spaces from finite sets to all the way upto bounded sets and also strengthens its conclusions. We need the following notions. DEFINITION 2.1. Let X be a Banach space and A ~ X. (a) We define a partial ordering on X as follows : for Xl, X2 EX, we say that Xl :SA X2 if IlxI - all IIx2 - all for all a E A. We will denote by mx (A) the set of points of X that are minimal with respect to the ordering :SA and often refer to them as :SA-minimal points of X. Note that :SA defines a partial order on any Banach space containing A and we will use the same notation in all such cases. f(x2) (b) A function f : X -4 IR+ is said to be A-monotone if f(xd whenever Xl :SA X2· (c) Let Y be a subspace of X and A ~ Y. Following [9], we say X E X is a minimal point of A with respect to Y if for any y E Y, Y :SA X implies y=x. We denote the set of all minimal points of A with respect to Y in X by Ay,x, Note Ay,x ;2 A. For A ~ X, the set Ax,x will be called minimal points of A in X, and will be denoted simply by min A. (d) For A ~ X bounded, the Chebyshev radius of A in X is defined by

:s

:s

r(A) = inf sup Ilx xEX aEA

all.

THEOREM 2.2. (a) If A ~ X is bounded and X ~ A + r(A)B(X), then there exists y E X such that y :SA x. (b) If X = Z· is a dual space and A is bounded, then every A-monotone and w*-lower semicontinuous (henceforth, lsc) f : X -4 IR+ attains its minimum. In particular, for every p, ¢ A,p attains its minimum.

CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS

45

(c) If X = Z* is a dual space, for every Xo E X, there is a Xl E mx(A) such that Xl ::;A Xo. In particular, the minimum in (b) is attained at a point ofmx(A). PROOF. (a). Let X ~ A + r(A)B(X). Then, there exists c > 0 such that IIx - all > r(A) + c for all a E A. By definition of r(A), there exists y E X such that sUPaEA Ily - all < r(A) + c. Clearly, y ::;A x. (b). By (a), if X ~ A + r(A)B(X), there exists y E X such that y ::;A X, and hence, f(y) ::; f(x). Thus, the infimum of f over X equals the infimum over A + r(A)B(X). Moreover, since X is a dual space and f is w*-lsc, it attains its minimum over any w*-compact set. Thus f actually attains its minimum over X as well. Since the norm on X is w*-lsc, so is cPA,p for every p. (c). Consider {x EX: x ::;A xo}. Let {Xi} be a totally ordered subset. Let z be a w*-limit point of Xi. Since the norm is w*-lsc, we have liz -

all ::; lim inf Ilxi - all =

inf Ilxi

- all

for all a E A.

Thus the family {Xi} is ::;A-bounded below by z. By Zorn's lemma, there is a Xl E mx(A) such that Xl ::;A Xo. Now let Xo be a minimum for f. There is a Xl E mx(A) such that Clearly, f attains its minimum also at Xl.

Xl

::;A Xo. 0

REMARK 2.3. (a) It follows that for any bounded set A, minA C A + r(A)B(X). This improves the estimates in [9] or [18]. (b) Apart from cPA,p, there are many examples of A-monotone and w*lsc f : X = Z* ---- lR+. One particular example that has been treated extensively in [4] is the function cPf.,l defined by cPf.,l(x) = fA Ilx - aI1 2 dJ.L(a), where J.L is a probability measure on a compact set A ~ X. (c) Observe that though minimal points of A are ::;A-minimal, there is some distinction between the two notions. The two notions coincide if X is strictly convex. See Proposition 3.1 below. Now, if A is a bounded subset of a Banach space X, then by Theorem 2.2, A has a weighted Chebyshev centre in X**. But what about a weighted Chebyshev centre in X? When A is finite, Vesely [18] has shown that the infimum of cPA,p over X and X** are the same, and A admits weighted Chebyshev centres in X for any weight p if and only if X satisfies Theorem 1.2(a). We now show that both of these are special cases of more general results. We need the following definition. DEFINITION 2.4. Let Y be a subspace of a Banach space X. Let A be a family of subsets of Y. (a) We say that Y is an almost A-C-subspace of X if for every A E A, X E X and c > 0, there exists y E Y such that

(1)

Ily - all::; IIx - all + c forall a E A. (b) We say that Y is an A-C-subspace of X if we can takec = 0 in (a). (c) If A is a family of subsets of X, we say that X has the (almost) A-IP if X is an (almost) A-C-subspace of X**.

BANDYOPADHYAY AND DUTTA

46

Some of the special families that we would like to give names to are : (i) F = the family of all finite sets, (ii) K = the family of all compact sets, (iii) B = the family of all bounded sets, (iv) 'P = the power set. Since these families depend on the space in which they are considered, we will use the notation F(X) etc. whenever there is a scope of confusion. REMARK 2.5. (a) Note that F-C-subspaces were called central (C) subspaces in [3], 'P-C-subspaces were called almost constrained (AC) subspaces in [1, 2]. Also if X has the F-IP, it was said to belong to the class (GC) in [18, 3], and the 'P-IP was called the Finite Infinite Intersection Property (IPj,oo) in [7, 2]. (b) The definition of almost A-C-subspace is adapted from the definition of almost central subspace defined in [17]. The exact analogue of the definition in [17] would have, in place of condition (1), sup Ily aEA

all::; sup Ilx - all + c. aEA

Clearly, our condition is stronger. We observe below (see Proposition 2.7) that this definition is more natural in our context. (c) By the Principle of Local Reflexivity (henceforth, PLR), any Banach space has the almost F-IP. More generally, if Y is a!l ideal in X (see definition below), then Y is an almost F-C-subspace of X. DEFINITION 2.6. A subspace Y of a Banach space X is said to be an ideal in X if there is a norm 1 projection P on X* with ker(P) = y.l. PROPOSITION 2.7. Let Y be a subspace of a Banach space X. Let A be a family of bounded subsets of Y. Then the following are equivalent: (a) Y is an almost A-C-subspace of X (b) for all A E A and p : A ---t lR.+, if naEABx [a, p(a)] ¥- 0, then for every e > 0, naEABy[a, p(a) + c] ¥- 0. (c) for every bounded p, the infimum of ¢A,p over X and Yare equal. PROOF. Equivalence of (a) and (b) is immediate and does not need A to be bounded. (a) ::::} (c). Let Y be an almost A-C-subspace of X, A E A and p : A ---t lR.+ be bounded. Let M = supp(A). Let e > O. By definition, for x E X, there exists y E Y such that lIy - all::; Ilx - all + e for all a E A. It follows that

p(a)lly - all ::; p(a)llx -

all + p(a)e ::; p(a)llx - all + Me for all a E A.

and hence,

¢A,p(Y) ::; ¢A,p(X)

+ Me.

Therefore, inf ¢A,p(Y) ::; inf ¢A,p(X)

+ Me.

As e is arbitrary, the infimum of ¢A,p over X and Yare equal.

CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS

47

(c) =} (a). Let A E A, x E X and e > O. We need to show that there exists y E Y such that Ily - all IIx - all + e for all a E A. If x E Y, nothing to prove. Let x E X \ Y. Let N = sUPaEA IIx - all. Let p(a) = l/llx - all. Since x r/: Y and A ~ Y, p is bounded. Then ¢A.p(X) = 1, and therefore, inf ¢A,p(X) S 1. By assumption, inf ¢A,p(Y) = inf ¢A,p(X) S 1, and so, there exists y E Y, such that ¢A,p(y) 1 + e/N. This implies Ily - all S Ilx - all + ell x - all/N S Ilx - all + e for all a E A. 0

s

s

As noted before, by PLR, any Banach space has the almost F-IP. And therefore, the result of [18J follows. PROPOSITION 2.8. Let A and Al be two families of subsets of Y such that for every A E A and e > 0, there exists Al E Al such that A ~ Al + cB(Y). If Y is an almost AI-C-subspace of X, then Y is an almost A-C-subspace of X as well. Consequently, any ideal is an almost K-C -subspace and any Banach space has the almost K-IP. In particular, if A is a compact subset of X and p : A ---+ lR+ is bounded, then the infimum of ¢A,p over X and X** are the same. PROOF. Let A E A and e > O. By hypothesis, there exist Al E Al such that Al + eB(Y). Let x E X. Since Y is an almost AI-C-subspace of X, there exists y E Y such that A

~

lIy - alii S Ilx - alII

+ e/3 for all al

E AI.

Now fix a E A. Then there exists al E Al such that lIa - alii < e/3. Then Ily - alii + lIa - alii S Ilx - alii + 2e/3 Ilx - all + Iia - alii + 2e/3 S IIx - all + e. Therefore, Y is an almost A-C-subspace of X as well. Since any Banach space has the almost F-IP, by the above, it has the almost K-IP too. The rest of the result follows from Proposition 2.7. 0 Ily - all

S S

EXAMPLE 2.9. Vesely [18J has shown that if A is infinite, the infimum of ¢A,p over X and X** may not be the same. His example is X = co, A = {en: n ;::: I} is the canonical unit vector basis of Co and p == 1. Then inf ¢A,p(X) = 1 and inf ¢A,p(X**) = 1/2. The example clearly also excludes countable, bounded, or, taking Au {O}, even weakly compact sets. Thus Co fails the almost B-IP, almost P-IP and if A is the family of countable or weakly compact sets, then Co fails the almost A-IP too. Stronger conclusions are possible for A-IP. LEMMA 2.10. Let Y be a subspace of a Banach space X. For A ~ Y, the following are equivalent : (a) For every A-monotone f : A ---+ lR+ and x E X, there exists y E Y such that f(y) S f(x). ( b) For every p : A ---+ lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X). (c) For every continuous p : A ---+ lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X). (d) For every bounded p : A ---+ lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X).

BANDYOPADHYAY AND DUTTA

48

(e) Any family of closed balls centred at points of A that intersects in X also intersects in Y. (f) for any x EX, ther·e exists y E Y such that y :SA x. It follows that whenever any of the above conditions is satisfied, for every Amonotone f : A --+ lR+, the infimum of f over X and Yare equal and if A has a weighted Chebyshev centre in X, it has a weighted Chebyshev centre in Y.

PROOF. (a) => (b) => (c), (b) => (d) and (e) ¢} (f) => (a) are obvious. (c) or (d) => (f). As in the proof of Proposition 2.7, let p(a) = l/llx-all. Then p is continuous and bounded and
:s

:s

We now conclude the discussion so far by obtaining the extension of Theorem 1.2. THEOREM 2.11. For a Banach space X and a family A of bounded subsets of X, the following are equivalent: (a) X has the A-IP. (b) For every A E A and every f : X** --+ lR+ that is A-monotone and w*-lsc, the infimum of f over X** and X are equal and is attained at a point of (c) For every A E A and every p, the infimum of
x.

We now study different aspects of A-C-subspaces. DEFINITION 2.12. Let Y be a subspace of a Banach space X. Let A x E X and X* E B(X*), define U(x,A,x*)

= inf{x*(y) + IIx - yll : yEA}

L(x, A, x*)

=

~

Y. For

sup{x*(y) -lix - yll : yEA}

The following lemma is in [1]. We include the proof for completeness. LEMMA 2.13. Let Y be a subspace of a Banach space X and A ~ Y. For Xl, X2 EX, X2 :SA Xl if and only if for all x* E B(X*), U(X2, A, x*) U(XI, A, x*).

:s

:s

PROOF. If X2 :SA Xl, then for all x* E B(X*), x*(y) + IIx2 - yll x*(y) + IIxI -YII. And therefore, U(x2,A,x*):S U(xI,A,x*). Conversely, suppose IIx2 - Yo II > IIxI - Yo II for some Yo E A. Then there exists c > 0 such that IIx2 - Yo II - c ~ IIxI - Yoli. Choose x* E B(X*) such that IIxI - Yo II IIx2 - yolI- c < X*(X2 - Yo) - c/2. Thus U(XI, A, x*) x*(yo) + IIxI Yoll < X*(X2) - c/2 < U(X2, A, x*). 0

:s

:s

REMARK 2.14. Instead of B(X*), it suffices to consider the unit ball of any norming subspace of X*. We compile in the following propositions several interesting facts about A-Csubspaces and the A-IP. PROPOSITION 2.15. Let Y be a subspace of a Banach space X. For a family A of subsets of Y, the following are equivalent: (a) Y is an A-C-subspace of X

CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS

49

(b) for every x E X and A E A, there exists y E Y such that U(y, A, x*) ::; U(x,A,x*) for every x* E B(X*). (c) for any A E A, Ay,x ~ Y.

PROOF. This follows from Lemma 2.13 and the definition of Ay,x.

0

COROLLARY 2.16. X has the P-IP if and only if for every x** E X**, there exists x E X such that x is dominated on B(X*) by the 1~pper envelop of x** considered as a function on B(X*) equipped with the w*-topology. PROOF. Observe that for any x E X, U(x,X,·) == x on B(X*) and for x** E X**, U(x**, X, x*) is the upper envelop of x** considered as a function on B(X*) equipped with the w*-topology (see [8]). 0 PROPOSITION 2.17. (a) Let X be a Banach space and let Y be a subspace of X. Let A be a family of subsets of Y and let Al be a subfamily of A. If Y is a A-C -subspace of X, then Y is a Al -C -subspace of X as well. In particular, P-IP implies B-IP implies /C-IP implies F-IP. (b) 1-complemented subspaces are A-C-subspaces for any A. (c) Let Z ~ Y ~ X and let A be a family of subsets of Z. If Z is an A-C-subspace of X, then Z is an A-C-subspace of Y. And, if Y is an A-C-subspace of X, then the converse also holds. PROOF. The proof follows the same line of argument as in [3, Proposition 2.2]. We omit the details. 0 PROPOSITION 2.18. For a family A of subsets of a Banach space X, the following are equivalent : (a) X has the A-IP (b) X is a A-C-subspace of some dual space. (c) for all A E A and p : A ---+ lR+, ni=l Bx [ai, p(ai) + c] =f. 0 for all finite subset {aI, a2, ... ,an} ~ A and for all c > 0 implies naEABx [a, p(a)] =f. 0. In particular, any dual space has the A-IP for any A. Let S be any of the families F, /C, B or P. The S-IP is inherited by S-C-subspaces, in particular, by 1-complemented subspaces. PROOF. Clearly, (a) =} (b), while (c) =} (a) follows from the PLR. (b) =} (c). Let X be an A-C-subspace of Z*. Consider the family {Bz. [a, p(a)+ c] : a E A, c > O} in Z*. Then, by the hypothesis, any finite subfamily intersects. Hence, by w*-compactness, naEABz • [a, p(a)] =f. 0. Since X is an A-C-subspace of Z*, we have naEABx[a,p(a)] =f. 0. 0 The following result significantly improves [3, Proposition 2.8] and provides yet another characterization of the A-IP. PROPOSITION 2.19. Let Y be an almost F-C subspace of a Banach space X. Let A be a family of subsets of Y. If Y has the A-IP, then Y is an A-C -subspace of X. In particular, the conclusion holds when Y is an ideal in X. PROOF. Let x EX, A E A. Since Y be an almost F-C subspace of X, for all finite subset {aba2, ... ,an} ~ A and for all c > 0, nb:IBy[ai, Ilx.,... aill + c] =f. 0. Since Y has the A-IP, by Proposition 2.18(c), naEABy[a, Ilx - alll =f. 0. 0 Since X is always an ideal in X**, the following corollary is immediate.

BANDYOPADHYAY AND DUTTA

50

COROLLARY 2.20. For a Banach space X and a family A of subsets of X, the following are equivalent : . (a) X has A-IP. (b) X is an A-C-subspace of every superspace Z in which X embeds as an almost F -C subspace. (c) X is an A-C-subspace of every superspace Z in which X embeds as an ideal.

3. Strict convexity and minimal points PROPOSITION 3.1. If a Banach space X is strictly convex, then for every A X, min A = mx(A).

~

PROOF. As we have already observed, min A ~ mx(A). Let Xo E mx(A) and Xo ~ minA. Then there is an x E X such that x f= Xo and x :S::A Xo. Since Xo E mx(A), we must have Ilx - all = Ilxo - all for all a E A. Since X is strictly convex, II (x + xo)/2 - all < IIxo - all for all a. This contradicts that Xo E mx(A). Hence Xo EminA. D REMARK 3.2. If X is strictly convex, by a similar argument, for every Xo EX, there is at most one Xl E mx(A) such that Xl :S::A Xo. Thus for a strictly convex dual space, for every Xo E X*, there is a unique xi E mx· (A) such that xi :S::A Xo' PROPOSITION 3.3. Let X be strictly convex. Let A be a compact subset of X. For each continuous p, A admits at most one weighted Chebyshev centre. PROOF. Suppose A admits two distinct weighted Chebyshev centres Xo, Xl E X. Then
3.4. Let X be a Banach space such that (i) X has the F-IP; and (ii) for every compact set A ~ X, mx(A) is weakly compact. Then X has the K-IP. Moreover, if X** is strictly convex, then the converse also holds. THEOREM

Let X have the F-IP and for every compact set A ~ X, let mx(A) be weakly compact. Observe that for any B ~ A, we have mx(B) ~ mx(A). Let A ~ X be compact and let X** E X**. By Lemma 2.10, it suffices to show that there is a Zo E X such that Ilzo - all :s:: Ilx** - all for all a E A. Let {an} be a norm dense sequence in A. Take a sequence Ck --+ O. By compactness of A, for each k, there is a nk such that A ~ U~k BX[an,ck]. Since X has the F-IP, there exists Zk E nlk Bx [an' Ilx** - anll] and Zk E mx( {al, a2 ... ank }) ~ mx(A). Then IIzk-all :s:: Ilx**-all+2ck for all a EA. Now, by weak compactness of m x (A), we have, by passing to a subsequence if necessary, Zk --+ Zo weakly for some Zo EX. Since the norm is weakly Isc, we have Ilzo -all :s:: lim inf IIZk -all :s:: Ilx** -all for all a E A. Conversely, let X have the K-IP and X** be strictly convex. Let A ~ X be compact. It is enough to show that any sequence {xn} ~ mx(A) has a weakly convergent subsequence. Without loss of generality, we may assume that {xn} are PROOF.

CHEBYSHEV CENTRES AND INTERSECTION PROPERTIES OF BALLS

51

all distinct. By Remark 2.3 (a), mx(A) ~ A + r(A)B(X) is bounded. Let x** be a w*-cluster point of {x n } in Xu. It suffices to show that x** EX. Suppose x** E X** \ X. Since X has the K-IP, there exists Xo E mx(A) such that Ilxo - all ::; Ilx** - all for all a E A. Since X** is strictly convex, lI(x** +xo)/2 - all < IIx** - all for all a E A. Since (x** +xo)/2 E X** \X, by K-IP again, there exists Zo E mx(A) such that Ilzo -all::; II(x** +xo)/2-all < IIx** -all for all a E A. Since A is compact, there exists c > 0 such that Ilzo - all < IIx** - all - c for all a E A. Observe that

Ilzo - all < Ilx** - all- c ::; liminf IIx n n

all- c for all a E A.

Therefore, for every a E A, there exists N(a) EN such that for all n

~

N(a),

IIzo - all < Ilxn - all- c. By compactness, there exists N E N such that IIzo - all < IIx n - all - c/4 for all n ~ N and a E A. Thus, Zo ::;A Xn for all n ~ N. Since Xn E mx(A) and X is strictly convex, Zo = Xn for all n completes the proof.

~

N. This contradiction

0

REMARK 3.5. In proving sufficiency, one only needs that {Zk} has a subsequence convergent in a topology in which the norm is lsc. The weakest such topology is the ball topology, bx . So it follows that if X has the F-IP and for every compact set A ~ X, mx(A) is bx-compact, then X has the K-IP. Is the converse true? COROLLARY 3.6. [4, Corollary 1] Let X be a reflexive and strictly convex Banach space. Let A ~ X be a compact set. Then min(A) is weakly compact. REMARK 3.7. Clearly, our proof is simpler than the original proof of [4]. If Z is a non-reflexive Banach space with Z*** strictly convex, then X = Z· is a non-reflexive Banach space with K-IP such that X** is strictly convex. Thus, our result is also stronger than [4, Corollary 1].

4. L l -preduals and PI-spaces Our next theorem extends [3, Theorem 7], exhibits a large class of Banach spaces with the K-IP and produces a family of examples where the notions of FC-subspaces and K-C-subspaces are equivalent. DEFINITION 4.1. (a) [12] A Banach space X is called an Ll-predual if X* is isometrically isomorphic to £1(J-t) for some positive measure J-t. (b) [11] A family {Bx[xi,ri]} of closed balls is said to have the weak intersection property iffor all x* E B(X*) the family {Ba[x*(xd,ri]} has nonempty intersection in JR. THEOREM 4.2. For a Banach space X, the following are equivalent: ( a) X is a K -C -subspace of every superspace (b) X is a K-C -subspace of every dual superspace ( c) X is a F -C -subspace of every superspace (d) X is an almost F -C -subspace of every superspace (e) X is a F-C-subspace of every dual superspace (f) X is an almost F-C-subspace of every dual superspace (g) X is an Ll-predual.

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PROOF. Observe that if X ~ Y ~ y** and X is a A-C-subspace of Y**, then X is a A-C-subspace of Y. Thus (a) {::} (b) and (c) {::} (e). And clearly, (a) ::::} (c) ::::} (d) ::::} (f). (f) ::::} (g). Since the definition of almost central subspaces in [17J is weaker than our definition of almost F-C-subspaces, this follows from [17, Theorem 1, 2::::} 3J (g) ::::} (a). Suppose X is an LI-predual, and let X ~ Y. Let A ~ X be compact with at least three points. Let Yo E Y. Then the family of balls {Bx[a, Ilyo - allJ : a E A} have the weak intersection property. Since X is an £l-predual and since the centres of the balls are in a compact set, by [14, Proposition 4.4J, naEABx [a, IlyoallJ ¥- 0. If A has two points, observe that two balls intersect if and only if the distance between the centres is less than or equal to the sum of the radii, it is independent of the ambient space. 0 COROLLARY 4.3. Every LI-predual has the /C-IP and hence also the F-IP.

Y

~

PROPOSITION 4.4. Suppose X is an LI -predual space. Then for a subspace X, the following are equivalent : (a) Y is an ideal in X (b) Y is a /C-C-subspace of X (c) Y is a F-C-subspace of X (d) Y is an almost F-C-subspace of X (e) Y itself is an LI -predual

PROOF. (e) ::::} (b) follows from Theorem 4.2 and (e) ::::} (a) follows from [16, Proposition IJ. And clearly, (b) ::::} (c) ::::} (d) and (a) ::::} (d). (d) ::::} (e). This again is an easy adaptation of the proof of [17, Theorem 1, 2 ::::} 3J. We omit the details. 0 The analog of Theorem 4.2 for 'P-C-subspaces involves 'PI-spaces. DEFINITION 4.5. Recall that a Banach space is a 'PI-space if it is 1complemented in every superspace. THEOREM 4.6. For a Banach space X, the following are equivalent: (a) X is a 'PI -space ( b) X is i-complemented in every dual space that contains it ( c) X is a 'P -C -subspace of every superspace (d) X is a 'P -C -subspace of every dual space that contains it ( e) X is isometric to C (K) for some extremally disconnected compact Hausdorff space K. PROOF. (a) {::} (b) and (c) {::} (d) follow as in the first paragraph of Theorem 4.2. And clearly, (a) ::::} (c). (d) ::::} (a). By Proposition 2.18 and Theorem 4.2, (d) implies X is an LI_ predual with 'P-IP. Recall that [12, Theorem 3.8J a Banach space X is a 'PI-space if and only if every pairwise intersecting family of closed balls in X intersects. And that X is a LI-predual if and only if X** is a 'PI-space. Now given a pairwise intersecting family of closed balls in X, since X** is a 'PI-space, they intersect in X**. And since X has 'P-IP, they intersect in X too. (a) {::} (e) is also observed in [12, Section 11J. 0

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PROPOSITION 4.7. Let A be a family of subsets of X such that F ~ A. Then, the following are equivalent : (a) X is an Ll -predual with A-IP (b) X is an A-C-subspace of every superspace (c) for every A E A, every pairwise intersecting family of closed balls in X with centres in A intersects. PROOF. (a) =? (b). Since X has the A-IP, it is an A-C-subspace of every superspace in which it is an ideal (Proposition 2.20) and since X is an L1-predual, it is an ideal in every superspace [16, Proposition 1]. Thus (b) follows. (b) =? (a). Since F ~ A, this is immediate. (a) =? (c). This is similar too the proof of Theorem 4.6 (d) =? (a). (c) =? (a). If every finite family of pairwise intersecting closed balls in X intersects, then X is an L1-predual. And that X has the A-IP follows from Propo0 sition 2.18 (c). Let C(T, X) be the space of all X-valued bounded continuous functions on a topological space T equipped with the sup norm. We now characterize when C(T, X) is a real L1-predual. First we need the following lemma. LEMMA 4.8. Suppose Y is a subspace of a Banach space X and Y is a real L1-predual. Let A ~ Y be a compact set and r : A -7 lR,+ be such that naEABx[a,r(a)] =I- 0. Let y E naEABy[a,r(a) + c] for some c > O. Then there exists z E naEABy [a, r(a)] such that Ily - zll S c. PROOF. Since naEABx [a, r(a)] =I- 0, and intersection of intervals is an interval, for any y* E B(Y*), naEABJR[y*(a), r(a)] =I- 0 and is a closed interval. As y*(y) E naEABJR[y*(a),r(a) + c] for any y* E B(Y*), the family {By[y,c].By[a,r(a)] : a E A} is a weakly intersecting family of balls in Y. Since Y is a L1-predual,

By[y,C]nnaEABy[a,r(a)] =1-0.

0

PROPOSITION 4.9. A Banach space X is a real Ll -predual if and only if for each paracompact space T, C(T, X) is a real L1-predual. PROOF. Since X is I-complemented in C(T, X), hence a K-C-subspace, by Proposition 4.4, if C(T, X) is an L1-predual, then so is X. Conversely, suppose X is a real L1-predual. Let Z = C(T, X), {!I, 12,···, fn} ~ Z and rl, r2,···, rn > 0 be such that the family {Bz[fi, ri] : i = 1, ... , n} intersects weakly. Then for each t E T, the family {B x [fi (t), r i] : i = 1, ... , n} intersects weakly, and since X is a real L I-predual, they intersect in X. Consider the multi-valued map F: T -7 X given by F(t) = nf=lBx[Ji(t),ri]. Note for each t, F(t) is a nonempty closed convex subset of X. CLAIM : F is lower semicontinuous, that is, for each U open in X, the set V = {t E T: F(t) n U =I- 0} is open in T. Let to E V. Let Xo E F(to) n U. Let c > 0 be such that lI:r - xoll < c implies x E U. Let W be an open subset of to such that t E W implies 111;(t)- fi(to) II < c/2 for all i = 1, ... , n. We will show that W ~ V. Let t E W. Then for any i = 1, ... , n, Ilxu - fi(t)11 S Ilxo - f.i(tO) II + 111;(to) fi(t)11 Sri + c/2. Therefore, Xo E nf=lBx[fi(t),rj + c/2]. By Lemma 4.8, there exists z E F(t) = nf=lBx[fi(t), ri] such that Ilxo-zll S c/2 < c. Then z E F(t)nU, and hence, t E V. This completes the proof of the claim.

BANDYOPADHYAY AND DUTTA

54

Now since T is paracompact, by l\Iichael's selection theorem, there exists 9 E Z such that g(t) E F(t) for all t E T. It follows that 9 E ni'=l Bz[f;, ril, 0 REMARK 4.10. For T compact Hausdorff, this result follows from [13, Corollary 2, p 43]. But our proof is simpler.

5. Stability Results In this section we consider some stability results. With a proof similar to [3, Proposition 14], we first observl' that PROPOSITION 5.1. K-IP is a separ-ably determined property. i.e .. if every sepamble subspace of a Banach space X have K-IP. then X also has K-IP. DEFINITION 5.2. [10] A subspace Y of a Banach space X is called a semi-Lsummand if there exists a (nonlinear) projection P : X ----> Y such that P(>..x

+ Py) Ilxll

+ Py, and IIPxl1 + Ilx - VI'II

>"P.r

for all x, y EX. >.. scalar. In [3]. it was shown that semi-L snmmands are .1'-C-subspaces. Basically the same proof actually shows that PROPOSITION 5.3. A semi-L-summand is an A-C-subspace for any A. Our next result concerns proximinal subspaces. DEFINITION 5.4. A subspace Z of a Banach space X is called proximinal if for every :r E X, there exists Zo E Z such that Ilx - zoll = d(J;, Z) = infzEz Ilx - zll· The map PZ(J:) = {zo E Z : Ilx - zoll = infzEz lI:r - zll} is called the metric projection. PROPOSITION 5.5. Let Z ~ Y ~ X, Z proximinal in X. (a) Let A be a family of subsets of Y / Z. Let A' be a family of subsets of Y s'ltch that for any :1; E X and A E A. there exists A' E A' such that for any a + Z E A. {a + Pz(x - a)} n A' i=- 0. S-uppose Y is a A'-C -subspace of X. Then Y/Z 'is a A-C-subspace of X/Z. Let S be any of the families .1', l3 or P. (b) IfY is a S(Y)-C-subspace of X. then Y/Z is a S(Y/Z)-C-8'ubspace of X/Z. (c) Suppose the metric projection has a continuous selection. Then. if Y is a K(Y)-C-subspace of X, Y/Z is a K(Y/Z)-C-subspace of X/Z. (d) Let Z ~ Y ~ X*, Z w*-closed in X*. 11' Y is a S(Y)-C-subspace of X*, then Y/Z is a S(Y/Z)-C-subspace of X* /Z. and hence. has the S(Y/Z)-IP. (e) Let X have the S(X)-IP. Let l'd ~ X be a reflexive subspace. Then X/M has the S(X/l'd)-IP. PROOF. (a). Let A E A and x + Z E X/Z. Choose A' as above. Then, for a + Z E A, there exists z E Pz(x - a) (depending on .1: and a) such that a + zEA'. Since Y is a A' -C-subspace of X, there exists Yo E Y snch that Ilyo - a - zll ~ Ilx - a - zll for all a + Z E A. Clearly then Ilyo - a + ZII ~ Ilyo - a - zll ~ 11:1; - a - zll = IIx - a + ZII·

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If S is the family under consideration in (b) and (c) above and A = S(Y/Z), then for any choice of A' as above, S(Y) ~ A'. Hence, (b) and (c) follows from (a). For (d), we simply observe that any w*-closed subspace of a dual space is proximinal. And (e) follows from (d). 0 As in [3, Corollary 4.6], we observe PROPOSITION 5.6. Let Z ~ Y ~ X, Z proximinal in Y and Y is a semi-Lsummand in X. Then Y/Z is a P-C-subspace of X/Z. Let us now consider the Co or fp sums. THEOREM 5.7. Let r be an index set. For all 0: E r, let Ya: be a subspace of Xa:. Let X and Y denote resp. the Co or fp (1 :::; p :::; 00) sum of Xa: 's and Ya: 'so (a) For each 0: E r, let Aa: be a family of subsets ofYa: such that {O} E Aa: and for any A E Aa:. there exists B E Aa: such that Au {O} ~ B. Let A be a family of subsets of Y such that for any 0: E r, the 0:section of any A E A belongs to Aa:. Then Y is an A-C-subspace of X if and only if for each 0: E r, Ya: is an Aa:-C-subspace of Xa:. Let S be any of the families F, 1C, B or P. (b) Y is a S(Y)-C-subspace of X if and only if for any 0: E r, Ya: is a S(Ya:)-C-subspace of Xa:. (c) The S-IP is stable under fp-sums (1 :::; p:::; 00). PROOF. (a). The proof is very similar that of to [3, Theorem 4.7]. We omit the details. (c). Xa: has S-IP if and only if Xa: is a S-C-subspace of some dual space Y';. Now the fp-sum (1 :::; P :::; 00) of Y';'s is a dual space. 0 REMARK 5.8. The result for F-IP has already been noted by [18] with a much different proof. The stability of the P-IP under f1-sums is noted in [15] again with a different proof. [18] also notes that F-IP is stable under co-sum. And Corollary 4.3 shows that Co has the IC-IP. However, we do not know if the IC-IP is stable under Cosums. As for the B-IP or P-IP, we now show that co-sum of any infinite family of Banach spaces lacks the B-IP, and therefore, also the P-IP. This is quite similar to Example 2.9. PROPOSITION 5.9. Let r be an infinite index set. For any family of Banach spaces Xa:, 0: E r, X = fficoXa: lacks the B-IP. PROOF. For each

0: E

r, let Xa:

be an unit vector in Xa: and define ea: E X by if f3 = 0: otherwise

Then the set A = {ea: : 0: E r} is bounded and the balls Bx--[ea:, 1/2] intersect at the point (1/2xa:) E X**, but the balls Bx[ea:, 1/2] cannot intersect in X. 0 REMARK 5.10. As before, taking AU{O}, it follows that X lacks the A-IP even for A = weakly compact sets. Coming to function spaces, we note the following general result.

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PRDPOSITION 5.11. Let Y be a subspace of a Banach space X and A be a family of subsets of Y. (a) For any topological space T, if C(T, Y) is a A-C-subspace of C(T, X), then Y is a A-C -subspace of X. Moreover, if C(T, X) has A-IP, X has A-IP. (b) Let (n, L-, fl) be a probability space. If for some 1 ~ P < 00, LP(fl, Y) is aA-C-subspace of LP(fl, X), then Y is aA-C-subspace ofX. Moreover, if LP(fl, X), has A-IP, then X has A-IP. PROOF. For (a) and (b), let F(X) denote the corresponding space of functions and identify X with the constant functions. In (a), point evaluation and in (b), integral over n gives us a norm 1 projection from F(X) onto X. Thus X inherits A-IP from F(X). Now suppose F(Y) is an A-C-subspace of F(X). Let P : F(Y) -+ Y be the above norm 1 projection. Let x E X and A E A. Then, there exists g E F(Y) such that IIg-ali ~ IIx-all for all a E A. Let y = Pg. Then, lIy-all ~ IIg-all ~ IIx-all for all a E A. 0 The following Proposition was proved in [3]. PROPOSITION 5.12. (a) Let X has Radon Nikodym Property and is 1complemented in Z* for some Banach space Z. Then for 1 < p < 00, LP(IL,X) is 1-complemented in U(fl, Y)* (l/p+ l/q = 1), and hence has the 'P-IP. (b) Suppose X is separable and 1-complemented in X** by a projection P that is w*-w universally measurable. Then for 1 ~ P < 00 LP(fl, X) is 1-complemented in Lq(fl,X*)* (l/p+ l/q = 1), and hence has the 'P-IP. Since the B-IP or 'P-IP is inherited by I-complemented subspaces and the B-IP, the next result follows essentially from the arguments of [17].

Co

lacks

PROPOSITION 5.13. (a) Let X be a Banach space containing Co and let Y be any infinite dimensional Banach space. Then X 181" Y fails the B-IP and 'P-IP. (b) If C(K, X) has the B-IP, then either K is finite or X is finite dimensional. C(K,X) has the 'P-IP if and only if either (i) K is finite and X has the 'P-IP or (ii) X is finite dimensional and K is extremally disconnected. (c) For· any nonatomic measure space (n,L-,/L) and a Banach space X containing co, £1 (fl, X) fails the B-IP. In the next Proposition, we prove a partial converse of Proposition 5.11(a) when Y is finite dimensional and K is compact and extremally disconnected. PROPOSITION 5.14. LetS be any ofthefamiliesF, IC, B or'P. LetY be a finite dimensional a S(Y)-C-subspace of a Banach space X. Then for any extremally disconnected compact space K, C(K, Y) is a S(C(K, Y))-C-subspace of C(K, X). PROOF. We argue similar to the proof of [3, Proposition 4.11]. Let K be homeomorphically embedded in the Stone-Cech compactification .B(r) of a discrete set r and let ¢ : .B(r) -+ K be a continuous retract. Let A E S(C(K, Y)) and g E C(K, X). Note that since Y is finite dimensional, by the defining property of .B(r), any Y-valued bounded function on r has a norm preserving extension

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57

in C(,B(r), Y). Thus C(,B(r), Y) can be identified with EBeoc(r) Y. Lift A to this space. In view of Theorem 5.7, this space is S(Y)-C-subspace of EBooX. This latter space contains C(,B(r) , X). Thus by composing the functions with 4>, we get a f E C(K, Y) such that Ilf - hll ~ Ilg - hll for all h E A. Hence the result. 0 And now for a partial converse of Proposition 5.11(b). THEOREM 5.15. Let Y be a separable subspace of x. IfY is a P-C-subspace of X, then for any standard Borel space 0 and any a-finite measure jl, Lp(jl, Y) is a P-C-subspace of Lp(jl,X). PROOF. Let f E Lp(jl, X). Since Y is a P-C-subspace of X, for each x E X, nyEyBy[y, IIx - yll] =1= 0. Define a multi-valued map F : 0 ----+ Y, by

F(t) = {

nyEy

By[y, Ilf(t)

{f(t)}

- ylll

if

f(t) EX \ Y

if

f(t) E Y

Let G = {(t, z) : z E F(t)} be the graph of F. Claim: G is a measurable subset of 0 x Y. To establish the claim, we show that GC is measurable. Since Y is separable, let {Yn} be a countable dense set in Y. Observe that z ~ F(t) if and only if either f(t) E Y and z =1= f(t) or f(t) E X \ Y and there exists Yn such that liz - Ynll > IIf(t) - Ynll· And hence,

G C = {f(t) E Y and z

=1=

f(t)} u

U {f(t) EX \ Y and liz - Ynll > Ilf(t) -

Ynll}

n;:::l

is a measurable set. By von Neumann selection theorem, there is a measurable function 9 : 0 ----+ Y such that (t,g(t)) E G for almost all tEO. Observe that Ilg(t)11 ~ IIf(t)1I for almost all t. Hence 9 E Lp(jl, Y). Also for any h E Lp(jl, Y) we have Ilg(t) - h(t)11 ~ Ilf(t) - h(t)1I for almost all t. Thus, Ilg - hll p ~ IIf - hll p for all h E Lp(jl, Y). 0 QUESTION 5.16. Suppose Y is a separable lC-C-subspace of X. Let (0, ~,jl) be a probability space. Is LP(jl, Y) a lC-C -subspace of LP(jl, X)? REMARK 5.17. This question was answered in positive in [3] for F-C-subspaces and we did it for P-C-subspaces. Both the proofs are applications of von Neumann selection Theorem. The problem here is for a compact set A in LP(jl, Y) and wE 0 the set {f(w) : f E A} need not be compact in Y. ACKNOWLEDGEMENTS. Partially supported by a DST-NSF grant no. RP041/2000. The first-named author availed this grant to visit Southern Illinois University at Edwardsville, USA in May-June 2002 and attended the Fourth Conference on Function Spaces, where he presented a talk based on this work. He would like to thank Professor K. Jarosz for the warm hospitality and a wonderful conference. We also thank the referee for suggestions that improved the paper.

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References [1] P. Bandyopadhayay, S. Basu, S. Dutta and B. L. Lin Very nonconstmined subspaces of Banach spaces, Preprint 2002. [2] P. Bandyopadhayay and S. Dutta, Almost constmined subspaces of Banach spaces, Preprint 2002. [3] Pradipta Bandyopadhyay and T. S. S. R. K. Rao, Centml subspaces of Banach spaces, J. Approx. Theory, 103 (2000),206-222. [4] B. Beuzamy and B. Maurey, Points minimaux et ensembles optimaux dans les espaces de Banach, J. Functional Analysis, 24 (1977), 107-139. [5] J. Diestel, Geometry of Banach Spaces, selected topics, Lecture notes in Mathematics, Vo1.485, Springer-Verlag (1975). [6] J. Diestel and J. J. Uhl, Jr., Vector' measures, Mathematical Surveys, No. 15, Amer. Math. Soc., Providence, R. 1. (1977). [7] G. Godefroy, Existence and uniqueness of isometric preduals: a survey, Banach space theory (Iowa City, lA, 1987), 131-193, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [8] G. Godefroy and N. J. Kalton, The ball topology and its applications, Banach space theory (Iowa City, lA, 1987), 195-237, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [9] G. Godini On minimal points, Comment. Math. Univ. Carotin., 21 (1980), 407-419. [10] P. Harmand D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, 1547, Springer-Verlag, Berlin, 1993. [11] O. Hustad, Intersection properties of balls in complex Banach spaces whose duals are L1 spaces, Acta Math., 132 (1974), 283-313. [12] H. E. Lacey, Isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. [13] J. Lindenstrauss, Extension of compact opemtors, Mem. Amer. Math. Soc., No. 48, 1964. [14] A. Lima, Complex Banach spaces whose duals are L1-spaces, Israel J. Math., 24 (1976), 59-72. [15] T. S. S. R. K. Rao, Intersection properties of balls in tensor products of some Banach spaces II, Indian J. Pure Appl. Math., 21 (1990),275-284. [16] T. S. S. R. K. Rao, On ideals in Banach spaces, Rocky Mountain J. Math., 31 (2001), 595-609. [17] T. S. S. R. K. Rao Chebyshev centers and centmble sets, Proc. Amer. Math. Soc., 130 (2002), 2593-2598. [18] L. Vesely, Genemlized centers of finite sets in Banach spaces, Acta Math. Univ. Comen., 66 (1997),83-115. (Pradipta Bandyopadhyay) STAT-MATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, E-mail: pradiptaClisical.ac.in (S Dutta) STAT-MATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, E-mail: sudipta.rGlisical.ac . in

Contemporary Mathematics Volume 328, 2003

The boundary of the unit ball in HI-type spaces Paul Beneker and Jan Wiegerinck ABSTRACT. This is a survey on the extreme, exposed and strongly exposed boundary points of the unit baH in various Hl-type spaces.

1. Introduction

Theorems like the Krein-Milman theorem and Phelps' theorem assert the existence of many extreme points in certain convex sets. However, to decide whether a particular point in the boundary of a convex set is extreme, exposed or strongly exposed is quite a different question. In this paper we survey what is known in the very special situation of the unit ball of certain Hardy spaces. Of course we refer to the literature for the majority of the proofs. But we have included some, and sketched others, hoping that this will clarify the exposition. After introducing extreme, exposed and strongly exposed points, we end the introduction with definitions of the Hardy-spaces in which we study our problem. These are: the standard Hl(]I))) of the disc ]I)) in e, HI(0.) of a domain of finite connectivity 0. c e, the Bergman space Al(]I))) of the disc and Hl(lB n ) of the unit ball in en. In Section 2 we will study the boundary points of the unit ball of HI (]I))). Here good function theoretic descriptions of extreme and strongly exposed points are possible. Exposed points seem to escape such a description. The ball of HI (0.) is the topic of Section 3. We will see that results for the disc in general do not extend to this case. In Section 4 we will turn to the Bergman space. Now all boundary points of the unit ball are exposed and many well-behaved functions, like polynomials, are strongly exposed. However, there are boundary points that are not strongly exposed. The final section is devoted to the little we know about the higher dimensional situation. The paper borrows heavily from the expository parts of the PhD-thesis of the first author, [3]. 1.1. Generalities. Let X be a Banach space with (only in this section) real dual space X* and let A be a bounded closed subset of X. We recall some notions that are connected with convexity properties at a boundary point of A. Research of the first author was supported by the Dutch research organization NWO.

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PAUL BENEKER AND JAN WIEGERINCK

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DEFINITION 1.1. A point x E A is called extreme if x = tp + (1 - t)q for some p, q E A and 0 < t < 1 implies that p = q = x. A point x E A is called exposed if there exists L E X* such that Lx = 1 while Ly < 1 for every yEA \ {x}. The functional L is called an exposing functional for x.

For L E X* we set

M(L, A) = sup{Lx : x E A}

(1.1) and (1.2) If IILII

(1.3)

1\1(A) = sup{llxll : x

= 1 and a > 0 we

E

A}.

define a slice of A as

S(L,a,A) = {x E A: Lx> 1\1(L, A) - a}.

Let B be the closed unit ball of X. DEFINITION 1.2. A subset A of X is called dentable if for every c > 0 there is a point x E A such that x is not in the norm-closed convex hull of A \ (x + cB). Equivalently, A is dent able if for every c > 0 there is a slice S(L, a, A) of diameter less than c. DEFINITION 1.3. A point x E A c X is called a denting point if for every c > 0 there exists a slice S(L, a, A) of diameter less than c that contains x. A point :1: E A c X is called a strongly exposed point if there exists L E X* such that for every c > 0 there exists a > 0 such that the slice S(L, a, A) contains x and has diameter less than c. In other words, there exists L E X* such that Lx = 1\1(L, A) and if LX n ---+ 1\1(L, A) for {x n } C A, then Xn ---+ x in the norm of X.

With these definitions at hand we make some observations. A strongly exposed point in A is exposed. Normalizing the functional L in the previous definition will yield an exposing functional for x. An exposed point of A is extreme and an extreme point of A belongs to the boundary of A. A theorem of Phelps, [38], states that if every bounded subset of X is dentable, then every bounded closed convex set is the convex hull of its strongly exposed points. It is known that separable dual spaces have the property that every bounded set is dentable, cf. [10], Chapter 6. However, neither Phelps' theorem nor its proof gives us any information about, say, which of the exposed points of a convex set C are strongly exposed points. In the rest of the paper it will be more convenient to use an equivalent, more analytic definition of strongly exposed point: An exposed point x of the set A C X with exposing functional L is strongly exposed if and only if every sequence (xn)in A with the property that LX n ---+ 1 converges strongly to x. Moreover, we will work over C, so that the definition of exposed point becomes slightly adapted. A point x E A is an exposed point of A, if there exists a (complex) functional L on X with the following property: Lx = 1 and for all yEA, Y =1= x ~ Re L(y) < 1. (For A E C, Re A denotes the real part of A.) Again we say that the functional L is an exposing fUIlctional for x, or simply that L exposes the point x.

THE BOUNDARY OF THE UNIT BALL IN HI_TYPE SPACES

61

1.2. Hardy spaces. Next we briefly recall the definition and some properties of Hardy spaces. Excellent introductions are [12, 16, 24], and for domains of finite connectivity [13]. We will denote the space of holomorphic functions on a domain D c en by H(D). Let !Dl be the unit disc in e and 'JI' its boundary. DEFINITION 1.4. Let 0 < p < 00. The Hardy space HP = HP(!Dl) consists of all holomorphic functions I on !Dl for which 11111it-v:= sup

OCrCl

1

0

2~

dO II(re i9 W - <

2w

00.

The Hardy space HOC(!Dl) consists of the bounded functions in H(!Dl). It is a closed subspace of LOC(!Dl) under the sup-norm 11.lIoc. If IE HP(!Dl), then

(1.4) exists for almost allO. The function 1* belongs to U('JI'). For alII::; p::; 00, II.IIHP is a norm, IIIIIHP = 111*IILp and HP(!Dl) becomes a Banach space that is a subspace of U('JI'). We will simply write 1IIIIp instead of 11111Hv. Alternatively, the Hardy space HP(!Dl) (1::; P < 00) may be described by

HP(!Dl) = {f E H(!Dl) : IIIP admits a harmonic majorant} = closure of holomorphic polynomials in £P ('JI'). The value at 0 of the least harmonic majorant of IIIP,

uf(z) = inf{h(z) : h harmonic and h> IIIP} equals the norm, 1IIIIp = u(O?/p. Equivalent norms are given by (uf(z)?/P, (z E

!Dl) . These descriptions are useful for defining HP-type spaces on a domain n in

e:

= {f E H(n) : IIIP admits a harmonic majorant}.

HP(n)

Norms, all equivalent, can be defined via the least harmonic majorant uf by (Uf(Z))l/P, (z En). If n has smooth boundary an then HP(n) is again a closed subspace of U(an). Similarly on the unit ball ~n of (n ~ 2):

en

HP(~n)

= closure of holomorphic polynomials in

LP(a~n).

We end this section by defining the Bergman space AP(!Dl). Let 1 ::; p < IE H(!Dl) is such that

00.

If

1111I~p:= i'D[11(zW dxdy < 00, W then we say that I belongs to the Bergman space AP(!Dl). In other words, AP(!Dl) = H(!Dl) n U(!Dl). It is a closed subspace of U(!Dl). One motivation for looking at the Bergman space Al(!Dl) is that

Al(!Dl) ~ {f E Hl(~2): I(Zl,Z2) = I(Zl'O)}. Therefore B(Al(!Dl)) may serve as a testing ground for B(H1(!Dl)).

PAUL BENEKER AND JAN WIEGERINCK

62

2. The boundary of unit ball in H1(]]))) 2.1. Extreme points. The subject started with a famous theorem of deLeeuw and Rudin, [26], that characterizes the extreme points of B(H 1 (]])))). It will be presented below. Recall the inner-outer decomposition for f in H1(]]))), cf. [16]:

f(z) = I(z)U(z). Here I E H(]]))) is an inner function, that is, I E HOC(]]))) with 1I*(ei9 )1 = 1 a.e. and U E H(]]))) is an outer function, i.e. log IUI(z) = P[log IUI](z), where P is the Poisson operator:

Pu z = [ ]()

1

211"

0

1 - Izl2 1 - 2Re (ze- i9 )

+ Izl2 u (e

.

t9

d() -. ) 27r

The inner factor thus contains all the zeros of f. In fact, every inner factor is of the form

I(z) = B(z)S(z),

(2.1)

with B a Blaschke product and S is the singular factor (2.2) where J.L is a positive, singular measure on 'JI' and - denotes harmonic conjugation. EXAMPLES 2.1. Outer functions are zero free, and e.g. a holomorphic function on]])) that is continuous and away from 0 on iij is outer. Taking a root and applying a limit argument shows then that f E HI (]]))) that assumes values only in C \ (-00,0) is outer too. EXAMPLE 2.2. If I is an inner function then (1

+ 1)2

is outer.

DeLeeuw and Rudin proved THEOREM 2.3 ([26]). A function f is an extreme point of the unit ball of HI (]]))) if and only if f is an outer function and IIfl11 = 1. PROOF. Suppose f is an outer function of unit norm and suppose g in H1 is such that IIf + gill = Ilf - gill = 1. Let dJ.L be the probability measure If I on the unit circle. Then, with k = gl f on 'JI', the relations Ilf + gill = Ilf - gill = 1 imply that J;11" 11 + kl + 11 - kldJ.L = 2. Because for all z E C: 11 + zl + 11 - zl 2 2, with equality if and only if z E [-1,1], we conclude that for almost dJ.L-every ~ E 'JI' : g(~)1 f(O E [-1,1]. Observe that this inclusion also holds for almost d()-every ~ E 'JI', because ~ = If I i:- 0 almost d()-everywhere on 'JI'. In particular, Igl :::; If I d()- a.e. on 'JI'. By the fact that f is outer, then also for all zED: Ig(z)1 :::; If(z)l. Hence gI f E HOC has real boundary values on 'JI'. The Poisson integral representation of glf yields that glf is constant. Finally, because Ilf + gill = 1, this constant is zero, i.e., g == 0, so f is extreme. In the other direction, suppose f = I· F is of unit norm, but I is a non-trivial inner function. Because for every ei9 E 'JI' \ {±1}:

g!

±(1 ± ei9 )2 e

.

---'-----:-.,,::--' --- = 2 ± 2Re (e t9 ) = 2 ± 2 cos( ()) > 0, t17

THE BOUNDARY OF THE UNIT BALL IN

Hi_TYPE

SPACES

63

the functions I and (1 ± I)2 have the same argument a.e. on'lI'. Let 9 = We have III ± gill =

dB

l±t .F.

dB

h[2'" 1F1(1 ± Re (I)) 2w = 1± Re (h[2'" IFI' I ~ ).

Now if we replace I by AI throughout the preceding, where A E 'lI' is arbitrary, then we obtain: 2", dB II/±glll=I±Re(A IFI·I-). o 2w

1

Therefore, we can choose A in such a way that Re (A J:'" IFI . I quently, III ± gill = 1, but 9 "¥= 0, so I is not extreme.

g!)

=

o.

ConseD

EXAMPLE 2.4. Normalized polynomials without zeros in j[)) are extreme points. 2.2. Exposed points in B(Hl(j[)))), function theoretic methods. As far as the authors know, there is no clearcut function theoretic characterization of exposed points of B«Hl(j[)))). The necessary and sufficient condition of Helson, [21] given below in Theorem 2.8 probably comes closest. The following lemma identifies the exposing functionals. LEMMA 2.5.

II I

is exposed in B(Hl(j[)))), then Lf:

9

~

J7 9

dB UT2w

is the unique exposing functional. PROOF. Suppose I is an exposed point of B(HI(j[)))) with exposing functional L. By the Hahn-Banach theorem there exists a function cp E V"', Ilcplloo=l, which represents the action of L: [21< dB L(g) = gcp 2w'

io

Because L(f) = 11/111 = 1, and because the function I has mass (almost) everywhere on 'lI', there is only one cp that has the desired properties, namely cp = 711/1 = exp( -i arg(f)). In particular we see that the exposing functional for I is unique and of the above form. D DEFINITION 2.6. A function I E HI (j[))) is called rigid if 9 E HI (j[))) and arg 9 = arg I implies that 9 = cl for some c > O. An HI-function is rigid if its argument on 'lI' determines the function up to a multiplicative constant. By the previous lemma exposed points are precisely the rigid functions of norm 1. Not all outer functions are rigid as the following example demonstrates. EXAMPLE 2.7. By the DeLeeuw-Rudin theorem (Theorem 2.3), no HI-function with non-trivial inner factor is rigid. Inspection of the proof of the theorem explicitly gives another HI-function with the same argument: if I = I· F with F outer, then 9 = (1 + 1)2. F is an outer function, cf. Example 2.2 that has the same argument as f. This argument can be reversed: suppose I "¥= 0 is divisible in HI by the outer function (1 + I)2, where I is any non-trivial inner function, then I

PAUL BENEKER AND JAN WIEGERINCK

64

and I I / (1 + I) 2 have the same argument, so non-trivial inner functions I we have (2.3)

I

If I is rigid, then 1/(1

is not rigid. In other words, for all

+ 1)2 rt HI

Nakazi [28, 29] conjectured that an outer function with fI(l + cz)2 rt HI for every c E 'JI' must be rigid. E. Hayashi [17] gave a counter example. Later Sarason [44, 45] conjectured that (2.3) characterizes rigid functions. Inoue [22] subsequently gave an example of a non-rigid function I E HI with the property that 1/(1 + 1)2 rt HI for all non-trivial inner functions I. Helson showed that a variant of (2.3) indeed characterizes rigid functions; THEOREM 2.8 ([21]). Let I be an outer lunction in HI. Then I is rigid il and only il lor all inner lunctions p and q, not both constants, I/(p + q)2 rt HI. The drawback of Helson's theorem is that his condition is difficult to check in practice. More user friendly are the following results of Yabuta. THEOREM 2.9. (1) II I E HI is invertible in HI, then I is rigid. [51] (2) Suppose ther'e exists h E Hoo \ {O} such that Re(h!) 2: 0 a.e. on 'JI'. Then I is rigid. [52] The proof uses a result of Neuwirth and Newman [36], that is interesting in its own right. LEMMA 2.10 ([20],[36]). Every lunction I E H 1 / 2 that is positive a.e. on'JI' is constant. Nakazi [29] gives another proof of Yabuta's results. EXAMPLE 2.11 ([23]). Normalized polynomials P without zeros on II)) and at most simple zeros on 'JI' are exposed. Indeed, the argument of such a function makes jumps of height 7r at the zeros in 'JI', hence by adding a smooth function

+iv> yields a function with positive real part and the second of Yabuta's criteria applies. 2.3. Strongly exposed points. Whereas it seems impossible to give a characterization of exposed points that allows to check if a given function in aB(HI (II)))) is an exposed point, it is possible to characterize the strongly exposed points. This is the content of work of Nakazi and, independently, the first author. Let us describe strongly exposed points in terms of properties of the exposing functional. THEOREM 2.12 ([2,4]). II alunction IE 8B(H 1 (1I)))) is strongly exposed, then d( Tfr, HOO) < 1. (d = Loo-distance). PROOF. Suppose that the LOO-distance of

g!

The space HOO + G('JI') that appears in the next theorem, has been studied extensively, see [42] and cf. [16]. It is a closed subalgebra of LOO('JI').

THE BOUNDARY OF THE UNIT BALL IN HI_TYPE SPACES THEOREM

2.13 ([47]). Let IE B(Hl(]])))) be exposed. Ild(

65

&!. Hoo+C(1I')) < 1,

then I is strongly exposed. We will explain how the proof goes if 'P := 111/1 E Hoo + C(1I'). The exposing 7r functional L for I is defined by L(g) = g'Pg!. Suppose that (fn)f is a sequence in the unit ball of Hl such that L(fn) --) 1 as n --) 00. Clearly this implies that II In I 1 --> 1. We claim that there is a subsequence (fnk) that converges to I in HI. By weak* compactness of the unit ball of (C(1I'))* the sequence of measures Ing! has a weak* convergent subsequence Ink g! with limit dJ.L({}) , and IIJ.LII :::; 1. For all r 27r em . (J n = 1,2,3, ... , we have Jo dJ.L({}) = O. Now the F. & M. Riesz theorem [16] precisely states that J.L is of the form dJ.L( (}) = F g! for some F in the unit ball of HI . Write 'P = 'Pl + 'P2 with 'Pl E H[J = zHoo and 'P2 E C. Because H[J annihilates HI, the weak* convergence of Ink g! implies that

fg

We conclude that F = I because I is exposed. Again, by weak* convergence Ink converges pointwise to I. The following result of D.J. Newman ensures that the convergence is in Hl with limit I. The argument shows that every subsequence of (fn) has a subsubsequence that converges in HI to I, and therefore In --) I in HI. 2.14 ([35]). Let (fn)i be a sequence 01 functions in the unit ball Suppose In converyes pointwise to I E Hl on ]])) and I In I 1 --> 11/111 as Then In converges to I in HI-norm.

PROPOSITION

01 Hl. n

--> 00.

The method for the general case in [47] is to work in the maximal ideal space of LOO(1I') and use the Helson-Lowdenschlager generalization of the F. & M. Riesz theorem. Cf. [30] for a related approach. 2.4. Strong exposedness and Helson-Szego weights. In this section we will explicitly describe the strongly exposed points in the unit ball of HI: they are the outer functions induced by so-called Helson-Szego weights on the unit circle. We will discuss three different (albeit related) roads to this result. The first one quickly establishes that strongly exposed points are induced by Helson-Szego weights and uses little background on exposed points and Hardy space theory. The second one uses function theory to show that Helson-Szego weights give rise to strongly exposed points and is essentially due to T. Nakazi [31]. The third approach (to be discussed in Section 2.5) uses operator theory and the relation between exposed points and Toeplitz operators [2] to prove that strongly exposed points and HelsonSzego weights are "the same". These last two proofs use the characterization of strong exposedness obtained in the previous section (Theorem 2.13). To put the results in perspective, we will mention some surprising properties of strongly exposed points. Let J.L be a finite (positive) measure on 1I'. Let P be the collection of polynomials in z and let Q be the space of polynomials in z vanishing at the origin. On the unit

PAUL BENEKER AND JAN WIEGERINCK

66

circle P and Q consist of the trigonometric polynomials of the form Ln>o ane inlJ and Ln
p:= sup I iPqd/-Ll, where the supremum is taken over all pEP and q E Q restricted by Ilpll£2(/J) ~ 1 and Ilqll£2(/J) ~ 1. By Cauchy-Schwarz, 0 ~ P ~ 1. We see that the spaces P and Q are orthogonal if and only if P = O. On the other hand, when the L 2 (/-L)-closures P of P and Q of Q have a non-trivial intersection, then clearly p = 1. The size of the number p is related to the question when the sum P + Q is closed in L 2 (/-L). Namely, when p < 1, then for all p E P, q E Q in L 2 (/-L)-norm, lip + ql12 ~ IIpl12

+ IIqll2 -

2p· Ilpll· Ilqll ~ (1- p)(llpl12

+ IlqIl2),

which implies that P + Q is closed in When p < 1 we say that P and Q are at the positive angle cos-I(p) > O. When the spaces P and Q are at positive angle, the projection

L 2 (/-L).

N

N

P+(Lane inlJ ) = LaneinlJ , -N

0

L 2 (/-L),

which is densely defined on extends to a bounded operator on L 2 (/-L). Conversely, if the intersection of P and Q in L 2 (/-L) is trivial, then the definition of P+ acting on trigonometric polynomials is well-defined and extends to a bounded operator on L 2 (/-L) if and only if P and Q are at positive angle. DEFINITION 2.15. We say that a function w ~ 0 on 'll' is a Helson-Szego weight (on 'll') ifthere exist real valued u,v E Loo('ll') with Ilvll oo < I such that w = eU+V. (Here v is the boundary function of the harmonic conjugate of the Poisson integral of v to JIll.) The following theorem of H. Helson & G. Szego elegantly describes all measures /-L for which P and Q are at positive angle. THEOREM 2.16 ([19]). The subspaces P and Q are at positive angle in L 2 (/-L) if and only if the measure /-L is of the form d/-L = wdO, for some Helson-Szego weight won'll'. COROLLARY 2.17. If the function f is strongly exposed in the unit ball of HI, then If I is a Helson-Szego weight on'll'. PROOF. Assume f is an exposed point, such that If I is not a Helson-Szego weight. We will show that f is not strongly exposed. Let /-L be the probability By the theorem of Helson and Szego the spaces P and Q are at measure If I zero angle (p = 1). Thus we can find sequences (Pn) and (qn) in the L 2(/-L)-unit balls of P and Q respectively, such that

g!.

1

211'

(2.4)

0

Pnqn d/-L

--+

1,

as n --+ 00. For n = 1,2, ... , let fn be the HI-function Pnqnf. These functions are contained in the unit ball of HI:

THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES

67

If we set

1

211"

L(g) =

0

dO g

Now (2.4) expresses that L(fn) ~ 1 as n ~ 00, yet the functions fn do not converge to f in HI because fn(O) = qn(O) = O. We conclude that f is not strongly exposed. D Already we can mention a surprising property of strongly exposed points. COROLLARY 2.18 ([2]). If the function f is strongly exposed in the unit ball of HI, then for all small c > 0, the functions f and 1/ f are contained in H1+ e . The corollary together with Theorem 2.13 show that if f E B(HI(][})) satisfies the distance condition in Theorem 2.13, then f is strongly exposed if and only if 1/ f E HI. (It is worth noting that there exist exposed points f E HI such that 1/ f is in no HP, p > 0, [37, 44].) Since f and l/f are outer functions, the corollary is immediate from the following lemma (which is actually an exercise in [16]). LEMMA 2.19. If the function w is a Helson-Szego weight on 'JI', then for all small c > 0, we have w E £1+ e ('JI') and l/w E L1+e('JI'). LEMMA 2.20. (Cf. [16], Lemma IV. 3. 3, p. 148.) If'lf; is a measurable real function, then the LOO-distance of e-i.p to Hoo is less than 1 if and only if there exist c > 0 and h E H oo such that 7r (2.5) Ihl 2:: c and I'lf; + arghl $ "2 - c (mod 27r)

almost everywhere on 'JI'. THEOREM 2.21 ([2, 31]). Let f be an extreme point of the unit ball of HI. Then f is strongly exposed if and only if If I is a Helson-Szego weight. PROOF. ([31]) By Corollary 2.17 we need only to prove the backward implication. Let us assume that the outer function f is such that If I is a Helson-Szego weight on 'JI', say, If I = exp(u + v), where Ilvll oo < l We remark that the function is exposed by Theorem 1 and Lemma 2.19. Next, again using the fact that f is outer, we notice that

f(z) = eu(z)+iu(z) . ev(z)-iv(z),

because the right hand side is an outer function with the appropriate absolute values on 'JI'. We set

O. Recall Corollary 2.18: if f is strongly exposed then for all small c > 0, the functions p+e and 1/ f1+ e are contained in HI. This result can be sharpened somewhat using Theorem 2.21:

68

PAUL BENEKER AND JAN WIEGERINCK

PROPOSITION 2.23 ([2]). Let f be a strongly exposed point in the unit ball of HI. Then for all sufficiently small c > 0, the (normalized) functions ce f1+E: and del f1+E: of unit norm are again strongly exposed in the unit ball of HI. The Proposition explains the following examples from [47]. EXAMPLES 2.24. A polynomial is rigid in HI if and only if its zeros on the unit circle are single zeros (and it is zero-free on llJ), obviously). Any normalized polynomial P with at least one single zero on '][' is not strongly exposed however, because 1/ P ¢ HI Let f#(z) be the extreme point c(1 + z) log2(1 + z) in the unit ball of HI. Because f# is outer and because 1/lf#1 E LI, we conclude that f# is exposed (Theorem 2.9. 1). However, 1/lf#1 ¢ L1+E:, so f# is not strongly exposed.

2.5. Toeplitz operators and De Branges-Rovnyak spaces. Let P+ be the orthogonal projection of L2('][') onto H2: 00 00 P+(Lane inll ) = Lane inll , -00 0 and let P_ be the orthogonal projection of L2 onto (H2).l. 00 -1 P_(Lane inll ) = Laneinll . -00 -00

=

H~:

DEFINITION 2.25. Given a bounded function '¢ E LOC the Toeplitz operator T", is the bounded map T", : H2 _ H2 given by T",(f) = P+('¢f).

We say that '¢ is the defining function of the Toeplitz operator T",. We see that the norm of the Toeplitz operator T", is at most 1I,¢1I00. It is not difficult to show that the norm of T", is in fact equal to 1I'¢1I00, but we will not need this result. Also, it is a routine exercise to verify that the adjoint of T", is the Toeplitz operator Ttji. Clearly, if'¢ E Hoo, then T",(f) = '¢f. Combining these two observations we have the following result: LEMMA 2.26. If cp or'¢ is contained in H oo , then T-qiT", = TVi",. Given a function '¢ E Loo, the Hankel operator H", (with defining function '¢) is the bounded operator H",(f)

= '¢f - T",(f) = (I - P+)(,¢f) = P_('¢f)

from H2 into (H2).l.. By the same reasoning the norm of H", is at most 1I,¢1I00. If two functions cp and '¢ in Loo differ by an element of H OO , then the associated Hankel operators coincide. Hence the operator norm of H", is at most Loo-dist('¢,HOO). The basic fact about Hankel operators, due to Z. Nehari, is that equality holds: THEOREM 2.27 ([34]). The operator norm of H", equals the Loo -distance of '¢ to HOO.

THE BOUNDARY OF THE UNIT BALL IN

HI_TYPE

SPACES

69

The following result (and its corollaries) will be of great importance. We only have as reference Sarason's notes [43]. THEOREM 2.28. (Devinatz-Rabindranathan) If'IjJ is unimodular, then T1fJ is left-invertible if and only if Loo-dist('IjJ,Hoo) < 1.

A bounded linear operator L : X -+ Y is said to be left-Fredholm if the range of X under L is closed and of finite codimension in Y. For left-Fredholm operators one has the following result. COROLLARY 2.29 ([11]). (Douglas-Sarason) If'IjJ is unimodular, then T1fJ is left-Fredholm if and only if the Loo -distance of'IjJ to HOO + C is less than 1. See [43], p. 119 ff. also for a proof of the next result COROLLARY 2.30 ([9, 49]). (Devinatz-Widom) If'IjJ is unimodular, then T1fJ is invertible if and only if'IjJ can be written as ei(uH), where u and v are real functions in Loo such that IIvll oc < 'IT". We will now explain the relation between Toeplitz operators and rigidity (exposedness) of functions in HI. LEMMA 2.31. Let f be an outer function in HI. Then f is rigid if and only if the Toeplitz operator with defining function

{af, ag)M := (f, g)2.

This makes M(a) a Hilbert space, and the Toeplitz operator Ta : H2 -+ M(a) unitary. Clearly, when lIall oo ::; 1, the inclusion i : M(a) -+ H2 is a contraction.

70

PAUL BENEKER AND JAN WIEGERINCK

Let b be a non-constant function in the unit ball of Hoo. We will construct the space 1t(b), the so-called complementary space to M(b), in a similar fashion; the narile reflects the fact that M(b) + 1t(b) = H2 (although the intersection M(a) n 1t(b) is trivial only when b is an inner function). Observe that I - TbTj) is a positive contraction on H2, hence the operator (I - nTj»)l/2 is well-defined on H2 and contractive. As a linear space 1t(b) consists of all H 2-functions in the range of the operator (I - TbTj»)l/2 on H2. We also use this map to give 1t(b) a Hilbert space structure. Namely, if lor 9 is orthogonal in H2 to ker(I -TbTj»)l/2 = ker(I -TbTj»), we set

((I - TbTj»)l/2 I, (I - TbTj»)l/2g)b:= (/,g)2. As a consequence (I -TbTj)?/2 is a co-isometry from H2 onto 1t(b) and the inclusion map i : 1t(b) --+ H2 is another contraction. Moreover, we see that if Ilbll oo < 1, then 1t(b) is all of H2 with an equivalent norm. Given an outer function I E Hl (not a constant) Sarason constructs three auxiliary holomorphic functions: 27T

(2.6)

(2.7) (2.8)

G(z) = {

io

ifJ e, + z I/(eifJ)1 dO, e,fJ - z 211'

( ) _ 2v'f(Z)

a z - G(z)

+ l'

G(z) - 1 b( z) = G ( z) + 1 '

where we may take any branch of -IT Observe that we can recover the function I from a and b: 1= F2, with F = a/(1 - b) E H2. The function G is reasonably well-behaved on JI): because ReG(z) ~ I/(z)1 ~ 0, G is contained in HP for all o < p < 1. The real part of G is positive and majorizes hence a and bare contained in the unit ball of Hoo. The fact that Re G = III a.e. on 'II' gives us that lal 2 + IW = 1 a.e. on 'II'. By a theorem in [26] the functions a and b are not extreme in the unit ball of Hoo. It is known in such cases that the polynomials are contained and dense in 1t(b) (cf. [46], IV-3).

Jill.

We shall also encounter Toeplitz operators on H2 with unbounded defining functions. For a function 't/J E L2, the Toeplitz operator T,p (with defining function 't/J) acts as follows. Take a function 9 E H2, then 't/Jg is an L l ('II')-function with Fourier series E~oo cn['t/Jg]einfJ and we set 00

T,p(g) =

L en ['t/Jg]zn

E

H(D).

n=O

For bounded 't/J this coincides with the Toeplitz operators as defined earlier. For unbounded 't/J, the map T,p is a densely defined operator from H2 to H2 and is continuous from H2 into the space of holomorphic functions on JI) with the topology of uniform convergence on compact subsets. In this light it is perhaps surprising that the product of the operators Tl-b and Tp (the former is ordinary multiplication by 1 - b) is bounded on H2. This is a consequence of the following theorem of Sarason. The proof uses that G = ~ is the Herglotz integral of an absolutely continuous measure (cf. (2.6)); it may be found in [46], IV-13, p. 30.

THE BOUNDARY OF THE UNIT BALL IN HI_TYPE SPACES

71

THEOREM 2.32. The operator T1-bTF is unitary from H2 onto 1i(b). Next one can show that TI-bTFTF/F = TI-bTFF/F = TI-bTF = Ta

(compare with Lemma 2.26). Therefore T1-bTF maps the range ofTF/F in H2 onto M(a) C 1i(b) (Theorem 2.32). Consequently, M(a) is dense in 1i(b) if and only if the range ofTF/F on H2 is dense in H2, or equivalently ifTF/F = T;/F is injective on H2. Combining this with Lemma 2.31 we find Sarason's result:

I

THEOREM 2.33 ([44]). Let I be an extreme point is exposed il and only il M(a) is dense in 1i(b).

01 the

unit ball in HI. Then

This fundamental result has several striking consequences, one of them being that if I is exposed, then so are the squares of the H 2-functions FA = a/(1 - )"b) ().. E 1l'), cf. [44, 46] As a next step one may even replace).. by any inner function u. Sarason proves that the square of Fu = a/(I- ub) is exposed in HI. We now formulate a variant on Theorem 2.33 that deals with strongly exposed points.

I

THEOREM 2.34 ([2]). Let I be an extreme point is strongly exposed il and only il M(a) = 1i(b).

01 the

unit ball

01 HI.

Then

PROOF. First observe that for any FE H2 \ {O}, the Toeplitz operator TF/F is injective. Sarason's reasoning preceding Theorem 2.33 shows that (with 1= F2 as before) (2.9)

M(a) = 1i(b) if and only if TF/F maps H2 onto H2.

Hence, (2.10)

M(a) = 1i(b)

¢:}

TF/F is invertible

¢:}

TF/F is invertible.

Suppose I is strongly exposed. By Theorem 2.13, the LOO-distance of F / F to H OO + C is less than 1. The Douglas-Sarason theorem (Corollary 2.29) implies that the operator TF/F has closed range in H2. As we have seen its adjoint TF/F is injective, hence the range of TF/F is dense in H2 too. Thus TF/F : H2 -+ H2 is surjective. The operator is also injective by the rigidity of I = F2 (Lemma 2.31). By (2.10), M(a) = 1i(b). To establish the other half of the theorem suppose M(a) = 1i(b). We observe that I is exposed by Theorem 2.33. By (2.10), the operator TF/F is invertible, in particular, it is left-invertible. Theorem 2.28 implies that the LOO-distance of 711/1 to HOC is less than 1. We conclude that I is strongly exposed by Theorem 2.13. 0 There is an alternative proof of the implication

"I strongly exposed =* 1/1 E

HI" that lies on the surface: if I is strongly exposed, then 1 E 1i(b) = M(a), because 1i(b) contains all polynomials. Thus l/a E H2 and 1/1= (l-b)2/ a2 E HI. lt should come as no surprise that l/a is actually in HP for slightly larger p > 2. Indeed, equality of the spaces M (a) and 1i(b) occurs if and only if the operator Ta/a

is invertible and the non-extreme pair (a, b) satisfies the so-called Corona condition: inf

zED

la(zW + Ib(zW > 0,

72

PAUL BENEKER AND JAN WIEGERINCK

see [44] and [46], IX-5, p. 66. We see in particular that for strongly exposed f, a2 Illall~ is also strongly exposed (cf. (2.10)), and that lla E H2+£, 1/(I-b) E Hi+£ for sufficiently small E > O. Let us go back to the distance condition used in Theorem 2.13 and assume f is an extreme point such that the LOO-distance of 7/1fl to HOO + C is less than 1. Starting with the Douglas-Sarason theorem (Corollary 2.29) and exploiting the full strength of the statement that TF/F is (left-)Fredholm it follows that the operator TF/F has closed range in H2 of finite codimension, say, N. Hence M(a) is closed and of finite co dimension N in 1i(b). Using [20], Theorem 6 or [46], X-18, p. 77, we conclude that f is of the form f = p2g, where 9 is strongly exposed in the unit ball of Hi and p is a polynomial of degree N that has all its zeros on 11'. Conversely, if f is of the above form, then

distallfl, H oo

+ C) = dist(z2N . gllgl, H oo + C) = dist(gllgl, H oo + C) < 1, by the algebra structure of Hoo + C, but dist(7/lfl,H OO ) = 1 when N > O. This calculation serves to illustrate that if for a given extreme point f the distance of 7Ilfl to H oo + C is less than 1, the ("only") thing that can prevent f from being exposed (and hence strongly exposed) is the divisibility of f in Hi by functions of the form (1 - U)2 with u(z) = >.z (>. E 11') a particularly simple inner function. Functions in Hi that lack this divisibility property are called strong outer functions. We see that for any strong outer function that is not exposed, like Inoue's example, [22], it is true that the distance of 7/1fl to H OO +C is 1. Alternatively, the strongly exposed points are the (normalized) strong outer functions f that satisfy distallfl, H oo + C) < 1.

3. The boundary of B(Hi(O)) Having studied the sets of exposed and strongly exposed points in the unit ball of the classical Hardy space Hi of the unit disc, we will investigate how these results hold up for the Hardy space Hi of a domain of finite connectivity ("finite domain") 0 c C. Such domains are e.g. conformally equivalent to domains with a smooth boundary consisting of finitely many components or to a disc from which a finite number of disjoint slits are deleted, cf. [33]. We defined Hi (0) in Section 1.2. There are two important differences with the classical Hardy space that make the analysis very different. On domains of finite connectivity, Hi-functions may not allow a classical factorization using Blaschke products, (singular) inner functions and outer functions. Indeed, the argument principle prevents the existence of a holomorphic function f on the annulus A = {I < Izl < 2}, such that If I = 1 a.e. on 8A and f has exactly one zero on A. Secondly, and in a way related, there now exist extreme points in the unit ball with (finitely many) zeros (Section 3.1). While such zero sets are somewhat generic for extreme points, their location plays a surprisingly crucial role in its being a (strongly) exposed point (Section 3.2). As in Section 2.1 we call a function I E HOO(O) an inner function if 11*1 = 1 almost everywhere with respect to arc length (dO') on 80. Consequently, II(z)1 ::; 1 for all z E O. Let w(·, z) denote harmonic measure on 80 at z. We call a function F E Hi(O) an outer function if for every z in 0 (equivalently, for at least one

THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES

73

zEn): log IF(z)1 = (

lao.

log IF*(~)I dw(~, z).

An outer function is zero free on n. Green's function G(z; zo) and the Blaschke factor Bzo(z) = t:z:oz for are related by G(z; zo) = log IBzo(z)l. This suggests the following definition.

-1:°1

]I))

DEFINITION 3.1. Let G(z; zo) be Green's function on a finite domain n with pole at Zo and let G(z; zo) denote its (multiple valued) harmonic conjugate. Suppose that Zl, Z2, . .. satisfy the Blaschke condition En G(z; Zn) < 00. Then the Blaschke product with zeros at Zl, Z2, ... is the multiple valued function

(3.1)

B(z) = exp( - L G(z; zn)"- iL G(z; Zn». n

n

DEFINITION 3.2. A multiple valued function F on n is modulus automorphic if • IF(z)1 is well-defined; • locally on n, IFI coincides with the absolute value of a holomorphic function. If in addition, IF(z)IP admits a harmonic majorant on n (0 < p < oo), then we say that F E MHP(n}. If IFI is bounded, we say that F E MHOO(n}. I E MHOO(n} is called innerif 11*1 = 1 a.e on r. I is singular inner if I is in addition zero-free. Blaschke products are modulus holomorphic inner functions. One can show that if FE MHP(n}, then IFI has non-tangential limits (denoted IF*!) a.e. on an, IF*I E p(an,da}, and unless F == 0, 10glF*1 E LI(an,da) and (3.2)

log IF*(z)l:::; (

lao.

log IF*(~}ldw(~, z),

for all zEn.

If equality holds in (3.2) at all points zEn (equivalently, for at least one zEn), we call F an outer function in MHI(n). While a single valued inner-outer decomposition is impossible in HP(n), a decomposition in modulus automorphic inner and outer factors is possible.

THEOREM 3.3 ([48]). Let f E HP(n) be not identically zero. Then there exist a Blaschke product BE MHOO(n), a singular inner function 8 E MHOO(n}and an outer function F E MHP (n) such that for all zEn

If(z}1 = IB(z)I·18(z}I·IF(z}l· This factorization is unique in the following sense: if B 1 , 8 1 and FI are a Blaschke product, a singular inner function and an outer function on n for which If(z)1 = IB1 (z}I·181 (z}I·IFI(z)l, then IBI = IBII, 181 = 181 1 and IFII = IFI· 3.1. Extreme points in HI(n}. The question arises which functions are extreme in the unit ball of Hl(n). After the deLeeuw-Rudin theorem 2.3, the following result is elementary: LEMMA 3.4. Iff E HI(n} is an outer function of unit norm, then f is extreme in the unit ball of HI(n}. Any attempt to copy the proof of the deLeeuw-Rudin theorem in the other direction will break down. For suppose f = I· F where I is a non-trivial inner function (in MHOO(n)}. Then deLeeuw and Rudin look at the function 9 = (1+12)F

74

PAUL BENEKER AND JAN WIEGERINCK

and show that IIf ± gill = 1. However, unless I is a single valued inner function, 1 + 12 is not a well-defined modulus automorphic function. There is no remedy for this problem, because as we have already mentioned, when m ~ 2 there exist extreme points with a non-trivial inner part. The following theorem of F. Forelli is crucial in understanding which inner functions can appear in the inner-outer factorizations of an extreme points. THEOREM 3.5 ([14]). Let f be an extreme point of the unit ball of H1(n). Then the codimension of the H 1-closure of f . Hoo in H1 is at most T' when n is bounded by m + 1 closed smooth curves. All functions in the H 1-closure of f . HOO(n) inherit the zeros of f, hence Forelli's theorem implies that an extreme point can only have a limited number of zeros. In fact one has: COROLLARY 3.6 ([14]). If f is an extreme point of the unit ball of H1 (n), then zeros. the inner part of f is a finite Blaschke product with at most

T

Inspection of the proof of the deLeeuw-Rudin theorem gives us the following criterion for extremity (where we identify H 1-functions with their boundary values): LEMMA 3.7. Let f E H1(n) be of unit norm. Then f is not extreme if and only if there exists a non-constant real function k E LOO(an) for which kf E H1(n). Suppose f = I . F is of unit norm where I is a finite Blaschke product, and F outer. Let us suppose that f is not an extreme point of the unit ball of H1(n). Then let k be as in the lemma, and let 9 E H1(n) have boundary values kf. Because F is an outer function, for all zEn: Ig(z)1 $ IIkll oo ·1F(z)l. Hence, because also III = 1 everywhere on an, the meromorphic function h = g/ f is bounded near an, real-valued (a.e.) on an, and has its poles in the zeros of f (with corresponding mUltiplicities). Conversely, if h is a meromorphic function on n with these three properties, then with k := h on an and 9 := hf E H 1(n), we have 9 = kf on an, so by the previous lemma, f is not an extreme point. We come to the following definition. DEFINITION 3.8. Let I be a finite Blaschke product. We say that I is an extremal Blaschke product if there exists no meromorphic function h on n that is bounded near an, real-valued on an and has its poles in the zeros of I, with no greater multiplicity than the zeros of I. The conclusions of the previous paragraphs may thus be summarized as follows: PROPOSITION 3.9. Let the norm of f E H1(n) be 1. Then f is an extreme point of the unit ball of H1(n) if and only if the inner part of f is an extremal Blaschke product. We wish to stress that Forelli's theorem also gives us an upper bound for the number of zeros of an extremal Blaschke product on n. Also, by the previous proposition we see that it is only the location of the zeros of a function in H1 (n) and not so much the outer factor that decides whether or not the function is extreme in the unit ball (after normalization). The problem of determining the extreme points of H1 (n) has thus been reduced to a problem on meromorphic functions on n with pre-described poles, that is: a problem concerning meromorphic divisors on n.

THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES

75

Let I be a finite Blaschke product with zeros Zl, Z2, ... , Zn repeated according to multiplicity. Thus I has n zeros on O. Let 8 := 1'ZI +1'Z2+' -+1,zn be the divisor on 0 associated with I. If 8' = L:zEf! d'(z) . Z is another divisor on 0 we say that 8' ~ 8 if at every Z E 0: 8'(z) ~ 8(z). The space of all meromorphic differentials w on 0 that are real-valued on 00 and for which the associated divisor (w) satisfies (w) ~ 8 is a real linear space of dimension MD(8). Using a theorem of H.L. Royden [39], based on the Riemann-Roch theorem, T.W. Gamelin & M. Voichick proved the following result: THEOREM 3.10 ([15]). The Blaschke product I with zeros Zl, Z2, ... , Zn and associated divisor 8 is extremal if and only if MD(8) + 2n = m. In particular, using only the fact that the inner factor of an extreme point is a finite Blaschke product (as shown by Forelli), Gamelin & Voichick also arrived at Forelli's upper bound (W-) for the number of zeros of an extremal Blaschke product. They proved that this upper bound is also sharp. THEOREM 3.11 ([15]). The HI-closure of the set of extreme points in the unit ball of HI(O) is the collection of all functions in HI(O) that have unit norm and no more than W- zeros. There is a special type of finite domains where Gamelin and Voichick described the zero sets of extreme points explicitly, namely the so called real slit domains. These are usually defined as the extended complex plane with a finite number of intervals deleted. In our situation we prefer a (conformly equivalent) definition. DEFINITION 3.12. We will call any domain n of the form JDl \ (It u ... U 1m), where It, 12', • .• ,Im are disjoint, bounded and closed intervals in (-1, 1) a real slit domain.

n

THEOREM 3.13 ([15]). Let be a real slit domain and let Zl, Z2, ... , Zn be points of (not necessarily distinct). Then the Blaschke product with zero set Zl, Z2, ... ,Zn is extremal if and only if:

n

• n::; W- and • for all i, j: Zi =I- Zj. (In particular, none of the Zi is real.)

We omit most of the proof, and only observe that because the meromorphic function Z~Zi + z~z; + l-~iz + l-kz is bounded and real-valued on an c JR., no zeros of an extremal Blaschke product are conjugated.

3.2. Strong exposedness and the location of zeros. In this section we investigate exposed and strongly exposed points in the unit ball of HI(O). We give several examples and criteria for (strongly) exposed points. Also we show that nontrivial properties of strongly exposed points in HI (JDl) (for example: Ll-invertibility on the boundary) have no analogue for finite domains. Finally, we again look at the zero sets of extreme points and the question of divisibility of extreme functions by functions of the form (1 + U)2, where u is a non-constant inner function. The Hahn-Banach theorem again gives that for f an exposed point of the unit ball of HI(O) the exposing functional L for f is unique and given by;

L : 9 E HI

1-+

fan 9 I~I da.

PAUL BENEKER AND JAN WIEGERINCK

76

Hence, like H1(1I))), a function I in the boundary of the unit ball of H1(n) is exposed if and only if it is rigid: apart from (positive) constant multiples of I there is no H 1-function with the same argument a.e. (dO") on an. The following criteria for rigidity of H 1 (1I)))-functions carryover to finite domains word for word: • If IE H1 and 1/1 E Hl, then I is rigid (Theorem 2.9. 1). • If there is agE Hoo such that Re(fg) > 0 a.e. on an, then I is rigid (Theorem 2.9. 2). • If u is a non-constant inner function such that 1/(1 + u)2 is in Hl, then I is not rigid (or I == 0). A priori the first two conditions can only be used to demonstrate rigidity of outer functions. In both cases III cannot be too small near the boundary of 0.: if I satisfies the second condition, then 1/1 E H1 -"'(an) for all E: > 0, so 1/1 is "nearly" in H1. The first condition can be modified to allow for exposed points with zeros on n. PROPOSITION

3.14 ([4]). II I is extreme in H1(n) and 1/1/1 E L1(an) then I

is exposed.

Similar to Theorem 2.13 is: THEOREM 3.15. Let I be a lunction in H1(n). Then I is strongly exposed in the unit ball 01 H1 il and only il I is exposed and L oo-dist(7 /1/1, Hoo+C(an)) < 1. Throughout the remainder of this section the domain R will be a real slit domain with m slits that contains the origin. Note that on 11'\ {i} the function (z+i)2 has the same argument as iz so (z+i)2 is not rigid in H1 (II))). LEMMA 3.16 ([4]). For all m ~ 2, the normalized lunction I(z) = c(z + i)2 is strongly exposed in the unit ball 01 H 1(R). The proof is an amusing exercise in elementary function theory. One supposes that 9 E H1 has the same argument as I a.e. on oR and sets h = g/ f. Schwarz's reflection principle eventually shows that h extends to a rational function which turns out to have no poles at all. Hence I is exposed and by Theorem 3.15 strongly exposed. REMARK 3.17. Similarly, if m > k + 1, then the normalized function hk(Z) = c(z + i)2k is strongly exposed in the unit ball of H 1 (R). In particular we see that there exist strongly exposed points in the unit ball of H1 (R) that are "small" on the boundary: 1/lhkl rt L 1 / 2k (OR). We recall that for I = I . F to be an extreme point the only requirement is that the inner part I of I is an extremal Blaschke product - a generic zero set; "most" of the properties of the function I then follow from its outer factor, i.e., the size of Ilion the boundary an. It is reasonable to ask whether exposedness is also essentially a property of the outer factor. We make this question precise in the following sense: if I E H1 is a rigid outer function, I is an extremal Blaschke product on 0., and 9 is invertible in MHOO(n) and such that Ig E HOO(n) (a single valued function), is the extreme point Ig· 1/IIIg . 1111 also exposed? (Compare

THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES

77

with the first example in Section 2.2.) Proposition 3.14 tells us the answer is yes if 11f E Ll(80). Theorem 3.18 below shows that in general the answer is no. We mentioned in Section 2.2 Helson's criterium for exposedness of outer functions, 2.8. It fails for finite domains: THEOREM

3.18 ([4)). For m = 3, there exists ~

f(z)

=

E

R such that the function

c(z - ~)(z + i)2

is extreme in the unit ball of Hl(R), but not exposed.

We will only describe a non-trivial HI (R)-function 9 with the same argument

as

f a.e. on an, for suitable ~ E R \ lR and refer to [4J for details.

Suppose the three slits are the intervals [Xl,YlJ, [X2,Y2J, [X3,Y3J. Let k(z) be a rational function with poles of order 2 in ~,~, 1/~, 1/~, and poles of order 4 in ±i l , and double zeros at ±1, while k(oo) = -1. and zeros at the 12 points Next, let q be the (well-defined) square root of k on R with q(oo) = -i. Now with a suitable choice of ~ and an appropriate polynomial p of degree 8 that is positive on lR U '][', one can show that the function

xtl, yt

g(z)

p(z) + q(z) (z - ~)(z - i)2

1 (1- ~z)(l- ~z) belongs to HOO(R) and has the same argument as f on 8R \ {-i}, which implies that f is not rigid. Because the inner part of f is the extremal Blaschke product with zero at ~, f is extreme, however. By Lemma 3.16, (z + i)2 is rigid. We conclude that if Helson's criterion were valid, for any two inner functions u, v not both constant on R, (z + i)2/(u(z) + v(z))2 (j. Hl(R), hence also (z-~)(z+i)2/(u(z)+v(z))2 (j. Hl(R), a contradiction! =

4. The Bergman space AI(JD)) Recall from Section 1.2 the definition of the Bergman space

AP(JD)) = H(JD)) n LP(JD)) In the Bergman space extreme and exposed points are extremely simple. Indeed, every f E 8B(Al(JD))) is exposed, and (hence) extreme. The exposing functional is

Lf :

gl---+

kgl~ldA(Z).

If Lfg = 1 for some 9 E 8B(Al(JD))), the argument of 9 would be equal to that of f, (a.e) and this clearly implies f = g. However, to study strongly exposed points we need some machinery.

4.1. Bergman projection and Bloch space. An important tool for the study of strongly exposed points will be the Bergman projection; this is the orthogonal projection It is given by an integral operator (4.1)

(4.2)

Pcp(z)

= =

(f, kz)

[

=

f(w)

iD (1 _ zw)2 dA(w)

fUn+ 1)· 1m f(w)'W'dA(w))zn. n=O

D

PAUL BENEKER AND JAN WIEGERINCK

78

Elementary properties are, see e.g. [18], • If cp has compact support in]l)), then Pcp is holomorphic on a neighborhood of Jij. • If cp is Coo on Jij, then Pcp is Coo on Jij. The effect of the Bergman projection on Loo(]I))) and on C(Jij) will be very important for us. We need the Bloch spaces. DEFINITION 4.1. The Bloch space B consists of all holomorphic functions f on ]I)) with the property that (1-lzI2)lf'(z)1 is bounded on]l)). Equipped with the norm

(4.3)

IlfllB := If(OI

+ sup (1 -lzI 2)1f'(z)l, zED

B becomes a Banach space. The set of all functions f in B for which the expression (1-lzI2)1f'(z)1 Izl --> 1 is a closed subspace of B, called the little Bloch space Bo.

-->

0 as

Let Co denote the continuous functions on Jij that are zero on T. We have the following theorem of R. Coifman, R. Rochberg and G. Weiss: THEOREM 4.2 ([8]). The Bergman projection P maps Loo(.]I))) boundedlyonto B. Furthermore, P maps both C(D) and Co boundedlyonto Bo. The result proved in [8] is much more general than Theorem 4.2. As we state it, the theorem may be found in [18], Theorem 1.12, with an elementary proof. There we also find the following results THEOREM 4.3. (1) The dual space of Al is the Bloch space B under the following pairing:

(4.4)

g E B : f E Al !--+lim ( fr(z)g(z)dA(z). rTl

JD

(2) The dual space of the little Bloch space Bo is the Bergman space A l under the pairing: f

E

Al : g

E

Bo

1--+

lim ( f(z)gr(z)dA(z). rTl

JD

REMARK 4.4. When one identifies (A1)* with the Bloch space in Theorem 1 B, the dual norm on B yields a norm that is equivalent with, but not equal to the norm 11.1113 that we have previously defined on B. Hence, there exists a norm 11.11. on the Bergman space A I that is equivalent to 11.111 and is such that the dual norm of 9 E B = (AI)* equals IIgIiB. The strongly exposed points in the unit ball of Al with the norm 11.11. have been described by C. Nara ([32]), who also showed that up to isometrical isomorphisms, Al with the 11.11. norm is the unique pre-dual of B. 4.2. Strongly exposed points of AI(.]I))). The following theorem is a consequence of Theorems 5.3 and 5.2, the fact that Al(]I))) may be identified with the subspace of HI (lI£2) consisting of functions that depend only on one variable, and the fact that these functions are exposed in B(HI(lI£2). We set

(4.5)

THE BOUNDARY OF THE UNIT BALL IN Hl-TYPE SPACES

79

THEOREM 4.5. Let I E Al be 01 unit norm. Then I is strongly exposed in Ball(AI) il and only il the LOO-distance 01 filII to the space (AI)1. + C(D) is less than one. Henceforth we will simply write (AI)1.

+C

instead of (AI)1.

+ C{D).

The question now is: how can we estimate the distance in Loo of cp = 1/111 to (AI)1. +C, where I is a given function in AI? (Clearly the distance cannot exceed one. ) Let us first look at polynomials of a particularly simple form: I (z) = c( z - 0:) n , where c is normalizing. We will assume n 2: 1 because the constant functions are strongly exposed by Theorem 4.5. We distinguish three cases in order of increasing difficulty: 10:1 > 1, 10:1 < 1 and 10:1 = 1. The case where 10:1 > 1 is very easy: 1/111 is continuous on D, so I is strongly exposed. In fact, we may even take non-integer powers n and products of such functions and we always obtain strongly exposed points after normalization. When 10:1 < 1, let us write cp = 1/111 = 'l/Jo + 'l/JI, where 'l/JI is compactly supported in 1D> and cp == 'l/JI on a neighborhood of 0:, while 'l/Jo is smooth on D. From (4.1) we see that P'l/JI is holomorphic across the unit circle because 'l/JI is compactly supported in 1D>. Next, because 'l/Jo is smooth on D, also P'l/Jo is smooth on D. Hence Pcp is continuous on D. Now cp - Pcp is bounded, so cp = (cp - Pcp) + Pcp is contained in (AI)1. + C. By Theorem 4.5 I is strongly exposed. Again, our reasoning readily shows that the normalized product I of functions (z - O:i)n;, for all ni and all O:i ft 'll', is strongly exposed. Now suppose 10:1 = 1; we may take 0: = 1. Let us write In{z) = cn (1- z)n and CPn = In/l/nl· Introducing polar coordinates and applying Cauchy's theorem one finds that the exposing functional L for h is given by

L(g) = [ g(z) ~1 -

J"D

1-

z~: dA{z) = Z

[ Ig{z) (1 -

JD -

z

z) dA{z) =

Cog{O)

+ CIg'{O).

But then there exists a polynomial P2 such that L(g) = fD gP2 dA. Therefore CP2 - P2 is contained in (A I) 1., hence CP2 E (A I) 1. + C so h is strongly exposed. Similarly, for all even n, CPn is contained in (A I) 1. + C and In is strongly exposed in AI. Again, we may introduce non-integer exponents. Let 1f3 = cf3(1 - z)f3, where (3 > -2 to ensure that I f3 E A I; the constant Cf3 > 0 is normalizing. Set

cpf3 = 1f3/I/f3I· PROPOSITION 4.6 ([5]). For all (3 > -1, the Loo-distance 01 CPf3 to (AI)1. + C is at most Isin( f327i") I· In particular, lor all (3 > -1, (3 =I- 1,3,5, ... , the lunction I f3

is strongly exposed in the unit ball 01 A I. The proof consists of observing that if 1(3 - 2nl < 1, then IIcpf3 - CP2nli00

=

IsinCf)1 < 1. We will come back to odd exponents in Section 4.4 and end this section with and example of a boundary points of B (A I (1D>)) that is not strongly exposed. 2

EXAMPLE 4.7 ([5]). The normalized function I(z) = (I-z)2~~g2(I--cz) is not strongly exposed in the unit ball of A I. The functions I f3 tend pointwise to O. However, limf3!-2 fD /{3"(j5 dA = 1.

PAUL BENEKER AND JAN WIEGERINCK

80

·4.3. The space (AI).1 + C. We have observed that (AI).1 + C plays the same role in Theorem 4.5 with respect to the Bergman space as (HI).1 + C('1I') = Hoo +C('1I') with respect to the Hardy space HI(JI))) (Theorem 2.13). We mentioned already in Section 2.3 that Hoo + C is closed in L oo . From this then it followed relatively easily that Hoo + C('1I') is in fact an algebra, cf. [41], Theorem 6.5.5. How far do these results extend to the space (A I ).1 + C? From [5] we quote (1) Al (JI))).1 + c(il~) is a closed subspace of LOO(JI))). (It equals P-I(80 )!) (2) AI(JI))).1 + C(ii)) is a C(ii))- module. (3) A I (JI))).1 + C(ii)) is not an algebra. (4) The space (A 1).1 + C is invariant under composition with holomorphic automorphisms of JI)). EXAMPLE 4.8. [5] As for (3), let J{3 = (1- z){3 and let 'P{3 = J{3/IJ{31 for {3 E R Then 'P2 and 'P-4 E (AI).1, but 'P-2 is not contained in (AI).1 + C. The space (A I ).1 + C is not an algebra. The properties of (A 1).1 proposition.

+C

lead in a straightforward way to the following

PROPOSITION 4.9. Let J be a strongly exposed point in AI. Then (a) iJu is an automorphism oJJI)), then the normalized Junction FI = CI(fou) is strongly exposed; (b) iJ v E A(JI))) is zero-Jree on the circle, then the normalized Junction F2 = C2!v is strongly exposed. Furthermore, the functions J IIJI, FdlFII and F2/1F21 have the same Loo-distance to (AI).1 + C. 4.4. Strong exposedness of (1 - z){3. We saw in Section 4.2 that the functions J{3 = c{3(1 - z){3 are strongly exposed in the unit ball of Al for all {3 > -1 except possibly when {3 = 1,3,5, .... This was deduced from rather straightforward estimates of the L 00 -distances of the functions 'P = ff3 I IJ{31 to the space (A 1).1 + C (Proposition 4.6). In [5] a much sharper result is proved. THEOREM 4.10. For all {3 ::::: 0, the Bloch distance oj the Junction P'P{3 to 8 0 4I sin({h)1

equals 1r

{3+T .

SKETCH OF PROOF. It is convenient to rewrite 'P{3 as 'P{3(w) = (1- w){3/2/(1w){3/2. Using the series expansions for the Bergman kernel 1/(1- zw)2 (see (4.2)), as well as for (1 - W){3/2, and 1/(1 - w){3/2, we evaluate the Bergman projection P'P{3' One obtains P'P{3 = E~=o c{3,nzn, where

c{3,n =

-(n + 1) sin(~) ~ r(m + ~)r(m + n - ~) 271" ~ m!(m + n + I)! .

It is proved in [5] that for fixed (3 > 0:

(4.6)

~r(m+~)r(m+n-~) =:0 m!(m + n + I)!

where the o(I)-term tends to zero as n

4

= n2{3({3 + 2) (1 + 0(1)),

-+ 00.

-2sin(~)

This implies that

c{3,n = 7I"({3 + 2)n (1

+ 0(1)),

THE BOUNDARY OF THE UNIT BALL IN where the o{I)-term vanishes as n

--> 00.

!::tv

(3,n

SPACES

81

But then,

so the Bloch distance of PCP(3 to Bo is at least large N, I{ ~ c

HI-TYPE

41:~~~1I.

On the other hand, for

zn)'1 < ~ nlc 1.l z ln-1 < 21 sin{~)ll + 0(1) - n~ (3,n - 7r(,B + 2) 1 - Izl '

where the o{I)-term tends to zero as N increases. Using the fact that the polynomials are contained in Bo it follows that the Bloch distance of PCP(3 to Bo is at most 4I sin (¥)1 o ... ((3+2) . COROLLARY 4.11. Let d(cp(3, (Al)1. (Al)1. + C. Then for all,B 2: 0, (4. 7)

+ C)

denote the LOO-distance of CP(3 to

~ 1sin( !?f) 1 < d( (A 1) 1. + C) < ~ 1sin( ~) 1 < ~ 2 ,B + 2 CP(3, - 7r ,B + 2 - 7r'

In particular, all f(3 are strongly exposed for ,B 2: O. SKETCH OF PROOF. Let q: B --> B/Bo be the quotient map. By Theorem 4.2, the map q 0 P : L OO --> B/Bo is continuous and surjective. The kernel of the map q 0 P is the space (Al)1. + C, cpo Property (I) from Section 4.3. Hence the derived map

P* : L oo / ((Al)1.

+ C)

-->

B /Bo

is bijective and bounded (by ~ as follows from the proof in [18] of Theorem 4.2). This gives the lower bound for d(cp(3, (Al)1.

+ C), because

IIP*cp(311 =

~ ISi~~y)l.

By the closed graph theorem, the inverse P* -1 of P* is also bounded. Actually, one can show directly that II P* -111 ::; 1, which in turn yields the upper bound for d{cp(3, (Al)1.+C). Supposing that FE B/Bo has norm 1, we will show that P*-I(F) has norm at most 1 in L oo /((Al )1.+C). For any c > 0, we can find a representative fEB of the coset F such that IIfl18 < 1 + c. In the proof of Theorem 4.2 in [18] one finds that

'¢(w) = (I -IWJ2) . f'(w) ~ f'(O)

E

L oo

w satisfies f(z) - P'¢{z) = f(O) + j'(O)z E Bo. Thus '¢ is a representative of P*-I(F) in LOO. As a consequence IIp*-l{F)IILOO/((AI)J.+C) ::; d(,¢, (Al)1.

=

+ C)

::; lim esssuPrl 2 lim sup 1(1 -lwI )f'(w)1 ::; IIfll8 < 1 + C. Iwl-->l

o The module structure of (Al)1.

+C

and the previous corollary lead to

82

PAUL BENEKER AND JAN WIEGERINCK

COROLLARY 4.12. Suppose that g E A(JI1» vanishes nowhere on'll'. Let Zl, Z2, E 'll' be distinct and let {31, {32, ... ,(3n be real numbers greater than - 2. Then the normalized Junction J(z) = cg(z) n~=l (1- ZZi),8i is strongly exposed in the unit ball oj A I iJ and only iJ all Junctions J,8i = C,8i (1 - z ),8i are strongly exposed. In particular, all choices oj {3i > -1 yield strongly exposed points and all normalized

•.. , Zn.

polynomials are strongly exposed in the unit ball oj A I. We end this section with a conjecture on the functions J,8 for -2 < {3 < 0, for which strong exposedness is already implied by Proposition 4.6 when -1 < {3 < O. In [5] it is shown that

~lsin(!?f)I
-2< (.J
with equality in the limit as {3 ! -2. This leads us to surmise the following: CONJECTURE 4.13. For all-2 < (3 < 0, d(cp,8,(AI)1. +C) = particular, the Junctions J,8 are strongly exposed Jor all said (3.

~lsi~~y)J. In

5. Scattered results for B(HI (lRn )) The geometry of B(HI(lR n )) is still quite out of reach. At one time it was conjectured that all boundary points would be extreme, [41], but this conjecture was refuted as soon as the existence of inner functions on lR n was proved, [1, 27]. Just as in Theorem 2.3 one finds that inner functions, or more generally products with an inner factor cannot be extreme. However, it is well known that there is no inner-outer decomposition possible for B(HI(lR n )), because the zero sets of functions in various HP classes are essentially different in size. Rudin already observed in [41] that a function J E 8B(H 1 (lR n )) which is continuous on an open set in 8lRn must be extreme. The proof goes by considering the intersection of lR n with appropriate complex lines and applying I-dimensional theory. The same idea can be used to show that such functions are in fact exposed, [50]. EXAMPLE 5.1. Not all extreme points are exposed: c(1 has the same argument as I.

+ 1)2

is extreme but

Example 4.7 also is an example of an exposed point in B(HI(lR2)) that is not strongly exposed. Let us write S = 8lRn , HI = HI(lR n ) and HJ for the functions in HI that vanish at o. We define

(HI)1. =

N

E Loo(S) :

Is

{JjdA = 0 for allJ E HI}

Analogously we define (HJ)1.. Let d denote the Loo-distance on S. Theorems 2.13 and 2.12 have a full analogue in B(H 1 ). THEOREM 5.2 ([50]). Let J be an exposed point in B(H 1 ). Then exposed iJ and only il d(J 11/1, C(S) + (HI)1.) < 1.

I

is strongly

The proof again uses methods from uniform algebra, but is more involved. Noticing that (HJ)1.) is a fairly small subspace of (HI)1. +C(S), this theorem can be strengthened in one direction. THEOREM 5.3. Suppose that d(J IIJI, (HJ)1.) < 1.

I

is a strongly exposed point in B(H 1 ). Then

THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES

83

EXAMPLES 5.4. Homogeneous polynomials in 8B(H 1 ) are strongly exposed, see [50J. Corollary 4.12 shows that polynomials of one variable in 8B(H 1 (B 2 )) are strongly exposed.

It seems natural to try to generalize ideas of Section 4 to HI. The Bergman projection should be replaced by the Szego projection P and the Bloch spaces 8 and 8 0 by BMOA, the space of analytic functions of bounded mean oscillation and VMOA, the space of functions of vanishing mean oscillation (with respect to the non-isotropic balls in S). It is known that P maps Loo(S) surjectively to BMOA and C(S) surjectively to VMOA, see [25J for an overview of these kind of results.

So we again have a map P*: Loo(S)/«H1)1. +C)

->

BMOA/VMOA. However,

we have no idea how to estimate (p*)-l, nor how to estimate the norm of P*cp for cp = p/ipi with p for example a polynomial. Answers to these questions would be very interesting. References [1] A.B. Aleksandrov, A. B. The existence of inner functions in a ball, (Russian) Mat. Sb. (N.S.) 118(160) (1982), no. 2, 147-163,287. (32A40 46J15) [2] P. Beneker, Strongly exposed points, Helson-Szego weights and Toeplitz operotors, J. Int. Eq. Oper. Th. 31 (1998), 299-306. [3] P. Beneker, Strongly exposed points in unit balls of Banach spaces of holomorphic functions, Academisch Proefschrift, Universiteit van Amsterdam, 2002. Electronically available at http://remote.science.uva.nl/ ~beneker /thesis.ps [4] P. Beneker, J. Wiegerinck, Exposedne'ss in Hardy spaces of domains of finite connectivity, Indag. Math., N.S., 11 (4),2000,487-497. [5] P. Beneker, J. Wiegerinck, Strongly exposed points in the ball of the Bergman space, http://arXiv.org/abs/math.CV /0208234 to appear. [6] L. de Branges, J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966. [7] L. de Branges, J. Rovnyak, Appendix on square summable power series, in "Perturbation Theory and its Applications in Quantum Mechanics", John Wiley and Sons, New York (1966), 347-392. [8] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in severol variables, Ann. of Math. (2), 103 (1976), 611-635. [9] A. Devinatz, Toeplitz operotors on H2 spaces, Trans. Amer. Math. Soc. 112 (1964), 304-317. [10] J. Diestel, Geometry of Banach paces - Selected topics LNM 485, Springer, Berlin etc. 1975. [11] R.G. Douglas, D.E. Sarason, Fredholm Toeplitz operotors, Proc. Amer. Math. Soc. 26 (1970), 117-120. [12] P.L. Duren, Theory of HP-spaces, Academic Press, 1970. [13] S.D. Fisher, Function Theory On Planar Domains, John Wiley & Sons, 1983. [14] F. Forelli, Extreme points in Hl(R), Can. J. Math. 19 (1967), 312-320. [15] T.W. Gamelin, M. Voichick, Extreme points in spaces of analytic functions, Can. J. Math. 20 (1968), 919-928. [16] J. Garnett, Bounded holomorphic functions, Pure & Applied Mathematics, 96, Academic Press, Inc., New York-London, 1981. [17] E. Hayashi, The solution of extremal problems in Hl, Proc. Amer. Math. Soc. 93 (1985), 690-696. [18] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer Graduate Texts in Mathematics 199, 2000. [19] H. Helson, G. Szego, A problem in prediction theory, Ann. Mat. Pura Appl. 51 (1960), 107-138. [20] H. Helson, D. Sarason, Past and future, Math. Scand. 21 (1967), 5-16. [21] H. Helson, Large analytic functions II, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York (1990), 217-220.

84

PAUL BENEKER AND JAN WIEGERINCK

[22] J. Inoue, An example of a non-exposed extreme function in the unit ball of Hl, Proc. Edinburgh Math. Soc. 37 (1993), 47-51. [23] J. Inoue; T. Nakazi, Polynomials of an inner function which are exposed points in Hl. Proc. Amer. Math. Soc. 100 (1987), no. 3, 454-456. [24] P. Koosis, Introduction to HP spaces, Cambridge University Press, 1980. [25] S.G. Krantz, Geometric Analysis and FUnction Spaces CBMS Regional Conference Series in Mathematics, 81. Amer. Math. Soc., Providence, RI, 1993. [26] K. de Leeuw, W. Rudin, Extreme points and extremum problems in Hl, Pacific J. Math. 8 (1958), 467-485. [27] E. Ll1lw,A construction of inner functions on the unit ball in CP. Invent. Math. 67 (1982), no. 2, 223-229. [28] T. Nakazi, Exposed points and extremal problems in Hl, J. Funet. Anal. 53 (1983), 224-230. [29] T. Nakazi, Exposed points and extremal problems in Hl, II, T6hoku Math. J. 37 (1985), 265-269. [30] T. Nakazi, Existence of solutions of extremal problems in Hl , Proc. Edinburgh Math. Soc. (2) 34 (1991), no. 2, 99-112. [31] T. Nakazi, personal communication. [32] C. Nara, Uniqueness of the predual of the Bloch space and its strongly exposed points, Illinois J. Math, 34 (1990), no. 1,98-107. [33] Z. Nehari, Conformal Mapping, McGraw-Hill, 1952. [34] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153-162. [35] D.J. Newman, Pseudo-uniform convexity in HI, Proc. Amer. Math. Soc. 14 (1963), 676-679. [36] J. Neuwirth, D.J. Newman, Positive Hl/2 functions are constant, Proc. Amer. Math. Soc. 18 (1987),958. [37] A. Nicolau, The coefficients of Nevanlinna's pammetrization are not in HP, Proc. Amer. Math. Soc 106 (1989), 115-117. [38] R.R. Phelps, Dentability and extreme points in Banach spaces, J. Funet. Anal. 17 (1974), 78-90. [39] H.L. Royden, The boundary values of analytic and harmonic functions, Math.Z. 78, (1962), 1-24. [40] W. Rudin, Analytic functions of class H p , Trans. Amer. Math. Soc. 78 (1955), 46-66. [41] W. Rudin, FUnction Theory in the Unit Ball of en, Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, New York-Berlin, 1980. [42] D.E. Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286-299. [43] D.E. Sarason, Function Theory on the Unit Circle, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1979. [44] D.E. Sarason, Exposed points in Hl, I, Operator Theory: Advances and Applications 41 (1989), Birkhiiuser Verlag Basel, 485-496. [45] D.E. Sarason, Exposed points in Hl, II, Operator Theory: Advances and Applications 48 (1990), Birkhiiuser Verlag Basel, 333-347. [46] D.E. Sarason, Sub-Hardy Hilbert Spaces in the unit disc, Univ. of Arkansas Leeture Notes in the Math. Sc. 10, Wiley-Interscience, 1995. [47] D. Temme, J. Wiegerinck, Extremal properties of the unit ball of Hl, Indag. Mathern., N.S., 3 (1) (1992), 119-127. [48] M. Voichick, L. Zalcman, Inner and outer functions on Riemann surfaces, Proc. Amer. Math. Soc. 16 (1965), 1200-1204. [49] H. Widom, Inversion of Toeplitz matrices III, Notices Amer. Math. Soc. 7 (1960), 63. [50] J. Wiegerinck, A chamcterization of strongly exposed points of the unit ball of Hl, Indag. Mathern., N.S., 4 (4) (1993), 509-519. [51] K. Yabuta, Unicity of the extremum problems in Hl(un), Proc. Amer. Math. Soc. 28 (1971), 181-184. [52] K. Yabuta, Some uniqueness theorems for HP(un) functions, T6hoku Math. J. 24 (1972), 353-357. FACULTY OF MATHEMATICS, UNIVERSITY OF AMSTERDAM, PLA:'ITAGE I\[UIDERGRACHT

1018

TV, AMSTERDAM, THE NETHERLANDS

E-mail

addre.~s:benekerClscience.uva.n1;janwiegClscience.uva.n1

24,

Contemporary Mathematics

Volume 328, 2003

Complete isometries - an illustration of noncommutative functional analysis David P. Blecher and Damon M. Hay ABSTRACT. This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from 'noncommutative functional analysis', more specifically the new field of operator spaces. In our illustration we show how the classical characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some advantages.

1. Introduction

The field of operator spaces provides a new bridge from the world of Banach spaces and function spaces, to the world of spaces of operators on a Hilbert space. For researchers in the new field, the philosophical starting point is the combination of the following two obvious facts. Firstly, by the Hahn-Banach theorem any Banach space X is canonically linearly isometric to a closed linear subspace of C(K), where K is the compact space Ball(X*). Secondly, C(K) is a commutative C*-algebra. Thus one defines a noncommutative Banach space, or operator space, to be a closed linear subspace X of a possibly noncommutative C* -algebra A. This simplistic idea becomes much more substantive with the addition of some additional metric structure. The point is that if A is any C*-algebra, then the *-algebra Mn(A) of nx 11, matrices with entries in A has a unique norm 11·lln making it a C*-algebra (this follows from the well known nnicity of C*-norms on a *-algebra). If X c A then Mn(X) inherits this norm II· lin, and more precisely we think of an operator space as the pair (X, {11·lln}n). We usually insist that maps between operator spaces are completely bounded, where the adjective 'completely' means that we are applying our maps to matrices too. Thus if T : X -4 Y, then T is completely contractive if Tn is contractive for all n E N, where Tn is the map [Xij] 1--+ [T(x;j )]. Similarly T is completely isometric if II[T(xij)]1I = I [Xij] I for all n E Nand [Xij] E 1I1n(X). It is an easy exercise (using one of the common expressions for the operator norm of a matrix in Mn = .Mn(C)) to prove that a linear map T : X -4 Y between 1991 Mat/;tematics Subject Classification. Primary 46L07, 46L05, 47L30; Secondary 46JlO. This research was supported in part by a grant from the National Science Foundation.. © 85

2003 American Mathematical

So(~iety

86

DAVID P. BLECHER AND DAMON M. HAY

subspaces of C(K) spaces is completely contractive if and only if it is contractive. Consequently such a T is isometric if and only if it is completely isometric. The identification of the term 'noncommutative Banach space' with 'operator space' may be thought of as a relatively recent entry in the well known 'dictionary' translating terms between the 'commutative' and 'noncommutative' worlds. We spend a paragraph describing some other entries in this dictionary. Although t.hese items are for the most part well known to the point of being tedious, it will be helpful to collect them here for the dual purpose of establishing notation, and for ease of reference later in the paper. The most. well known item is of course the fact that the noncommutative version of a C(K) space is a unital C*-algebra B. The noncommutative version of a unimodular function in C(K) is a unitary u E B (Le. u*u = uu* = 1). The noncommutative version of a function algebra A C C(K) 'containing constant functions' is a closed subalgebra A of a C* -algebra B, with 18 E A. We call such A a unital operator algebra. For a unital subset S of a C*algebra B, we will take as a simple noncommutative version of the assertion'S C C(K) separates points of K', the assertion 'the C*-subalgebra of B generated by S (namely, the smallest C*-subalgebra of B containing S) equals B'. The analogue of a closed subset E of a compact set K is a quotient B / I, where I is a closed twosided ideal in a unital C* -algebra B. More generally, unital *-homomorphisms 7r between unital C* -algebra.'i are the noncommutative version of continuous functions T between compact spaces. Indeed clearly any such T : Kl -> K2 gives rise to the unital *-homomorphism C(K2 ) -> C(Kd of 'composition with T', and conversely it is not much harder to see that any unital *-homomorphism C(K2 ) -> C(Kd comes from a continuous T in this way. Moreover such 7r is 1-1 (resp. onto) if and only if the corresponding T is onto (resp. 1-1). Thus the noncommutative version of a homeomorphism between compact spaces is a (surjective 1-1) *-isomorphism between unital C*-algebras. Coming back to 'noncommutative functional analysis', it is convenient for some purposes (but admittedly not for others) to view 'complete isometries' as the noncommutative version of isometries. It is very important in what follows that a 1-1 *-homomorphism 7r : A - B between C*-algebras, is by a simple and well known spectral theory argument, automatically an isometry, and consequently (by the same principle applied to 7r n ), a complete isometry. Similarly, a *-homomorphism 7r : A - B (which is not a priori assumed continuous) is aut.omatically completely contractive, and has a closed range which is a C* -algebra *-isomorphic to the C* -algebra quotient of A by the obvious two-sided ideal, namely the kernel of the *-homomorphism. The entries we have just described in this 'dictionary' are all easily justified by well known theorems (for example Gelfand's characterization of commutative C*algebras). That is, if one applies the noncommutative definition in the commutative world, one recovers exactly the classical object. Similarly one sometimes finds oneself in the very nice 'ideal situation' where one can prove a theorem or establish a theory in the noncommutative world (i.e. about operator spaces or operator algebras), which when one applies the theorem/theory to objects which are Banach spaces or function algebras, one recovers exactly the classical theorem/theory. An illustration of this point is the Banach-Stone theorem. The following is a much simpler form of Kadison's characterization of isometries between C*-algebras [17]:

COMPLETE ISOMETRIES - AN ILLUSTRATION

87

THEOREM 1.1. (Folklore) A surjective linear map T : A --+ B between unital C* -algebras is a complete isometry if and only if T = U7r('), for a unitary u E B and a *-isomorphism 7r : A --+ B. PROOF. (Sketch.) The easy direction is essentially just the fact mentioned earlier that 1-1 *-homomorphisms are completely isometric. The other direction can be proved by first showing (as with Kadison's theorem) that T(I) is unitary, so that without loss of generality T(I) = 1. The well known Stinespring theorem has as a simple consequence the Kadison-Schwarz inequality T(a)*T(a) ~ T(a*a). Applying this to T- 1 too yields T(a)*T(a) = T(a*a), and now the result follows immediately from the 'polarization identity' a*b = ~ E~=o(a + ikb)*(a + ikb). 0 Note that if one takes A = C(Kd and B = C(K2 ) in Theorem 1.1, and consults the 'dictionary' above, then one recovers exactly the classical Banach-Stone theorem. Indeed as we remarked earlier, in this case complete isometries are the same thing as isometries, unit aries are unimodular functions, and a *-isomorphism is induced by a homeomorphism between the underlying compact spaces. Indeed consider the following generalization of the Banach-Stone theorem: THEOREM 1.2. [15,22, 1, 20] Let fl be compact and Hausdorff, and A a unital function algebra. A linear contraction T : A --+ C(fl) is an isometry if and only if there exists a closed subset E of fl, and two continuous functions "f : E --+ '][' and r.p: E --+ 8A, with r.p surjective, such that for all y E E T(f)(y) = "f(y)f(r.p(y))·

Here 8A is the Shilov boundary of A (see Section 2). We have supposed that. T maps into a 'selfadjoint function algebra' C(fl); however since any function algebra is a unital subalgebra of a 'selfadjoint.' one, the theorem also applies to isometries between unital function algebras. If A is a C(K) space too, then 8A = K and then t.he theorem above is called Holsztynski's theorem. We refer the reader to [16] for a survey of such variants on the classical Banach-Stone theorem. Often the transition from the 'classical' to the 'noncommutative' involves the introduction of much more algebra. Next we appeal to our dictionary above to give an equivalent restatement of Theorem 1.2 in more algebraic language. THEOREM 1.3. (Restatement of Theorem 1.2) Let A, B be unital function algebras, with B selfadjoint. A linear contraction T : A --+ B is an isometry if and only if (A) there exists a closed ideal I of B, a unitary u in the quotient C*-algebra B / I, and a unital 1-1 *-homomorphism 7r : A --+ B / I, such that qI (T( a)) = u7r(a) for all a E A. Here qI is the canonical quotient *-homomorphism B

--+

B/I.

In light of Theorems 1.1 and 1.3 one would imagine that for any complete isometry T : A --+ B between unital operator algebras, the condition (A) above should hold verbatim. This would give a pretty noncommutative generalization of Theorem 1.3. Indeed if Ran T is also a unital operator algebra, then this is true (see ego B.l in [3]). However, it is quite easily seen that such a result cannot hold generally. For example, let Mn = Mn(C); for any x E Mn of norm 1, the map A 1---4 Ax is a complete isometry from C into Mn. Now Mn is simple (Le. has no

88

DAVID P. BLECHER AND DAMON M. HAY

nontrivial two-sided ideals), and so if the result above was valid then it follows immediately that x = u. This is obviously not satisfactory. To resolve the dilemma presented in the last paragraph, we have offered in [5] several alternatives. For example, one may replace the quotient B / I by a quotient of a certain *-subalgebra of B. The desired relation qJ(T(a)) = U7r(a) then requires u to be a unitary in a certain C* -triple system (by which we mean a subspace X of a C* -algebra A with X X* X c X). Or, one may replace the quotient B / I by a quotient B / (J + J*), where J is a one-sided ideal of B. Such a quotient is not an algebra, but is an 'operator system' (such spaces have been important in the deep work of Kirchberg (see [18, 19] and references therein). Alternatively, one may replace such quotients altogether, with certain subspaces of the second dual B** defined in terms of certain orthogonal projections of 'topological significance' (Le. correspond to characteristic functions of closed sets in K if B = C(K)) in the second dual B** (which is a von Neumann algebra [25]). The key point of all these arguments, and indeed a key approach to Banach-Stone theorems for linear maps between function algebras, C* -algebras or operator algebra.-" is the basic theory of C* -triple systems and triple morphisms, and the basic properties of the noncommutative Shilov boundary or triple envelope of an operator space. These important and beautiful ideas originate in the work of Arveson, Choi and Effros, Hamana, Harris, Kadison, Kirchberg, Paulsen, Ruan, and others. Indeed our talk at the conference spelled out these ideas and their connection with the Banach-Stone theorem; and the background ideas are developed at length in a book the first author is currently writing with Christian Le Merdy [7] (although we do not characterize non-surjective complete isometries there). Moreover, a description of our work from this perspective, together with many related results, may be found in [12]. Thus we will content ourselves here with a survey of some related and interesting topics, and with a new and self-contained proof of some characterizations of complete isometries between unital operator algebras which do not appear elsewhere. This proof has several advantages, for example the projections arising naturally with this approach seem to be more useful for some purposes. Also it will allow us to avoid any explicit mention of the theory of triple systems (although this is playing a silent role nonetheless). We also show how such noncommutative results are generalizations of the older characterizations of into isometries between function algebras or C(K) spaces. We thank A. Matheson for telling us about these results. In the final section we present some evidence towards the claim that (general) isometries between operator algebras are not the correct noncommutative generalization of isometries between function algebras. For the reader who wants to learn more operator space theory we have listed some general texts in our bibliography.

2. The noncommutative Shilov boundary At the present time the appropriate 'extreme point' theory is not sufficiently developed to be extensively used in noncommutative functional analysis. Although several major and beautiful pieces are now in place, this is perhaps one of the most urgent needs in the subject. However there are good substitutes for 'extreme point' arguments. One such is the noncommutative Shilov boundary of an operator space. Recall that if X is a closed subspace of C(K) containing the identity fUIlction lK on

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K and separating points of K, then the classical Shilov boundary may be defined to be the smallest closed subset E of K such that all functions J E X attain their norm, or equivalently such that the restriction map J f-+ JIE on X is an isometry. This boundary is often defined independently of K, for example if A is a unital function algebra then we may define the Shilov boundary as we just did, but with K replaced by the maximal ideal space of A. In fact we prefer to think of the classical Shilov boundary of X as a pair (aX, i) consisting of an abstract compact Hausdorff space ax, together with an isometry j : X -+ C(aX) such that j(IK) = lax and such that j(X) separates points of ax, with the following universal property: For any other pair (0, i) consisting of a compact Hausdorff space 0 and a complete isometry i : X -+ C(O) which is unital (i.e. i(IK) = IA), and such that i(X) separates points of 0, there exists a (necessarily unique) continuous injection r : ax -+ 0 such that i(x)(r(w)) = j(x)(w) for all x E X, w E ax. Such a pair (aX, i) is easily seen to be unique up to an appropriate homeomorphism. The fact that such ax exists is the difficult part, and proofs may be found in books on function algebras (using extreme point arguments). Consulting our 'noncommutative dictionary' in Section 1, and thinking a little about the various correspondences there, it will be seen that the noncommutative version of this universal property above should read as follows. Or at any rate, the following noncommutative statements, when applied to a unital subspace X c C(K), will imply the universal property of the classical Shilov boundary discussed above. Firstly, a unital operator space is a pair (X, e) consisting of an operator space X with fixed element e EX, such that there exists a linear complete isometry Il, from X into a unital C*-algebra C with Il,(e) = Ie. A 'noncommutative Shilov boundary' would correspond to a pair (B,j) consisting of a unital C*-algebra B and a complete isometry j : X -+ B with j(e) = IB, and whose range generates B as a C* -algebra, with the following universal property: For any other pair (A, i) consisting of a unital C* -algebra and a complete isometry i : X -+ A which is unital (Le. i(e) = IA), and whose range generates A as a C* -algebra, there exists a (necessarily unique, unital, and surjective) *-homomorphism IT : A -+ B such that IT 0 i = j. Happily, this turns out to be true. The existence for any unital operator space (X, e) of a pair (B,j) with the universal property above is of course a theorem, which we call the Arveson-Hamana theorem [2, 13] (see [3] for complete details). As is customary we write C;(X) for B or (B,j), this is the 'C*-envelope of X'. It is essentially unique, by the universal property. If X = A is a unital operator algebra (see Section 1 for the definition of this), then j above is forced to be a homomorphism (to see this, choose an i which is a homomorphism, and use the universal property). Thus A may be considered a unital subalgebra of C;(A). If A is already a unital C* -algebra, then of course we can take C; (A) = A. To help the reader get a little more comfortable with these concepts, we compute the 'noncommutative Shilov boundary' in a few simple examples. Example 1. Let Tn be the upper triangular n x n matrices. This is a unital subspace of M n , and no proper *-subalgebra of Mn contains Tn. Let (B,j) be the C* -envelope of Tn. By the universal property of the C* -envelope, there is a surjective *-homomorphism IT : Mn -+ B such that IT(a) = j(a) for a E Tn. The kernel of IT is a two-sided ideal of Mn. However Mn has no nontrivial two-sided ideals. Hence IT is 1-1, and is consequently a *-isomorphism, and we can thus identify Mn with B. Thus Mn is a C*-envelope of Tn.

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Example 2. Consider the linear subspace X of M3 with zeroes in the 1-3, 2-3, 2-1, 3-1 and 3-2 entries, and with arbitrary entries elsewhere except for the 3-3 entry, which is the average of the 1-1 and 2-2 entries. It is easy to see that the C* -algebra generated by X inside M3 is M2 EB C. However this is not the C*-envelope. Indeed the 3-3 entry here is redundant, since the norm of x E X is the norm of the upper left 2 x 2 block of x. The canonical *-homomorphism from M2 EB C onto Nh when restricted to X is a unital complete isometry from X onto T2 (see Example 1). Thus if one takes the quotient of M2 EB C by the kernel of this homomorphism, namely the ideal 02 EB C, then one obtains M 2 , which by Example 1 is the C* -envelope. Indeed this is typical when calculating the C* -envelope of a unital subspace X of Mn. The C*-algebra generated by X is a finite dimensional unital C*-algebra. However such a C* -algebras is *-isomorphic to a finite direct sum B of full 'matrix blocks' M nk • Some of these blocks are redundant. That is, if p is the central projection in B corresponding to the identity matrix of this block, then x t-+ x(IBp) is completely isometric. If one eliminates such blocks then the remaining direct sum of blocks is the C* -envelope. Example 3. Let B be a unital C*-algebra. Consider the unital subspace S(B) of the C*-algebra M 2 (B) consisting of matrices

[~;

:1]

for all x,y E B and >.,Ji, complex scalars. We claim that M2(B) is the C*-envelope C of S(B), and we will prove this using a similar idea to Example 1 above. Namely, first note that M 2(B) has no proper C*-subalgebra containing S(B), Thus by the Arveson-Hamana theorem there exists a *-homomorphism 11' : M 2 (B) ---+ C which possesses a property which we will not repeat, except to say that it certainly ensures that 11' applied to a matrix with zero entries except for a nonzero entry in the 1-2 position, is nonzero. It suffices as in Example 1 to show that Ker 11' = {a}. Suppose that 11'(x) = 0 for a 2 x 2 matrix x E M 2(B). Let Eij be the four canonical basis matrices for M 2, thought of as inside M2(B). Then 11'(E1i XEj2 ) = 11'(Eli)11'{X)11'(Ej2) = 0 for i,j = 1,2. Thus by the fact mentioned above about the 1-2 position, we must have EliXEj2 = O. Thus x = O. In fact a variant of the C*-envelope or 'noncommutative Shilov boundary' can be defined for any operator space X. This is the triple envelope of Hamana (see [14]). This is explained in much greater detail in [3], together with many applications. For example it is intimately connected to the 'noncommutative .M-ideals' recently introduced in [4]. This 'noncommutative Shilov boundary' is, as we mentioned in Section 1, a key tool for proving various Banach-Stone type theorems. However in the present article we shall only need the variant described earlier in this section. 3. Complete isometries between operator algebras We begin this section with a collection of very well known and simple facts about closed two-sided ideals] in a C* -algebra A, and about the quotient C* -algebra AI]. We have that ]1.1. is a weak* closed two-sided ideal in the von Neumann algebra A**, and there exists a unique orthogonal projection e in the center of A** with ]1.1. = A**(1 - e). The projection 1 - e is called the support projection for I, and

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1 - e may be taken to be the weak* limit in A ** of any contractive approximate identity for I. Thus it follows that A ** / I J..J.. ~ A ** e as C* -algebras. Therefore also A/Ie (A/ 1)** ~ A** / IJ..J.. ~ A**e

as C*-algebras. Explicitly, the composition of all these identifications is a 1-1 *homomorphism taking an a + I in A/I, to ae = eae in A **. Here' is the canonical embedding A -+ A** (which we will sometimes suppress mention of). Thus A/I may be regarded as a C*-subalgebra of A**, or of the C*-algebra eA**e. We next illustrate the main idea of our theorem with a simple special case. (The following appeared as part of Corollary 3.2 in the original version of [5], with the proof left as an exercise). Suppose that T : A -+ B is a complete isometry between unital C*algebras, and suppose that T is unital too, that is T(I) = 1. Let C be the C*-subalgebra of B generated by T(A). Applying the Arveson-Hamana theorem 1 we obtain a surjective *-homomorphism () : C -+ A such that (}(T(a)) = a for all a E A. If I is the kernel of the mapping (), then C / I is a unital C* -algebra *-isomorphic to A. Indeed there is the canonical *-isomorphism 'Y : A -+ C / I induced by (), taking a to T(a) + I. The next point is that C/I may be viewed as we mentioned a few paragraphs back, as a C* -subalgebra of C** , and therefore also of B**. Indeed if e is the central projection in C** mentioned there, then C / I may be viewed as a C* -subalgebra of eC** e C eB** e C B**. In view of the last fact, the map 'Y induces an 1-1 *-homomorphism 7r : A -+ B** taking an element a E A to the element of B** which equals (1)

-- -- --

T(a)e = eT(a)

eT(a)e

--

(these are equal because e is central in C**). Conversely, if T : A -+ B is a complete contraction for which there exists a projection e E B** such that eT(a)e is a 1-1 *-homomorphism 7r, then for all a E A,

IIT(a)1I

~ Ile~ell

=

117r(a)11 = Iiall

using the fact mentioned earlier that 1-1 *-homomorphisms are necessarily isometric. Thus T is an isometry, and a similar argument shows that it is a complete isometry. Thus we have characterized unital complete isometries T : A -+ B. If H is a Hilbert space on which we have represented the von Neumann algebra B** as a weak* closed unital *-subalgebra, then B may be viewed also as a unital C*-subalgebra of B(H), whose weak* closure in B(H) is (the copy of) B**. In this case we shall say that B is represented on H universally. (The explanation for this term is that the well-known 'universal representation' tru of a C*-algebra is 'universal' in our sense, and conversely if 7r is a representation which is 'universal' in our sense then 7r(B)" is isomorphic to 7ru(B)" ~ B**. See [27] Section 1.) If, further, e E B** is a projection for which (1) holds, then with respect to the splitting H = eH EEl (1- e)H we may write T(a) =

[7ro(·)

0]

SO '

for all a E A. We will see that this is essentially true even if T(IA) -lIB: 1We remark in passing that one does not need the full strength of the Arveson-Hamana theorem here, one may use the much simpler [8] Theorem 4.1.

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THEOREM 3.1. Let T : A --> B be a completely contractive linear map from a unital operator algebra into a unital C* -algebra. Then the following are equivalent:

(i) T is a complete isometry, (ii) There is a partial isometry u E B** with initial projection e E B**, and a (completely isometric) 1-1 *-homomorphism 1r : C;(A) 1r(1) = e, such that for all a E A T(a)e

= tL1r(a)

and 1r(a)

-->

eB**e with

-

= u*T(a).

Moreover e may be taken to be a 'closed projection' (see [25] 3.11, and the discussion towards the end of our proof). (iii) If H is a Hilbert space on which B is represented universally, then there exist two closed subspaces E, P of the Hilbert space H, a 1-1 *-homomorphism 1r : C;(A) --> B(E) with 1r(1) = IE, and a unitary u : E --> P, such that T(a)IE = u1r(a), and T(a)IE.L C p.l., for all a E A. Here E.l. for example is the orthocomplement of E in H. (iv) If H is as in (iii), then there exists two closed subspaces E, P of H, a unital 1-1 *-homomorphism 1r : C;(A) --> B(E), a complete contraction S : C; (A) --> B( E.l. , p.l.), and unitary operators U : E EEl p.l. --> Hand V : H --> E EEl E.l., such that T( ) - U [ 1r(a)

a -

0

0

S(a)

] V

for all a E A.

(v) There is a left ideal J of B, a 1-1 *-homomorphism 1r from C;(A) into

a unital subspace of B I (J + J*) which is a C* -algebra, and a 'partial isometry' u in B I J such that qJ(T(a)) = u1r(a)

&

for all a E A, where qJ is the canonical quotient map B

-->

BIJ.

Before we prove the theorem, we make several remarks. First, we have taken B to be a C* -algebra; however since any unital operator algebra is a unital subalgebra of a unital C* -algebra this is not a severe restriction. We also remark that there are several other items that one might add to such a list of equivalent conditions. See [5, 6]. Items (ii)-(iv), and the proof given below of their equivalence with (i), are new. We acknowledge that we have benefitted from a suggestion that we use the Paulsen system to prove the result. This approach is an obvious one to those working in this area (Ruan and Hamana used a variant of it in their work in the '80's on complete isometries and triple morphisms [28, 14]). However we had not pushed through this approach in the original version of [5] because this method does not give several of the results there as immediately. Statement (v) above has been simply copied from [5, 6] without proof or explanation. We have listed it here simply because Theorem 1.3 may be particularly easily derived from it as the special case when A and B are commutative (see comments below). Note that (iii) above resembles Theorem 1.2 superficially.

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PROOF. The fact that the other conditions all imply (i) is easy, following the idea in the paragraph above the theorem, namely by using the fact that a 1-1 *-homomorphism is completely isometric. In the remainder of the proof we suppose that T is a complete isometry. We view A as a unital subalgebra of C;(A) as outlined in Section 3. We define a subset S(B) of M 2(B) as in Example 3 in Section 2. Similarly define a subset S(T(A)) of S(B) using a similar formula (note that S(T(A)) has 1-2 entries taken from T(A) and 2-1 entries taken from T(A)*). Similarly we define the subset S(A) of the CO-algebra M2(C;(A)) (Le. S(A) has scalar diagonal entries and off diagonal entries from A and A*). We write 1 EEl 0 for the matrix in S(A) with 1 as the 1-2 entry and zeroes elsewhere. Similarly for 0 EEl 1. We also use these expressions for the analogous matrices in S(B). The map q, : S(A) -+ S(T(A)) c M 2(B) taking

[~!

:1] ~

[T~)* ~~~)]

is well known to be a unital complete isometry (this is the well known Paulsen lemma, see the proof of 7.1 in [23]). Let C be the C' -subalgebra of M2 (B) generated by S(T(A)). The CO-envelope of S(A) is well known to be M2(C;(A)) (see Example 3 in Section 2 where we proved this in the case that A is already a C* -algebra, or for example [3] Proposition 4.3 or [30]). Thus by the Arveson-Hamana theorem we obtain a surjective *-homomorphism () : C -+ 1V/z(C;(A)) such that () 0 q, is simply the canonical embedding of S(A) into M2(C;(A)). As in the special case considered above the theorem, we let 10 be the kernel of the mapping (), then C /10 is a unital CO-algebra *-isomorphic to M2(C;(A)). Indeed there is the canonical *-isomorphism "(: M2(C;(A)) -+ Clio induced by (), taking

[~!

:1] ~

[T~:)* T~~)]

+

10 ,

As in the simple case above the theorem, C /10 may be viewed as a C* -subalgebra of PoC"Po, for a central projection Po E C** (namely, the complementary projection to the support projection of 10)' Now PoC**Po C C** C M2(B)**, and it is well known that M2(B)** ~ M2(B**) as CO-algebras. Thus we may think of C** as a C* -subalgebra of .1112 (B**). Also, C'* contains C as a C* -subalgebra, and the projections 1 EEl 0 and 0 EEl 1 in C correspond to the matching diagonal projections 1 EEl 0 and 0 EEl 1 in M2(B**). These last projections therefore commute with Po, since Po is central in C**, which immediately implies that Po is a diagonal sum fEEl e of two orthogonal projections e, f E B**. Thus we may write the C* -algebra POM2(B**)po as the C*-subalgebra [ fB** f eB** f

f B**e ] eB*'e

of M2(B**). We said above that Clio may be regarded as a C*-subalgebra of the subalgebrapoM2(B**)po of M2(B*'). Thus the map "( induces a 1-1 *-homomorphism III : M2(C;(A)) -+ .!I1z(B**). It is easy to check that 1lI(1 EEl 0) = fEEl 0 and III (0 EEl 1) = 0 EEl e. Since III is a *-homomorphism it follows that III maps each of the four corners of M 2(C;(A)) to the corresponding corner of POM2(B**)po C M2(B**). We let R: C;(A) -+ fB**e be the restriction of III to the '1-2-corner'. Since III is 1-1, it follows that R is 1-1. If 7r is the restriction of III to the '2-2-corner', then 7r is a *-homomorphism C;(A) -+ eB**e taking lA to e. Applying the *-homomorphism

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DAVID P. BLECHER AND DAMON M. HAY

111 to the identity

[~ ~][~ ~]=[~ ~] we obtain that u = R(I) is a partial isometry, with u*u = 71"(1) = e. Similarly uu* = f. A similar argument shows that R(a) = R(I)7I"(a) for all a E C;(A). Thus u* R(a) = u*u7l"(a) = 7I"(a) for all a E C;(A). Next, we observe that 111 takes the matrix z which is zero except for an a from A in the 1-2-corner, to the matrix w = Pocp(z)Po. Since cp(z) E C** and Po is in the center of that algebra, we also have w = cp(z)Po = Pocp(z). Also w viewed as a matrix in M2(B**) has zero entries except in the .1-2-corner, which (by the last sentence) equals

---- ----

ff(;;) e = f(;;) e = ff(;;). Also using these facts and a fact from the end of the last paragraph we have --...

----* -

---*-

u*T(·) = R(I)*T(·) = (fT(I)e)*T(·) = eT(I) fT(·) = eT(I) T(·)e = u* R(·) = 71". Thus

-

-

-

T(·)e = fT(·) = uu*T(·) = U7l"(')' We have now also established most of (ii). One may deduce (iii) from (ii) by viewing B c B** c B(H), and setting E = eH, and F = (uu*)H. We also need to use facts from the proof above such as u*u = e. Clearly (iv) follows from (iii). As we said above, we will not prove (v) here. Claim: if e is the projection in (ii) above, then 1 - e is the support projection for a closed ideal I of a unital *-subalgebra D of B. Equivalently (as stated at the start of this section), there is a (positive increasing) contractive approximate identity (bd for I, with bt ~ 1 - e in the weak* topology. This claim shows that 1 - e is an 'open projection' in B**, so that e is a closed projection, as will be obvious to operator algebraists from [25] section 3.11 say. For our other readers we note that for what comes later in our paper, one can replace the assertion about closed projections in the statement of Theorem 3.1 (ii) with the statement in the Claim above. To prove the Claim, recall from our proof that Po = fEB e = Ie - Pi, where Pi is the support projection for a closed ideal 10 of C. Thus Pi = (1 - f) EB (1 - e). As stated at the start of Section 3, Pi is the weak* limit in C**, and hence also in M 2(B**), of a contractive approximate identity (et) of 10, By the separate weak* continuity of the product in a von Neumann algebra, it follows that the net bt = (OEB l)et(OEB 1) has weak* limit (OEB I)Pl (OEB 1) = OEB (1- e). Viewing these as expressions in B, the above says that bt ~ 1 - e weak* in B* *. View (0 EB 1) C (0 EB 1) as a *-subalgebra D of B, and view (0 EB 1)10(0 EB 1) as a two sided ideal I in D. It is easy to see that (bd is a contractive approximate identity of I. Thus it follows that 1 - e is the support projection of the ideal I. 0 Some applications of results such as Theorem 3.1 may be found in [6]. Next we discuss briefly the relation between our noncommutative characterization of complete isometries (for example Theorem 3.1 above), and Theorem 1.3. Our point is not to provide another proof for Theorem 1.3 - the best existing proof is certainly short and elegant. Rather we simply wish to show that the noncommutative result contains 1.3. Indeed Theorem 1.3 quite easily follows from Theorem

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95

3.1 (v). Since however we did not prove Theorem 3.1 (v), we give an alternate proof. COROLLARY 3.2. Let A, B be a unital function algebras, with B selfadjoint. Then condition (ii) in Theorem 3.1 implies condition (A) in Theorem 1.3. PROOF. By hypothesis, T(·)e = U1l"('), and u*u = e = 11"(1) so that u = u1l"(1) = T(I)e. Thus eT(I)*T(·)e = u*U1l"(') = 11"(1)11"(') = 11", so that Ran 11" C eBe = Be (note B** is commutative in this case). From [25] 3.11.10 for example, the 'closed projection' e in B** corresponds to a closed ideal J in B whose support projection is 1 - e. Alternatively, to avoid quoting facts from [25], we will also deduce this from the 'Claim' towards the end of the proof of Theorem 3.1. If I is the ideal in that Claim, let J be the closed ideal in B generated by I. Since J = Bl, the contractive approximate identity of I is a right contractive approximate identity of J. Thus J has support projection 1 - e too, by the first paragraph of Section 3 above. By facts in the just quoted paragraph, we have a canonical unital 1-1 map '11 : B / J ~ B** taking the equivalence class b + J of b E B to ebe. Indeed in this commutative case we see by inspection that '11 is a *-homomorphism from the C*algebra B/J onto the C*-subalgebra M = eBe of B**. Define O(a) = '11-1(1I"(a)), this is a 1-1 *-homomorphism A ~ B/J. Since 11"(1) = e, 0 is a unital map too. Since uu* = u*u = e, u is unitary in M, and so 'Y = '11-1(U) is unitary in B/J. Note also that T(a)e = '11(T(a) + J). Applying '11- 1 to the equation T(·)e = U1l"('), we obtain qJ(T(a)) = 'Y O(a), that is, condition (A) in Theorem 1.3. D If one attempts to use the ideas above to find a characterization analogous to condition (A) from Theorem 1.3 but in the noncommutative case, it seems to us that one is inevitably led to a condition such as (v) in Theorem 3.1. We address a paragraph to experts, on generalizations of the proof of Theorem 3.1. Consider a complete isometry between possibly non-unital C* -algebras. Or much more generally, suppose that T is a complete isometry from an operator space X into a C* -triple system W. One may form the so called 'linking C* -algebra' of W, with the identities of the 'left and right algebras of W' adjoined. Call this C'(W). As in the proof of Theorem 3.1 we think of S(W) c .c'(W). Similarly, if Z is the 'triple envelope' of X (or if X = Z is already a C*-algebra or C*-triple system), then we may consider S(X) C S(Z) c .c'(Z). As in the proof of Theorem 3.1 we obtain firstly a unital complete isometry : S(X) ~ S(T(X)) c .c'(Z), and then a unital 1-1 *-homomorphism 11" : .c'(Z) ~ .c'(W)**. By looking at the 'corners' of 11" we obtain projections e, f in certain second dual von Neumann algebras, so that fT( ·)e is (the restriction to X of a completely isometric) a 1-1 triple morphism into W**. In fact we have precisely such a result in [5] (see Section 2 there), but the key point is that the new proof gives different projections e, f, which are more useful for some purposes.

4. Complete isometries versus isometries Finally, as promised we discuss why we believe that in this setting of nonsurjective maps between C* -algebras say, general isometries are not the 'noncommutative analogue' of isometries between function algebras. The point is simply this. In the function algebra case we can say thanks to Holsztynski's theorem that the isometries are essentially the maps composed of two disjoint pieces Rand S, where R is

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DAVID P. BLECHER AND DAMON M. HAY

isometric and 'nice', and S is contractive and irrelevant. However at the present time it looks to us unlikely that there ever will be such a result valid for general nonsurjective isometries between general C* -algebras. The chief evidence we present for this assertion is the very nice complementary work of Chu and Wong [9] on isometries (as opposed to complete isometries) T : A -+ B between CO-algebras. They show that for such T there is a largest projection p E B** such that T(·)p is some kind of Jordan triple morphism. This appears to be the correct 'structure theorem', or version of Kadison's theorem [11], for nonsurjective isometries. However as they show, the 'nice piece' R = T(·)p is very often trivial (Le. zero), and is thus certainly not isometric. Thus this approach is unlikely to ever yield a characterization of isometries. A good example is A = M 2 , the smallest noncommutative C* -algebra. Simply because A is a Banach space there exists, as in the discussion in the first paragraph of our paper, a linear isometry of A into a C (K) space. However it is easy to see that there is no nontrivial *-homomorphism or Jordan homomorphism from A into a commutative C* -algebra. Such an isometry is uninteresting, and this is perhaps because the interesting 'nice part' is zero. Thus we imagine that the 'good noncommutative notions of isometry' are either complete isometries or the closely related class of maps for which the piece T(·)p from [9] is an isometry. This leads to three questions. Firstly, can one independently characterize the last mentioned class? Secondly, if T is a complete isometry, then is the projection p in the last paragraph equal (or closely related) to our projection e above? Finally, H. Pfitzner has remarked to us, there is already a gap between the isometry and the 2-isometry cases (not only isometries and complete isometries). It would be interesting if there were a characterization of 2-isometries.

References [1] J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413-428. [2] W. B. Arveson, Subalgebms ofC*-algebms, Acta Math. 123 (1969), 141-224; II, 128 (1972), 271-308. [3] D. P. Blecher, The Shilov boundary of an opemtor space, and the chamcterization theorems, J. Funct. An. 182 (2001),280-343. [4] D. P. Blecher, E. G. Effros and V. Zarikian, One-sided M-Ideals and multipliers in opemtor spaces. 1. To appear Pacific J. Math. [5] D. P. Blecher and D. Hay, Complete isometries into C* -algebms, http://front.math.ucdavis.edu/math.OA/0203182, Preprint (March '02). [6] D. P. Blecher and L. E. Labuschagne, Logmodularity and isometries of opemtor algebms, To appear, Trans. Amer. Math. Soc .. [7] D. P. Blecher and C. Le Merdy, Opemtor algebms and their modules - an opemtor space approach, To appear, Oxford Univ. Press. [8] M. D. Choi and E.G. Effros, The completely positive lifting problem for C·-algebms, Ann. Math. 104 (1976), 585-609. [9] C-H. Chu and N-C. Wong, Isometries between C· -algebms, Preprint, to appear Revista Matematica Iberoamericana. [10] J. B. Conway, A Course in Opemtor Theory, AMS, Providence, 2000. [11] E. G. Effros and Z. J. Ruan, Opemtor Spaces, Oxford University Press, Oxford (2000). [12] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Book to appear, CRC press. [13] M. Hamana, Injective envelopes of opemtor systems, Pub!. R.I.M.S. Kyoto Univ. 15 (1979), 773-785.

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COMPLETE ISOMETRIES - AN ILLUSTRATION

[14] M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 77-93. [15] W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 24 (1966), 133-136. [16] K. Jarosz and V. Pathak, Isometries and small bound peturbations of function spaces, In "Function Spaces", Lecture Notes in Pnre and Applied Math. Vol. 136, Marcel Dekker (1992). [17] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325-338. [18] E. Kirchberg, On restricted peturbations in inverse images and a description of normalizer algebras in C*-algebras, J. Funct. An. 129 (1995), 1-34. [19] E. Kirchberg and S. Wassermann, C*-algebras generated by operator systems, J. Funct. Analysis 155 (1998), 324-351. [20] A. Matheson, Isometries into function algebras, To appear. [21] M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodia Math. Sem. Rep. 11 (1959), 182-188. [22] W. P. Novinger, Linear isometries of subspaces of continuous functions. Studia Math. 53 (1975), 273-276. [23] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., Longman, London, 1986. [24] V. I. Paulsen, Completely bounded maps and operator algebras, To appear Cambridge University Press. [25] G. Pedersen, C*-algebras and their automorphism groups, Academic Press (1979). [26] G. Pisier, Introduction to operator space theory, To appear Camb. Univ. Press. [27] M. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. [28] Z. J. Ruan, Subspaces ofC*-algebras, Ph. D. thesis, U.C.L.A., 1987. [29] E. L. Stout, The theory of uniform algebras, Bogden and Quigley (1971). [30] C. Zhang, Representations of operator spaces, J. Oper. Th. 33 (1995), 327-351. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON.

E-mail address, David P. Blecher: dblecher~ath. uh. edu E-mail address, Damon Hay: dhayOmath.uh.edu

TX 77204-3008

Contemporary Mathematics Volume 328, 2003

Some Recent Trends and Advances in Certain Lattice Ordered Algebras Karim Boulabiar, Gerard Buskes, and Abdelmajid Triki ABSTRACT. In this paper we give a survey, intended as both a supplement as well as an update to a survey by Huijsmans [57], with results that have been obtained in the last ten years on Archimedean lattice ordered algebras. Special attention is paid to I-algebras, almost I-algebras and d-algebras and problems that were posed in the survey by Huijsmans about these special classes of lattice ordered algebras.

CONTENTS

l. Introduction 2. Definitions and elementary properties 3. i-algebra multiplications in C (X) 4. Multiplication by an element as an operator 5. Uniform completion and Dedekind completion 6. Powers in i-algebras 7. Functional Calculus on f-algebras 8. Relationships between i-algebra multiplications 9. Connection between algebra and Riesz homomorphisms 10. Positive derivations 11. Cauchy-Schwarz inequalities 12. Order biduals 13. Ideal theory 14. Representation of f-algebras 15. Linear biseparating maps on f-algebras References 1991 Mathematics Subject Classification. 06F25, 13J25, 16W80, 46A40, 46840, 46842, 46E25, 47L07, 47847, 47865. Key words and phrases. almost I-algebra, algebra homomorphism, I-algebra, d-algebra, lattice ordered algebra, order ideal, orthomorphism, representation theory, Riesz homomorphism, ring ideal, space of continuous functions, uniformly complete Riesz space. The second named author gratefully acknowledges support from an Office of Naval Research Grant with number N00014-01-1-0322. Part of this survey was written while the first named author was visiting the University of Mississippi in the Spring of 2002. © 2003 American Mathematical Society

99

~!3QtJLA8lAR,

GERARD BUSKES, AND ABDELMAJID TRIKI

1. Introduction

ftehistory of lattice ordered vector spaces (so called Riesz spaces or vector

lattices) goes back to Riesz and the International Congress of Mathematicians in Bologna in 1928. A study of the most important class of lattice ordered algebras (but not their name), J-algebras, was initiated by Nakano in [75] for a-Dedekind complete ordered vector space in 1951, subsequently in 1953 by Amemiya in [3], and finally with its present definition and name in 1956 by Birkhoff and Pierce in [22]. A precise date for the very first definition of lattice ordered algebras in general is very hard to pinpoint, but originated at around the same time as the previous three references. Indeed, in his review [44] of Birkhoff's 1950 address to the International Congress of Mathematicians in Cambridge, Massachusetts, Fl.·ink observed that a general study of lattice ordered rings seems to be needed to study what are now called averaging or Reynolds operators. A call for lattice ordered rings also had gone out by Birkhoff himself in the form of a listed problem at the end of his seminal 1942 paper [20] on lattice ordered groups. Thus lattice ordered algebras and J-algebras seem to have multiple origins, including a study of averaging operators, which themselves sprang forth from problems in fluid mechanics. An appearance at about the same time of J-rings and J-algebras has not resulted in an historical development on complete common ground for these objects. This is not unlike the development of lattice ordered groups versus the development of Riesz spaces. Where the latter have attracted attention from researchers in analysis, the former have been more widely investigated by algebraists. A similar divided attention from analysis versus algebra seems to underlie the connected but somewhat separate tracks of lattice ordered rings versus lattice ordered algebras. Though this separation of tracks is to some extent unavoidable, where each track does have ground that is truly its own, some overlap in results does exist, resulting in difficulties making accurate literature attributions in a survey like ours. We are grateful to two referees for pointing at some references that were missing in our manuscript, though we take full responsibility for possible remaining omissions in the reference list. This survey places itself almost completely on the track of lattice ordered algebras and our only apology for not linking algebra facts in a systematic way to ring results is that all three of us authors were trained as analysts. There is a natural back and forth between the two theories, in one direction by forgetting some of the structure, and in the other by finding, so to speak adjointly, an enveloping algebra. A nice survey on J-rings was written by Henriksen in 1995 (see [50]), to which we refer the interested reader for linkage to some of what follows in this survey. Historically, a lot of the credit for a revival of the theory of J-algebras points to the highly motivating Arkansas Lecture Notes by Luxemburg [67] and the 1982 Ph.D. thesis of de Pagter [76], who systematically explored both the existing literature as well as new directions. Another impetus to research in the area of J-algebras derived from the desire of Zaanen, who in the late seventies started to develop a program to prove many of the elementary results in the theory of Riesz spaces without using representation theorems for vector lattices. This desire is directly linked with a preference not to use the Axiom of Choice unnecessarily. The present survey is intended as an update to the one by Huijsmans [57]. We hasten to point out that we do not intend this survey to replace the one by Huijsmans, but rather that we think of it as augmenting part of it. Since [57] appeared, much progress has been made and several of the problems explicitly phrased in

LATTICE ORDERED ALGEBRAS

101

[57] have been solved. At the same time, some topics like ideal theory, connections between Riesz homomorphisms and algebra homomorphisms, and representations of f-algebras were absent in [57]. Thus an update as well as a supplement was needed. However, some sections of [57] receive no attention at all in this survey. We have not included important topics like the role of f-algebras in positive operator theory (e.g., we do not even include results on the previously mentioned averaging operators) and probability theory. We are rather focused on placing this update, as much as possible, in the setting of spaces of functions, hoping to interest as large an audience as possible via this approach. Moreover, we feel that the great source of inspiration was and continues to be the beautiful book by Gillman and Jerison [47], which certainly inspired and continues to inspire a large part of the research in lattice ordered algebras and rings. Last but not least we focus on what could be called distortions of f-algebras, in particular on Archimedean almost falgebras and d-algebras. Though these distortions have the potential to be seen as aberrations by some, we believe they point the way to techniques that are needed for the broader theory of lattice ordered algebras, as well as for illumination of various aspects in the theory of f-algebras. Note that the distortions disappear if the lattice ordered algebra under consideration has a multiplicative identity which is a weak order unit. Indeed, such algebras are automatically f-algebras. It should be mentioned that classes of lattice ordered algebras other than the ones that appear in this survey have been studied. Notably, the papers [85], [86], [87], and [88] by Steinberg discuss lattice ordered algebras in which every square is positive, a class of algebras that includes all almost f-algebras. We also pay no attention at all to non-Archimedean lattice ordered algebras. Finally, we point out that several results in this survey rely heavily on the (relative) uniform topology on Riesz spaces. In particular, since in Archimedean Riesz spaces uniform limits are unique, we shall include the 'Archimedean' property in the definition of uniformly complete Riesz spaces. A complete investigation of that topology can be found in Sections 16 and 63 of [69]. For terminology and concepts not explained or proved in this survey we refer the reader to the standards books [2], [47], [69], [72], [92] and [93].

2. Definitions and elementary properties A (real) Riesz space A is called a lattice ordered algebra (briefly, an i-algebra) if A also is an algebra and the positive cone A+ = {J E A : f ~ O} is closed under multiplication, that is, if f,g ~ 0 then fg ~ 0 (equivalently, if Ifgl :::; Ifllgl for all f,g E A).

We make the following blanket assumption: all Riesz spaces under consideration in this paper are assumed to be Archimedean (however, the latter blanket assumption has not stopped us to explicitly add the word Archimedean to the list of conditions in various results below). After B.irkhoff and Pierce (see [22, p. 55]), we define an i-algebra A to be an f-algebra if for every f, 9 E A, the condition f /\g = 0 implies Uh) /\g = (hi) /\g = 0 for all h E A+

102

KARIM BOULABIAR. GERARD BUSKES, AND ABDELMA.lID TRIKI

holds. We call the E-algebra A an almost f-algebra after Birkhoff in [21, Section 6] if f /\ g

= 0 ill A implies

fg

= O.

An E-algebra A for which f /\ 9 = 0 in A and h E A+ imply (.fh) /\ (gh)

=

(hf) /\ (hg) = 0

is called a d-algebra. The notion of d-algebra goes back to Kudlacek in [65]. Our focus in this survey on E-algebras is almost exclusively on f -algebras, almost falgebras and d-algebras. In this paragraph, we recall some properties of f-algebras. Using the Axiom of Choice, Birkhoff and Pierce in [22, Theorem 13] proved that any f-algebra is commutative. A constructive proof of this fact, due to Zaanen, can be found in [61, Theorem 2.1] or [92, Theorem 140.10]. All squares in an f-algebra are positive. Also, Ifgl = Ifllgl for all f,g in an f-algebra A. The multiplication by an element in the f -algebra A is order continuous, i.e., if inf {fT : T} = 0 in A then inf {g fT : T} = o for all 9 E A +. Phrased more generally, the multiplication 7rf by an element f E A (7rf (g) = f 9 for all 9 E A) is an orthomorphism and all orthomorphisms are order continuous. Recall that an orthomorphism on a Riesz space L is an order bounded linear operator 7r such that 17r (f)I/\ Igl = 0 whenever Ifl/\ Igl = 0 in L (the reader is referred to [2] or [92] for elementary properties of orthomorphisms). There is another important relationship between orthomorphisms and f-algebras, which we mention next. Indeed, let Orth (L) be the set of all orthomorphisms on a Riesz space L. Under the operations and the ordering inherited from £b (L), the ordered algebra of all order bounded operators on L, and under composition as multiplication, Orth (L) is an Archimedean f-algebra with the identity map h on L as unit element. The details of the facts recalled above can all be found in [2], [76] or [92]. Next we present some properties of almost f-algebras. Almost f-algebras, like f-algebras, are commutative too. The latter fundamental property was first established by Scheffold in [80, Theorem 2.1] for almost f-algebras that are Banach lattices. Using both Scheffold's result and the Axiom of Choice, Basly and Triki were the first to prove commutativity for arbitrary almost f-algebras [10, Thorme 1.1]. The first proof of the commutativity for almost f-algebras within ZermeloFraenkel set theory was given by Bernau and Huijsmans in their paper [13, Theorem 2.15]. Recently, a shorter constructive proof was published in [34, Corollary 3] by Buskes and van Rooij. Another property of f-algebras holds for almost f-algebras, namely the positivity of squares. Also, if A is an almost f-algebra then f2 = Ifl2 for all f E A. However, contrary to the order continuity of the multiplication in falgebra..'l, the multiplication by a fixed element in an almost f-algebra is not always order continuous as is shown in the following example. EXAMPLE 2.1. Write A = C ([0,1]), the vector space of all real-valued continuous functions on [0, 1]. With respect to the pointwise ordering (i. e., f :::; 9 in A if an only if f (x) :::; 9 (x) for all x E [0,1]), A is an Archimedean Riesz space. Define a multiplication • in A by

(f. g)(x)

=

{

f(x)g(x)

f (1/2) 9 (1/2)

(0 :::; x :::; 1/2) ; (1/2 :::; x :::; 1)

LATTICE ORDERED ALGEBRAS

103

lor all I, 9 E A. Then A is an almost I-algebm with respect to the multiplication •. For every natuml number n :::: 1, define the function In E A by (O :::; x :::; 1/2 - l/n) ; (1/2 - l/n :::; x :::; 1/2) ; (1/2:::; x :::; 1/2 + l/n) ; (1/2+1/n:::;x:::;1).

I f (:1") _ { n/2 - n:1: " ,-n/2 + nx 1

Then sup Un : n = 1,2, ... } exists in A and equals the function e defined bye (x) = 1 lor all x E [0,1]. On the other' hand,

(O :::; x :::; 1/2) ; (1/2:::;x:::;1)

lor all n E {I, 2, ... }, and clearly the set {e. In : n = 1,2, ... } does not have a supremum in A. We conclude that. is not order continuous. For more information about elementary theory of almost f-algebras, the reader is encouraged to consult [13], [23], [34], and [35]. At this point, we turn our attention to some properties of d-algebras. It follows directly from the definition of d-algebras that an i-algebra A is a d-algebra if and only if the multiplication map induced by any fixed element in A + is a Riesz (or lattice) homomorphism. It follows that a necessary and sufficient condition for an i-algebra A to be a d-algebra is that the identity If gl = 1/IIgi holds for all f, 9 in A. Contrary to (almost) I-algebras, d-algebras need not be commutative nor have positive squares. Next we give an example of a non-commutative d-algebra in which not all squares are positive. EXAMPLE

2.2. Let in this example A be the algebra of real {2 x 2)-matrices of

the form

(~

g)

with the usual addition, scalar multiplication, matrix product and partial or·dering. It is not hard to see that A is an Archimedean d-algebra. But A is not commutative and not all squares in A are positive. Indeed, if p=

(~ ~)

and q =

(~ ~)

then pq

=q

and qp = O.

Moreover, the square

is not positive. As for almost f-algebras, multiplication by a fixed element in a d-algebra is, in general, not order continuous. Point in case is the almost I-algebra that we considered in Example 2.1, which also is a d-algebra. Our main reference about d-algebras is [13]. In what follows, we look at some of the connections between the three kinds of i-algebras that we consider in this paper. It is immediate that any I-algebra is both, an almost f-algebra and a d-algebra. Almost f-algebras need not be d-algebras as we see in the next example.

KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

104

EXAMPLE 2.3. Take A as in Example 2.1, and let () E A be the function () (x) = { 1/2 - x x - 1/2

(0::; x ::; 1/2) ; (1/2::; x ::; 1).

For f, 9 E A, define (f

-9

)( ) _ {

x -

/,1-X f(s)g(s)ds

(0::; x ::; 1/2);

1/2

(1/2::; x ::; 1).

() (x) f (x) 9 (x)

Then A is an almost f -algebra under the multiplication -. However, A is not a d-algebra. Indeed, let e, f in A be defined by e(x)=l and

={

f (x)

forallxE[O,l]

-4x + 1 4x-3

(0::; x ::; 1/2) ; (1/2::;x::;1).

If - el (0) = 0 and (If I- e) (0) = If I- Igl fails in A.

Observe that

If - gl =

/,1

If(s)1 ds -; O. Thus the property

1/2

Since every almost f -algebra is commutative and has positive squares, Example 2.2 shows that d-algebras need not be almost f-algebras. However, if a d-algebra A is commutative or has positive squares then A is automatically an almost f-algebra [22, p. 60]. Summarizing part of the relations, we have the following diagram f-algebra::::} commutative d-algebra ::::} almost f-algebra. For more detail, see [22], [13] and [57]. The next lines deal with nilpotent element in i-algebras. The set of all nilpotent elements in the i-algebra A is denoted by N (A). In other words,

N (A)

= {f E A : r = 0 for some n =

1, 2, ... } .

Given a natural number p, we define Np (A)

= {f

E A : fP

= O} .

The i-algebra A is said to be semiprime if 0 is the only nilpotent element in A, that is, if N (A) = {O} . If A is an f-algebra then the following equalities hold N (A)

= N2 (A) = {f E A

: fg

= 0 for all 9 E A}.

(see [76, Proposition 10.2] or [92, Theorem 142.5]). If A is an almost f-algebra then N (A)

= N3 (A) = {J E A

: fg2

= 0 for all 9 E A}

= {f E A : fgh = 0 for all g, hE A}

(see [51, Theorem 3.11]) and, as for f-algebras, N2 (A)

= {f

E A : fg

= 0 for

all 9 E A}

(see [23, Lemma 5.3]). If A is a d-algebra then N (A)

= N3 (A) = {f E A

: gfh

= 0 for all 9

• A}

(see [13, Theorem 5.5] or [29, Theorem 5]). However, and co we documented for f-algebras and almost f-algebras, in d-algebl

ry to what he equality

LATTICE ORDERED ALGEBRAS

105

N2 (A) = {J E A: fg = 0 for all 9 E A} does not necessarily hold as is illustrated by the following example.

EXAMPLE 2.4. Take A as in Example 2.1 and define the multiplication. in A by (f. g)(x) = f (0) 9 (1) for all f, 9 E A. Let f E A be the function defined by f (x) = 1 - x for all x E [0,1]. Clearly, f. f = 0 but f. e i= 0 where e E A is defined bye (x) = 1 for all x E [0,1].

Finally, note that any f-algebra with multiplicative identity is semiprime and any semiprime almost f-algebra or semiprime d-algebra is automatically an falgebra (see Section 1 in [13]). As a final comment we remark that an l-algebra which has positive squares and has a multiplicative identity need not be an falgebra.

3. i-algebra multiplications in C (X) Let C (X) be the set of all real-valued continuous functions on a compact Hausdorff' topological space X. Under pointwise addition and scalar multiplication, C (X) is a real vector space. Moreover, C (X) is an Archimedean Riesz space with respect to the pointwise ordering (i.e., f ~ 9 in C (X) if and only if f (x) ~ 9 (x) for all x E X). By defining the multiplication in C (X) pointwise as well (Le., (fg) (x) = f(x)g(x) for all f,g E C(X) and all x E X), the space C(X) is easily seen to have the structure of an f-algebra with e as unit element, where e (x) = 1 for all x E X. Now consider another associative multiplication. in C (X). The main topic of this section is to produce necessary and sufficient conditions for C (X) to be an f-algebra (respectively, an almost f-algebra, a d-algebra) with respect to this new multiplication •. The first theorem in this direction goes back to Conrad (see [38, Theorem 2.2]), who obtained the following. THEOREM 3.1. Let. be an associative multiplication in C (X). Then C (X) is an f -algebra with respect to • if and only if there exists a positive function w E C (X) such that (f.g)(x) =w(x)f(x)g(x) for all f,g E C (X) and all x EX.

In fact, Conrad established the theorem above for any Archimedean f-ring with unit element. The representation formula given in Theorem 3.1 above was obtained in an alternative way by Scheff'old in [80, Korollar 1.4]. While Conrad's proof is purely algebraic and order theoretic, the proof presented by Scheff'old relies on analytic tools like the Riesz representation theorem. With the same analytic tools Scheff'old also obtained the following representation theorem for almost f-algebra multiplications in C (X) (see [80, Theorem 1.2]). THEOREM 3.2. Let. be an associative multiplication in C (X). Then C (X) is an almost f -algebra with respect to • if and only if there exists a family (/Jx : x E X) of positive measures such that (f. g)(x) for all f, 9 E C (X) and all x E

x.

=

Ix

f(s) 9 (s) d/Jx (s)

106

KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI

Since every commutative d-algebra is an almost I-algebra, the previous theorem remains valid for commutative d-algebra multiplications in C (X) as well. Recently in [25, Corollary 3.2], Boulabiar proved the following representation formula for any (not necessarily commutative) d-algebra multiplication in C (X). THEOREM 3.3. Let. be an associative multiplication in C (X). Then C (X) is a d-algebra with respect to • if and only if there exist (i) a positive function wE C (X), and (ii) functions h,k: X -+ X (continuous on coz(w) = {x EX: w(x)"! O}) such that (J. g) (x) = w (x) I (h (x)) 9 (k (x)) for all I,g E C(X) and all x E X.

Notice that if C (X) is a d-algebra with respect to the multiplication. then • is commutative if and only if the functions hand k coincide on coz (w) (where 11" k and ware as in Theorem 3.3). The latter observation yields. in addition to the formula cited in Theorem 3.2 above, another representation for commutative d-algebra multiplications on C (X) . More abstract versions of the results above will be given in Section 8 below. 4. Multiplication by an element as an operator Let A be an (-algebra and recall that C b (A) denotes the ordered algebra of all order bounded operators on A. For every f E A, we define the map 7rf on A by 7rf (g) = f 9 for all 9 E A. Clearly, 7rf is an order bounded operator on A for all lEA. The map p : A -+ Cb (A) defined by p (J) = 7rf for all I E A is obviously an algebra homomorphism, that is, p (J g) = p (J) p (g) for all I, 9 E A. Hence the range p (A) of p is a subalgebra of Cb (A). In this section, we will see that if A is an almost f-algebra then p (A) can canonically be equipped with an ordering, under which p (A) is an Archimedean I-algebra. A corresponding result will also be given for commutative d-algebras and f-algebras. Let A be an almost I-algebra. Since

N2 (A) = {f E A : Ig = 0 for all 9 E A}, = 7r9 if and only if f by putting

7rf

-

9 E N 2 (A). This allows us to define an ordering on p (A)

(0) The ordering defined by (0) coincides with the ordering inherited from Cb (A), namely, 7rf is positive with respect to (0) if and only if 7rf is a positive operator on A. Under the usual addition and composition of operators, and with the ordering defined by (0), p (A) is an Archimedean ordered subalgebra of Cb (A). In fact, we have the following theorem (see Theorem 4.2 and Theorem 4.4 in [23]). THEOREM 4.1. Let A be an Archimedean almost I -algebra. Then p (A) is an Archimedean I -algebra with respect to the addition and composition of operators, and the ordering inherited from Cb (A). The lattice operations in p (A) are given by 7r f

V 7r9

= 7rfV 9'

7r f 1\ 7r9

= 7rf /1.9

for all f, 9 EA.

In particular, (7rf)+

= 7rf+,

(7rf)-

= 7rf-'

l7rfl

= 7rlfl

for all lEA.

LATTICE ORDERED ALGEBRAS

107

In other words, p defines a surjective Riesz homomorphism from A onto p (A).

Theorem 4.1 of course holds in commutative d-algebras. Moreover, let A be a commutative d-algebra. For f E A and 9 E A + the equalities 17I'fl (g)

= 71'tft (g) = Iflg = sup{lfllhl : Ihl

::; g}

= sup {Ifhl : Ihl ::;g} = sup {17I'f (h)1 : Ihl ::;g} imply that 71'f has an absolute value in Lb (A), which coincides with its absolute value in p (A). We collect the latter observations for commutative d-algebras. THEOREM 4.2. Let A be an Archimedean commutative d-algebra. Then p (A) is an Ar'chimedean f -algebra when equipped with the addition and composition of operators, and the ordering inherited from Lb (A). Moreover, the absolute value 71'tft of 71'f in p (A) coincides with the absolute value of 71' f in Lb (A) for all f E A. that is, 17I'fl (g) = 71'tft (g) = sup {17I'f (h)1 : Ihl ::; g} for all 9 E A+.

We obtain the f-algebra case as a corollary. COROLLARY 4.3. Let A be an Archimedean f-algebra. Then p (A) is an fsubalgebra of the Archimedean f-algebra Orth (A) of all orthomoTphisms of A.

The fact that the range of p in Corollary 4.3 is an f -algebra was first proved in [22, Corollary 3, p. 57] by Birkhoff and Pierce, while the fact that Orth (A) itself is an f-algebra has been proved in [18] by Bigard and Keimel and in [39] by Conrad and Diem. This topic was also discussed in great detail by de Pagter in his thesis [76, Proposition 12.1]. Note that if A is an f-algebra then A is semiprime if and only if p is one-to-one as a map from A into Orth (A). In this case, A and p (A) are isomorphic a.'! f-algebras. Also, if A is an f-algebra then A has a multiplicative identity if and only if the map p is one-to-one and onto as a map A ---> Orth (A), and consequently A and Orth (A) are isomorphic as f-algebra.'!.

5. Uniform completion and Dedekind completion Let A be an Archimedean i-algebra. The closure A1'U of A in its Dedekind completion A8 with respect to the uniform topology is a uniformly complete Riesz space. Using Quinn's Definition 2.12 in [79], A1'U is the uniform completion of A. The following theorem was obtained by n'iki in [91]. THEOREM 5.1. Let A be an Archimedean f-algebra (respectively, almost falgebra, d-algebra, f -algebra). Then the multiplication in A extends uniquely to a multiplication in A1'U such that A1'U is a uniformly complete i-algebra (respectively, almost f -algebra, d-algebra, f -algebra) with respect to this extended multiplication. Moreover, if A is semiprime (respectively, has a unit element e) then ATU is semiprime (respectively, has e as unit element).

We now turn our attention to the Dedekind completion of the i-algebra A. Johnson in his paper [64] proved that if A is an f-algebra (or even an Archimedean f-ring), then the multiplication in A extends uniquely to an f-algebra multiplication in A8. The uniqueness of such an extended multiplication in A8 of course arises from the order continuity of the multiplication in the f-algebra A. Alternative proofs of this extension can be found in [76, pp. 66-67] and [59, p. 166].

108

KARIM BOULABIAR. GERARD BUSKES. AND ABDELMAJID TRIKI

THEOREM 5.2. Let A be an Archimedean f -algebra. Then the multiplication in A extends uniquely to a multiplication in A" such that Ad 'is a Dedekind complete falgebra with respect to this extended multiplication. FUrthermore, if A is semiprime (respectively, A has a unit element e) then A" is semiprime (respectively. has e as unU element).

The corresponding results for (I-algebras in general, or almost f-algebras and d-algebras in particular, is much harder because of the absence of order continuity of the multiplication. Nonetheless, extensions of the multiplication to the Dedekind completion often exist, though such extensions are no longer necessarily unique. For almost f-algebras Buskes and van Rooij proved the following (see [35, Theorem 10]). THEOREM 5.3. Let A be an Archimedean almost f-algebra. Then the multiplication in A extends to a multiplication 'in A" such that A" is a Dedekind complete almost f -algebra with respect to that extended m'ultiplication.

Using the previous result as a starting point, Boulabiar and Chil in [28, Corollary 3] proved that from amongst the extensions provided, A" can be equipped with a commutative d-algebra multiplication whenever A is a commutative d-algebra. Then in [37, Theorem 7], Chil wa.c; able to drop the commutativity condition and prove the following theorem. THEOREM 5.4. Let A be an Archimedean d-algebra. Then the multiplication in A extends to a multiplication in Ad such that A" is a Dedekind complete d-algebra with respect to that extended multiplication.

In summary, all but one of the problems concerning Dedekind completions that Huijsmans raised in his survey paper [57] have now been solved. The remaining problem, though admittedly outside the scope of this survey, is the following. PROBLEM 5.5. Let A be an Ar'chimedean (I-algebra. Does the multiplication in A extend to a multiplication in A" so that A" is a Dedekind complete (I-algebra?

6. Powers in i-algebras Let A be a uniformly complete i-algebra and let P E lR+ [Xl, ... , Xn] be a homogeneous polynomial of degree a non zero natural number p. In their paper [16]' Beukers and Huijsmans considered the following problem: does there exist in A a 'p-th root' of P(iI, ... ,fn) for iI, ... ,fn in A+? They gave an affirmative answer in the case where A is a semiprime f-algebra. More precisely, they prowd the following theorem (see [16, Theorem 5]). THEOREM

6.1. Let A be a uniformly complete semiprime f-algebra and let

P E lR+ [Xl, ... , Xn] be a homogeneo1Ls polynomial of degree a non zero natural numbe1'p. Thenfo1' every fl, ... ,fn E A+ there exists a 'unique f E A+ such that fP = P (iI, ... , fn). As a consequence, one has the following corollary (see Corollary 6 in [16]). COROLLARY 6.2. Let A be a uniformly complete f -algebra with unit element and p E {I, 2, ... }. Then for each f E A +, there exists a unique 9 E A + such that

gl'

= f.

LATTICE ORDERED ALGEBRAS

109

Note that the previous result was first proved for p = 2 in [17, Theorem 4.2 and Cororllary 4.3] by Beukers, Huijsmans and de Pagter. Also, it should be noted that Theorem 6.1 above is proved alternatively by Buskes, de Pagter, and van Rooij in [31, Corollary 4.11], a paper that deals with a more general functional calculus on Riesz spaces and f-algebrar; to which we will return in the next section. The problem corresponding to Theorem 6.1 for almost f-algebras was considered by Boulabiar and follows next (see Theorem 3 in [24]). jR+

THEOREM 6.3. Let A be a uniformly complete almost f -algebra and let P E [Xl, ... , Xn] be a homogeneous polynomial of degree a natural number p. Then

for every II, ... , fn E A+ there exists a (not necessarily unique) f E A+ such that fP = P (II, ... , fn)·

Observe that, where roots are unique in semiprime f-algebras, this is no longer always the case for almost f-algebras. We illustrate this with an example. EXAMPLE 6.4. Let A = C ([-1,1]) be the uniformly complete Riesz space of all real-valued continuous functions on [-1, 1] and define w E A by

W(x)={ O-x

(-1::;x::;0); (0::;x::;1).

For every f, g E A, we put

(f. g) (x) =

{

w(x)f(x)g(x)

(-1::; x::; 0);

lO/(S)g(S)dS

(0::; x::; 1).

Then A is an almost f -algebra with respect to the multiplication.. h, g, Ct, {3 E A defined by g(x) =

Ixl

h (x)

Consider

= exp (x),

and Ct

(x)

= Jx 2 + exp (2x)

, and {3 (x)

for all x E [-1,1]' where X[O,I] (x) Then Ct • Ct

=1

= X.X[O,I] (x) + Jx 2 + exp (2x)

if x E [0,1] and X[O,I] (x)

=0

if x E [-1,0).

= {3 • {3 = 9 • g + h • h.

At this point, we define for each non zero natural number p, Ap =

{II ... fp : II, ... , fp

E A}.

In what follows, we will investigate the order structure as well as the algebra structure of Ap (since Al = A, we suppose that p ~ 2). The sets A2 and A3 were first considered in [35] by Buskes and van Rooij and then in great detail by Boulabiar in [24] from which we summarize the results in the following theorem (see Theorems 4, 5, and 6 in [24]). THEOREM 6.5. Let A be a uniformly complete almost f -algebra and let p ~ 3 be a natural number. Then Ap is a uniformly complete semiprime f -algebra under the ordering and multiplication inherited from A. The positive cone At of Ap is defined by

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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

The lattice operations I\p and V p in Ap are given by

fP I\p gP = (f 1\ g)P and the absolute value

1.l p

and

fP

Vp gP

= (f V g)P

for all 0 ~ f,g E A,

in Ap is defined by If Pip

=

Ifl P

for all f E A.

Contrary to Ap (p 2: 3), A2 need not be a Riesz space under the ordering inherited from A as is proved by the next example. EXAMPLE 6.6. Consider A = C ([0,1]) with the pO'intwise addition, scalar rrmltiplication and partial ordering. For f, 9 E A, define

l

(f _ g)(x) = {

(0 x - 1/2

~

x ~ 1/2);

0

f(s)g(s)ds (1/2 < x ~ 1). o Then A is a uniformly complete almost f -algebra under the multiplication - and h is an element of A2 if and only if h(x) = 0 for all x E [0,1/2] and the restriction of h to [1/2,1] belong to C 1 ([1/2,1]). Hence A2 is not a Riesz space under the order inherited from A.

The following example proves that though Ap (p 2: 3) is a Riesz space, in general it is not a Riesz subspace of A. EXAMPLE 6.7. Take A = C ([-1, 1]) with the pointwise addition, scalar multiplication and ordering, and define w EA· by

w (x)

=

{ -x 0'

(-l~x~O);

(0 ~ x ~ 1) .

For f, 9 E A, define

(f - g) (x) =

{

w(x)f(x)g(x)

(-l~x~O);

10/(S)g(S)ds

(O~x~l).

Clearly, A is a uniformly complete almost f -algebra under the multiplication _. Define 0 E A by o (x)

= 2x + 1

for all x E [-1, 1].

It follows that

10 - 0 - 01 (1) =

1/10

=110 - 0 - ob (1) = (101-101-101) (1) =

1/8.

If, however, A is a commutative d-algebra then some of the unpleasantness of the preceding example disappears. COROLLARY 6.8. Let A be a uniformly complete commutative d-algebra and p 2: 2 be a natural number. Then Ap is a uniformly complete f -subalgebra of A. If in addition p 2: 3 then Ap is semiprime.

In spite of the improvement in the conclusion of Corollary 6.8 over the conclusion for the more general situation of almost f-algebras, A2 still need not be semiprime. This is illustrated in the next example.

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111

EXAMPLE 6.9. Let A be the coordinatewise ordered vector space R3 with the multiplication defined by:

n; R}

Then A is a uniformly complete commutative d-algebm and

A,

~{(

X,Y E

Dbuiou,ly, A, is an f-""balg,bm of A. FUrth,""ore, (

~ ) ' ~ 0 and A, is not

semiprime. For f -algebras we have the following corollary. COROLLARY 6.10. Let A be a uniformly complete f -algebm and let p 2: 2 be a natuml number. Then Ap is a uniformly complete semiprime f -subalgebm of A.

There is a universal way in which A 2 , or more generally Ap for any p 2: 2 can be described. We provide the details of that description for A2 next (see [36]). Let E and F be Riesz spaces. A bilinear map q> : E x E ~ F is called orthosymmetric if whenever f I\g = 0 for f,g E E we have q>(f,g) = 0 (the notion of orthosymmetric bilinear map was introduced by Buskes and van Rooij in [34]). The bilinear map q> is a Riesz bimorphism if it is a Riesz homomorphism in each variable separately (more about Riesz bimorphisms can be found in [29]). Let E be a Riesz space. The pair (E8, 8) is called a square of E, if E8 is a Riesz space and if (1) 8: E x E ~ E8 is an orthosymmetric Riesz bimorphism, and (2) for every Riesz space F, whenever q> : E x E ~ F is an orthosymmetric Riesz bimorphism there exists a unique Riesz homomorphism q>8 : E8 ~ F such that q>8 08 = q>. The existence and uniqueness of squares for any Riesz space follows easily from the Riesz space tensor product as constructed by Fremlin in [43]. To understand the structure of the square of a Riesz space is best not done via this tensor product. The set A2 described above is often more helpful. The connection between semi prime f-algebras and squares of uniformly complete Riesz spaces is described in the next theorem. After reading that theorem the reader might feel like moving the lower index 2 in A2 to an upper index. THEOREM 6.11. Let E be a uniformly complete Riesz subspace of an Archimedean semiprime f -algebm G whose multiplication is indicated by a period e. Put E2 := {x e y : x,y E E} as before. Then E2 is a Riesz subspace of G and (E2,e) is a square of E.

7. Functional Calculus on f-algebras The theorem that we presented in Section 6 on the existence of p-th roots of homogeneous polynomials in f-algebras is a very special case of a rich functional calculus on uniformly complete f-algebras. The idea behind functional calculus

112

KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

for Riesz spaces in general is straightforward. For elementary functions on JRN one ought to be able to simply substitute elements of the Riesz space into these functions and get elements of the Riesz space as output. The idea of how to execute this substitution of elements in sufficiently simple functions essentially goes back to Yudin and in the form that we represent it to Lozanovsky [70]. The technical problem surmounts to what the class of sufficiently simple functions really looks like. Let n EN. We denote by 1i(JRN) the Riesz space of all continuous functions r.p : JRN ---+ JR for which

r.p(tx) = tr.p(x) for all x E JRN and all t ~ O. Let E be a Riesz space, r.p E 1i(JRN) and

iI, ... , fn

E E. We say that

r.p(iI, ... '/n) exists in E if there is an element 9 of E such that

w(g) = r.p(w(iI), ... ,w(fn)) for every real-valued Riesz homomorphism w on the Riesz subspace of E generated by iI, ... , f n, g. For any given E, r.p and iI, ... ,/n there exists at most one 9 with this property. This 9 is also indicated by

r.p(iI, ... , fn). In this situation we have the following theorem (see Lozanovsky [70]). THEOREM 7.1. Let E be a uniformly complete Riesz space and Then r.p(iI, ... , fn) exists fOT every r.p E 1i(JRN). The map

iI, ... , fn

E

E.

r.p(iI, ... , fn) (r.p E 1i(JRN)) is a Riesz homomorphism from 1i(JRN) into E. r.p

---+

Remark. In a way, r.p(iI, ... , fn) is independent of E. Indeed, if D is any Riesz subspace of E that is uniformly complete and contains iI, ... , fn then r.p(iI, ... , fn) relative to D means the same as r.p(iI, ... , fn) relative to E. In particular, every Riesz subspace of E that is uniformly complete and contains iI, ... , fn must also contain r.p(fl, ... , fn). By A(JR N) we denote the set of all continuous functions r.p : JRN ---+ JR that are of polynomial growth and for which limt!o rlr.p(tx) exists uniformly on bounded subsets of JRN (the latter condition is equivalent to the existence of a 't/J E 1i(JRN) such that r.p(x) = 't/J(x) + 0(11 x II) (x ---+ 0)). Observe that A(JRN) is an f-algebra. Let E be a semiprime f-algebra, r.p E A(JRN) and iI, ... , fn E E. We say that

r.p(iI, ... ,fn)exists in E if there is agE E with

w(g) = r.p(w(iI), ... ,w(fn)) for every real-valued multiplicative Riesz homomorphism w defined on the f-subalgebra of E generated by iI, ... , fn,g. There exists only one such g, which is then called r.p(iI, ... , fn). This definition is in accordance with the one we gave for 1i(JRN) if r.p E 1i(JRN). For 1i(JRN) we have the following theorem (see [31, Theorem 4.10]).

LATTICE ORDERED ALGEBRAS THEOREM

ft, ... , fn

113

7.2. Let E be a uniformly complete semiprime f-algebra and let

E E. Then cp(ft, ... , fn) exists for every cp E A(~JII). The map

cp-+cp(ft, ... ,fn) (cpEA(~JII)) is a multiplicative Riesz homomorphism from A(~JII) into E.

8. Relationships between i-algebra multiplications

Let A be an i-algebra with multiplication denoted by juxtaposition, and assume that A is equipped with another associative multiplication e. In the first theorem of this section, we present a relationship between the two multiplications in A, under the conditions that A is a unital f-algebra with respect to the initial multiplication and an (almost) f-algebra with respect to the other multiplication e. For proofs, see [38, Theorem 2.2] and [23, Theorem 5.2]. THEOREM 8.1. Let A be an Archimedean f -algebra with identity element e and assume that A is furnished with another associative multiplication e. Then (i) A is an f-algebra under e if and only if

feg=(eee)fg

for allf,g E A, and

(ii) A is an almost f-algebra under e if and only 'if f e9

= e e (I g)

for all

J, 9 E A.

The corresponding problem in the case where A is d-algebra with respect to e is rather more difficult. Indeed, since then e need not be commutative, one cannot write the product f e 9 as a function of the product fg (I, 9 E A). However, there exists another way (involving f, 9 and the initial multiplication in A) to express the product f e g. This is the subject of the next result. First recall that the maximal ring of quotients Q (A) of the Archimedean f -algebra A with unit element e is again an Archimedean J-algebra with the same e as multiplicative identity. Moreover, A is an f-subalgebra of Q (A), a fact proved by Anderson in [4] (see also the recent paper [71, Cororllary 2.7.1] by Martinez). For the definition of the maximal ring of quotients of a ring, the reader can consult e.g. [66]. The proof of the following theorem can be found in [25, Theorem 4.3]. THEOREM 8.2. Let A be an Archimedean f-algebra with 'identity element e and let e be another associat'ive multiplication in A. Then A is a d-algebra with respect to e if and only if there exist two algebra and Riesz homomorphisms cp and 'l/J from A into its maximal ring of quotients Q (A) such that

f e 9 = (e e e) cp (I) 'l/J (g)

for all f, 9 E A.

As mentioned at the end of Section 3, the two preceding theorems are abstract versions of the corresponding results, given in that section, for the C (X)-case. In the second part of this section, we are interested in A being a commutative dalgebra with respect to the initial multiplication rather than an f-algebra with unit element. However, we will impose the additional assumption that A is uniformly complete. Uniform completeness is not needed for all our results but we will use the set A2 and remind the reader of the special nature of that set under the extra condition of uniform completeness (see Corollary 6.8). If there exists a positive operator T OIl A2 such that

(T)

f e9 =

T (lg)

for all

f, 9 E

A

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KARIM BOULABIAR, GERARD BUSKES. AND ABDELMA.HD TRIKI

then e is an almost f-algebra multiplication and N2 (A) c N; (A), where

N; (A)

= {f

E A :

f e f = O} .

In what follows, we show in detail what happens if we assume that A with the initial multiplication is a (uniformly complete) commutative d-algebra, and the inclusion N2 (A) c N; (A) holds. Under those circumstances, we then relate a necessary and sufficient condition for the new multiplication to be an almost f-algebra, a d-algebra or an f-algebra to the existence of some posit.ive operator T satisfying the relation (T). The details follow in the next theorem, the proof of which can be found in [23, Theorems 5.4 and 5.5]. THEOREM 8.3. Let A be a uniformly complete commutative d-algebra and assume that A is an I!-algebra with respect to another associative multiplication e such that p = 0 implies f e f = O. Then the following statements hold.

(i) A is an almost f -algebra under

e if and only if ther-e exists a positive operator T from A2 into A such that

feg=T(fg)

forallf,gEA.

(ii) A is a commutative d-algebra under

e if and only if there exists a Riesz homomorphism T from A2 into A such that

f e 9 = T (fg)

for all f, 9 E A.

(iii) A is an f-algebra under e if and only if there exists an operator T from A2 into A such that To 7rf E Ort.h (A) for all f E A+, where 7rf (g) = fg for all 9 E A, and f e9

=T

(f g)

for all f, 9 E A.

We remark that A2 in the previous theorem is a Riesz space (see Section 6). Next, we produce an example which shows that in Theorem 8.3 above, the hypothesis 'A is a commutative d-algebra' cannot be replaced by 'A is an almost f-algebra'. EXAMPLE 8.4. Take A = C ([-1,1]) with the usual operations and order and define a, (3 E A by

-(4x+l) a(x)= { 0 4x -1

(-I~x~-1/4); ~ x ~ 1/4); (1/4 ~ x ~ 1)

(-1/4

and (3 (x) = {

~4x + 1

(-1 ~x~ 1/4); (1/4 ~ x ~ 1).

For f, 9 E A, define a(x)f(x)g(x) (f x g)(x) =

{

(-1~x~1/4); ~ 3/4);

(1/4 ~ x °J(X-3/4)

f(s)g(s)ds

(3/4~x~1)

-(x-3/4)

and (feg) (x) =f3(x)f(x)g(x)

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115

for all x E [-1,1]. Then A is an almost f-algebra (respectively, an f-algebra) with respect to the multiplication x (respectively, .). It follows that N; (A)

= N 2x

(A)

= {f

E A : f (x)

= 0 for

all x E [-1, I/4]}.

Consider <.p, w E A, defined by

"'(X)~{ and oJ (x)

~

-3x-I x -3x+I

(-1 ::=;x::=; -1/4); (-I/4::=; x::=; 1/4); (i/4::=;x::=;I)

0 4x+ 1 { -4x+I 0

(-I::=;x::=;-I/4); (-I/4::=; x::=; 0); (0::=;x::=;I/4); (i/4::=;x::=;I).

Then and 'Pxw=O 'P. W =f:. O. Therefore, there is no operator T satisfying the condition f • g = T (f x g)

for all f, 9 E A.

We notice that the kind of relationships between two f-algebra multiplications via an operator T as above were first studied in [35] by Buskes and van Rooij. In particular part (i) of the theorem above has its origins in [35, Theorem 1], where a general representation theorem for almost f-algebras (without any extra conditions) was proved by first dividing out the radical and then showing the existence of an operator T a.'l above on the square (see the end of Section 6 above) of the resulting f-algebra, thus giving quantified credence to the almost part in the name almost f-algebras. They also used this representation to discover that the Dedekind completion of an almost f-algebra is an almost f-algebra and to prove (CSP) for almost f-algebras. Finally, we should point out that the Buskes-van Rooij representation theorem for almost f-algebras itself is, in a way, an abstract reformulation of Theorem 3.2 by Scheffold that we discussed earlier. We end this section with the following problem, left open in the above discussion. PROBLEM 8.5. Let A be a uniformly complete commutative d-algebra and assume that A is a non comm'utative d-algebra with respect to another multiplication •. Does there exists a relationship between the two multiplications in A?

9. Connection between algebra and Riesz homomorphisms Since in any f-algebra there is an order structure as well as an algebra structure, it is natural to compare the operators that preserve the lattice operations, namely the Riesz homomorphisms, with those that preserve the algebra structure, the algebra homomorphisms. Such a comparison was initiated by Ellis [41], who considered the problem for operators between spaces of continuous functions on compact Hausdorff spaces, and his central result is then one of equivalence: a Markov operator (Le., a positive operator preserving the identity) between such spaces is a Riesz homomorphism if and only if it is an algebra homomorphism. Some years. after Ellis's paper was published, Hager and Robertson presented in [48] a more abstract version of Ellis's theorem. More precisely, Hager and Robertson proved that any

116

KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI

Riesz homomorphism between two Archimedean f-algebras with unit elements that preserves the identity is an algebra homomorphism. Later, van Putten established the converse of the result by Hager and Robertson in his thesis [78]. The aforementioned results were generalized by Huijsmans and de Pagter in [59, Theorem 5.4] as follows. THEOREM 9.1. Let A be an Archimedean f-algebra with unit element e, B be an Archimedean semiprime f-algebra. and T : A ---> B be a positive operator. Then the following are equivalent (i) T is an algebra homomorphism. (ii) T is Riesz homomorphism with (Te)2 = Te.

Notice that the previous theorem generalizes all of the facts cited above. In the same paper [59], Huijsmans and de Pagter proved that., if A is an Archimedean falgebra with unit element and B is an Archimedean semiprime f-algebra then any order bounded algebra homomorphism from A into B is automatically a Riesz homomorphism [59, Theorem 5.3]. Very recently in [91, Theorem 4.3], Triki obtained this result in the more general setting of almost f-algebras. THEOREM 9.2. Let A be an Archimedean almost f-algebra and let B be an Archimedean semiprime f -algebra. Then any order bounded algebra homomorphism from A into B is a Riesz homomorphism.

Since any conmmtative d-algebra is automatically an almost f-algebra, Theorem 9.2 holds if one replaces 'A is an Archimedean almost f-algebra' by 'A is an Archimedean commutative d-algebra'. However, in the recent. work [89], Toumi proved that we have the same conclusion even if the d-algebra under consideration is not commutative. THEOREM 9.3. Let A be an arbitrary Archimedean d-algebra and let B be an Archimedean semiprime f-algebra. Then any order bounded algebra homomorphism from A into B is a Riesz homomorphism.

Even for f-algebras, the condition of order boundedness in Theorems 9.2 and 9.3 cannot be dropped as is shown in the following example. EXAMPLE 9.4. Consider the set A of all real sequences u = {un} n>l for which there exists a polynomial Pu E IR [X] and a natural number N such that Un = Pu (n) for all n 2: N. Under the usual operations and partial ordering, A is an Archimedean f-algebra (and thus an almost f-algebra and a d-algebra) and A is not relatively uniformly complete. Define the algebra homomorphism T : A ---> IR by T(u) = Pu (-1) (u E A). For every A E [l,+po[, we define u), = {U),.n}n~l E A by

u)',n =

{ 0An

(n ~ A);

(n> A) .

lfu = {n 2 } n>l then 0 ~ u), ~ u for all A E [1, +00[. Observe now that T (1L),) Therefore T 1:s not order bounded and hence not a Riesz homomorphism.

= -A.

In [59, Theorem 5.1], Huijsmans and de Pagter further illustrated the close ties between Riesz homomorphisms and algebra homomorphisms on f-algebras as follows. If A and Bare semiprime f-algebras and A is uniformly complete, then any algebra homomorphism from A into B is a Riesz homomorphism. Their theorem was generalized by Boulabiar in [27, Theorem 3] to the setting of almost f-algebras.

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117

THEOREM 9.5. Let A be a uniformly complete almost f -algebra and let B be an Archimedean semiprime f-algebra. Then any algebra homomorphism from A into B is a Riesz homomorphism.

Trivially, one can replace in Theorem 9.5 above 'almost f-algebra' by'commutative d-algebra'. The case of d-algebras that are not necessarily commutative was considered by Toumi in [89, Theorem 3]. THEOREM 9.6. Let A be an arbitrary uniformly complete d-algebra and let B be an Archimedean semiprime f-algebra. Then any algebra homomorphism from A into B is a Riesz homomorphism.

The condition that A is uniformly complete is not redundant in Theorems 9.5 and 9.6, even if A is an f-algebra. Indeed, A in Example 9.4 above is not uniformly complete. The following theorem by Triki in [91, Theorem 4.4] is an f-algebra version of the well known Nagasawa's theorem (see [74, Theorem 1]). First, recall that an operator T between two unital f-algebras A and B is said to be contractive if IT (J)I ~ eB in B whenever If I ~ eA in A, where eA and eB are the unit elements of A and B, respectively. If moreover T is bijective and the inverse T- 1 of T is also contractive then we say that T is bicontractive. THEOREM 9.7. Let A and B be Archimedean f-algebras with identity elements eA and eB, respectively. For an order bounded bijection T : A ----t B such that T (eA) = eB, the following are equivalent. (i) T is an algebra homomorphism. (ii) T is a Riesz homomorphism. (iii) T is bicontractive.

The last result of this section again deals with operators between two unital falgebras that preserve identities, so called Markov operators. Denote by M (A, B) the set of all Markov operators from an f-algebra A with unit element eA into an f-algebra B with unit element eB, that is, M (A,B)

= {T:

A

----t

B: T linear positive with T(eA)

= eB}.

Obviously, M (A, B) is a convex set. In [77, Theorem 2.1], Phelps proved that a Markov operator T from C (X) into C (Y) (where X and Yare compact Hausdorff topological spaces) is an algebra homomorphism if and only if T is an extremal point in M (C (X) ,C (Y)). This connection to extreme points predates the paper by Ellis and we refer the reader to Arens and Kelley [8] and A. and C. Ionescu Thlcea [62]. Combining the aforementioned Phelps's theorem and Ellis's result cited above we get that for T in M (C (X), C (Y)) the following are equivalent. (i) T is a Riesz homomorphism. (ii) T is an algebra homomorphism. (iii) T is an extremal point in M (C (X) , C (Y)). As a generalization, van Putten in his thesis obtained the following result [78, Theorem 18.8]. THEORSM 9.8. Let A and B be Archimedean f-algebras with unit elements, and let T E M (A, B). Then the following are equivalent. (i) T is a Riesz homomorphism.

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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

(ii) T is an algebra homomorphism, (iii) T is an extremal point in M (A, B). An elementary proof of van Putten's theorem above, due to Huijsmans and de Pagter, can be found in [59, Theorem 5.7].

10. Positive derivations We recall that an operator D on a commutative algebra A is said to be a derivation if for all f,g EA. D (1g) = f D (g) + gD(1) In this section, we investigate positive derivations on f-algebras as well as on almost f-algebras. Positive derivations on f-algebras were first considered in great detail by Colville, Davis and Keimel in [40]. Their main result provides the following necessary and sufficient condition for a positive operator on an f-algebra to be a derivation (see Theorem 5 in [40]), where we recall that if p is a non zero natural number, and A is an algebra then Ap = {11 ... fp : 11, ... , fp E A}. THEOREM 10.1. Let A be an f-algebra and D be a positive operator on A. Then D is a derivation if and only if

D (1) = 0 for all f E A2

and

D (1)2 = 0 for all f E A.

Positive derivations on f-rings were considered by Henriksen and Smith in [54]. It straightforwardly follows from Theorem 10.1 above that, if A is in addition semiprime, then there exist no non trivial positive derivations on A. We turn our attention now to positive derivations on almost f-algebras. The result corresponding to Theorem 10.1 for almost f-algebras is the following (for a proof, see [26, Theorem 3]). THEOREM

10.2. Let A be an almost f-algebra and D be a positive derivation

on A. Then D (1)

= 0 for

all f E A3

and

D (1)3

= 0 for

all f E A.

Contrary to f-algebras, Theorem 10.2 does not produce a characterization of positive derivations on almost f-algebras. Also, the third power in Theorem 10.2 is the best possible. The next example illustrates these facts. EXAMPLE 10.3. Consider A the Cartesian product IR x IR with coordinatewise addition, scalar multiplication and ordering. Define the multiplication. on A by

for all 0,/3,0',/3' E R

(0,/1). (0',/3') = (0,00')

Then A is an Archimedean almost f -algebra with respect to.. Observe that A3 fA satisfies

=

{o}. Hence the identity map fA

(1) = 0 for all f E A3

and

fA

(1) 3 = 0 for all f E A,

but fA is not a derivation on A. On the other hand, let D be the positive operator defined on A by for all a, /3 E R D(o,/3) = (0,2/3) Clearly, D is a derivation on A. However, D ((1, 0) • (1,0))

= (0,2)

=1=

(0,0)

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119

and

D (1, 0) • D (1, 0) = (0,1) =I- (0,0) . Theorem 10.2 above of course holds for commutative d-algebras as well. Moreover, for commutative d-algebras the third power is the best possible too. Indeed, the almost f-algebra considered in Example 10.3 is in fact a commutative d-algebra.

11. Cauchy-Schwarz inequalities We say that an i-algebra A possesses the Cauchy-Schwarz property (abbreviated as (CSP)), if for every vector space V and every bilinear map 'l/J : V x V -> A such that (i) 'l/J is symmetric, and (ii) 'l/J (I, f) E A+ for all f E V, we have (CSI)

'l/J(I,g)2 S'l/J(I,f)'l/J(g,g)

for all f,g E V

In [61, Corollary 3.5], Huijsmans and de Pagter proved that any Archimedean semiprime f-algebra has (CSP). Later, Bernau and Huijsmans in [15, Theorem 2.6] generalized (CSP) to arbitrary Archimedean f-algebras. THEOREM 11.1. Let A be an Archimedean f-algebra, V a vector space and 'l/J : V x V -> A a bilinear map such that (i) 'l/J is symmetric, and (ii) 'l/J (I, f) E A+ for all f E V . Then

'l/J (I, g)2 S 'l/J (I, f) 'l/J(g, g)

for all f,g E V.

Some years after Theorem 11.1 was published, Buskes and van Rooij established the corresponding inequality for Archimedean almost f-algebras and therefore for Archimedean commutative d-algebras (see [34, Corollary 4]). THEOREM 11.2. Let A be an Archimedean almost f-algebra, V a vector space and'l/J : V x V -> A a bilinear map such that (i) 'l/J is symmetric, and (ii) 't/J (I, f) E A + fOT all f E V . Then

'1/) (I, g)2 S .t/J (I, f) ,¢'(g, g)

for all f, g E V.

An alternative proof of the previous inequality was given by Boulabiar in [23, Theorem 3.9]. Not every d-algebra A has (CSP). An example illustrating this situation is the following. EXAMPLE 11.3. Let A be the set of all real-valued functions defined on [0,1] equipped with the usual operations and order. Consider the multiplication. defined in A by (I )() - { f (0) g (1) (0 S x S 1/2) ; .g x f(l)g(l) (1/2SxS1)

for all f, g E A. Then A is an Archimedean non-commutative d-algebra with r-espect to the multiplication •. At this point, let V be the f-algebra C ([0,1]) of all real-valued continuous functions on [0, 1] , provided with the pointwise addition, multiplication, scalar multiplication and partial ordering. We define a positive operator T from V into A by T (I) (x) = f (1 - x) for all f E V and x E [0,1]. Finally, let f E V such that f (0) = 2, f (1) = 1 and e E B such that e (x) = 1 for all x E [0,1]. Then (T (Ie) • T (Ie)) (0)

= (T (I) • T

(I)) (0)

=f

(0) f (1)

=2

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KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI

and Hence the inequality

T (fe). T (fe) :s T (J2) • T (e 2) does not hold in A. By defining '!/J(f,g) = T(fg) (f,g E V), we find a bilinear map V x V --t A for which 1jJ (f, e)2 :s 1jJ (f, f) 1jJ(e, e) does not hold. Though (CSP) does not hold for all non commutative d-algebras, Boulabiar and Toumi proved, in essence, the following variant of (CSP) in such algebras (see [29, Theorem 6]). THEOREM 11.4. Let V be a vector space and let A be an Archimedean d-algebra. Consider a bilinear map '!/J : V x V --t A such that (i) 1jJ is symmetric, and (ii) 1jJ (f, f) E A+ for all f E V. Then

11,b(f,g)21:s

~[(1jJ(f,f)1jJ(g,g))+('!/J(g,g)1jJ(f,f))]

and hence 11jJ(f,g)21:s (1jJ(f,f)'!/J(g,g)) V (l/J(g,g)'!/J(f,f)) for all f,g E V. 12. Order biduals We refer the reader to [2] for terminology and notations not explained below. For an Archimedean C-algebra A, the order dual is denoted by A~ and the order bidual is denoted by A~~. Recall here that the order dual A~ of A is the Dedekind complete Riesz space of all real-valued order bounded functionals on A. A multiplication can be introduced in A~~ in three steps as follows: for all u, v E A, f E A~ and cp, '!/J E A~~, we define f.u E A~, '!/J.f E A~ and cp.1jJ E A~~ by the following equations

(1)

(f:u) (v) = f (uv)

(2)

('!/J.f) (u) = '!/J (f.u)

(3)

(cp.'!/J) (f) = cp ('!/J.f)

The multiplication defined by the equation (3) is called the Arens multiplication in A~~ (see [6] and [7]). The order continuous order bidual (A~);: of A is the projection band of all order continuous elements in A~~. Moreover, (A~);: is closed under the Arens multiplication, that is, cp ..l/J E (A~);: whenever cp, '!/J E (A~);: . The following theorem was established by Huijsmans and de Pagter (see [60, Theorem 4.1]). 12.1. Let A be an Archimedean C-algebra. Then A~~ (and hence is a Dedekind complete C-algebra with respect to the Arens multiplication.

THEOREM

(A~);:)

e

If the f-algebra A has, in addition, a unit element e then E (A~);: (defined by e(f) = f (e) for all f E A~) is the unit element of A~~. Generally, A~~ need not be commutative even if A is commutative. An example was provided by Arens in [7]. However, commutativity does carryover from A to (A~);:, which was first established for normed C-algebras by Scheffold in his paper [82] and generalized by Grobler in [46, Theorem 4] to the more general setting of Archimedean C-algebras.

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121

THEOREM 12.2. Let A be an Archimedean i-algebra. Then the order continuous (A~);: , provided with the Arens multiplication, is commutative.

orde1' bidual

The first results concerning the order bidual of f-algebras are due to Huijsmans and de Pagter (see [60]). They proved, among other results, that if A is an f-algebra with unit element such that A~ separates the points of A (i.e., if u E A and f (u) = 0 for all f E A~ then u = 0) then A~~ = (A~);:. They deduced that A~~ then again is an f-algebra (of course, with unit element) with respect to the Arens multiplication (note that if A~ separates the points of A then A is Archimedean). Later Huijsmans was able to do away with the condition that A have a unit element, and he showed that the order bidual of an f-algebra with separating order dual is also an f-algebra with respect to the Arens multiplication (see [56, Theorem 2.8]). Then, without assuming any additional 'separation' condition, Bernau and Huijsmans proved in [15] that the order bidual of an Archimedean falgebra, equipped with the Arens multiplication, is again an f-algebra (see Theorem 3.2 and Theorem 3.5 in [15]). THEOREM 12.3. Let A be an Archimedean f-algebra. The order bidual A~~ of A, equipped with the Arens multiplication is an f-algebra and hence 'is commutative.

In the paper [14], Bernau and Huijsmans also dealt with the case of almost f-algebras. THEOREM 12.4. Let A be an Archimedean almost f -algebra. Furnished with the Arens multiplication, the order bidual A~~ of A is an almost f-algebra and hence is commutative.

Combining Theorem 12.4 and the fact that commutative d-algebras are almost f-algebras, Bernau and Huijsmans deduced the following result (see [14, Theorem 4.2]). COROLLARY

order bidual plication.

A~~

12.5. Let A be an Archimedean commutative d-algebra. then the of A is a commutative d-algebra with respect to the Arens multi-

It is interesting to observe that the square of the singular part of the bidual (the orthogonal complement of the order continuous part) vanishes in the case of an almost f-algebra. Contrary to the case of almost f-algebras, the corresponding problem for non commutative d-algebras remains open and only a partial result has been obtained, again by Bernau and Huijsmans in [14, Theorem 4.1]. Their result is the following. PROPOSITION 12.6. Let A be an Archimedean d-algebra. Then the order continuous order bidual (A~);: is a d-algebra with respect to the Arens multiplication.

As just observed, the question whether the full order bidual of a non commutative d-algebra is a d-algebra with respect to the Arens multiplication is still open. PROBLEM 12.7. Let A be an Archimedean d-algebra. Is the order bidual of A a d-algebra with respect to the Arens multiplication?

A~~

At the end of this section, we notice for the sake of completeness that all of the results above were obtained independently by Scheffold under the additional . assumption of A being a Banach lattice (including the result on d-algebras) (see [81] and [83]).

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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

13. Ideal theory In this section, we follow [69J in our terminology and notations. Recall that an order ideal in a Riesz space L is a vector subspace I of L with the extra condition III ::; Igl in Land gEl imply I E I. We call a ring ideal I in an i-algebra A an i-ideal if I is simultaneously an order ideal in A. Ideal theory in lattice ordered algebras comprises an investigation of ring ideals, order ideals and the connection between them. In the first part of this section, we focus on the following problem: (P 1) Under what conditions is any order ideal in an I -algebra a ring ideal? In the second part we consider the converse problem: (P2) Under what conditions is any ring ideal in an I -algebra an order -ideal? The first problem was initiated by Henriksen in [49J, who studied the ideal theory in I-rings by means of representations. In their work [58J on ideal theory in I-algebras, Huijsmans and de Pagter proved the next result by a representation-free approach (see [58, Proposition 3.1]). PROPOSITION 13.1. Let A be an Archimedean I-algebra. Then any unilormly closed order ideal is a ring ideal.

The problem (PI) for non uniformly closed order ideals was considered cxtensively by Basly and Triki in [l1J and [12J. In what follows, we present the major results they obtained in that direction. First, we need to recall some prerequisites. A ring ideal I in a commutative algebra A is said to be modular whenever there exists h E A such that I - I h E I for all I E A. In particular, any ring ideal in A is modular if A is in addition unital. From now on A is an I-algebra. We know that the map p: A ~ Orth(A)

I

I--->

7rf

(where 7rf (g) = I g for all I, g E A) is an algebra and Riesz homomorphism (see Section 4 above). An element I in A is said to be bounded if p (a) = 7ry. E Z (A), where Z(A) = {7r E Orth(A): 17r1::; >.IA for some real number A} is the centre of A (Le., the principal order ideal generated by IA in Orth (A)). We denote, after Triki (see [90]), the set of all bounded elements in A by Ab. Note that rather than A b , Henrikscn and Johnson use A* in [53J after a similar usage in Gillman and Jerison's book. Clearly, Ab is an I-subalgehra of A. The I-algebra A is said to be bounded if A = Ab, that is, if p (A) is a subset of Z (A). In particular, a unital I-algebra A is bounded if and only if its multiplicative identity is also a strong order unit, and in this situation A and Z (A) are isomorphic as f-algebras. Here we recall that the identity element in an Archimedean unital f-algebra is a weak order unit (see [22, p. 60]). The equivalence of (i) and (ii) in the next theorem is an immediate consequence of the definitions of an order ideal and a bounded Archimedean I-algebra, while for the proof of the rest of tllis theorem, we refer to Theorem 3 in [l1J and Theorem 4 in [12J. THEOREM 13.2. Let A be an Archimedean I-algebra. Then the following are equivalent. (i) Every order ideal is a ring ideal. (ii) A is bounded.

LATTICE ORDERED ALGEBRAS

123

If A is in addition uniforrn,ly complete then (i) above is equivalent to each of the following.

(iii) Every maximal modular ring ideal in A is uniformly closed. (iv) Every maximal modular ring ideal is the kernel of a Hiesz and algebra homomorphism from A onto JR. (v) Every maximal modular ring ideal is a maximal or'der ideal. We turn our attention to semiprime f-algebras. Let L be a Riesz space. A norm on a L is called a Riesz norm if IIfil ::; IIgil whenever If I ::; Igl in L. If a Riesz norm on L exists then L is said to be a normed Riesz space. If the normed Riesz space L is a Banach space as well, we say that L is a Banach lattice. We call the Riesz norm 11.11 on L an AI-norm if IIf V gil = max {IIfil , IIgll} for all f,g E L. An Ai-space is an At-normed Banach lattice. We can now state the following theorem (see [11, Theorem 5] and [12, Corollary 5]). 11.11

THEOREM 13.3. Let A be an Archimedean semiprime f -algebra. The following are equivalent.

(i) (ii) (iii) (iv)

Every order ideal in A is a ring ideal in A. A is a isomorphic as an f -algebra to a subalgebra of Z (A). There exists an M -norm in A. There exists a Riesz norm in A.

If, in addition, A is uniformly complete then each of (i), (ii), (iii) and (iv) above is equivalent to

(v) Every maximal modular ring ideal in A is uniformly closed. We note that Problem (PI) was extensively studied by Henriksen, Larson and Smith [52] in the context of f-rings. We move on to discuss Problem (P2). To this end, we need the notion of a normal Riesz space. The Riesz space L is said to be normal if L = {f+} d+{f-} d for allf E L, where {f+}d = {g E L: Igl.l\f+ = O} and {f_}d = {g E L: Igl.l\f- = O}. For a completely regular topological space X, the Riesz space C (X) is normal if and only if the sets P(f) = {x EX: f(x) > O} and N(f) = {x EX: f(x) < O} are completely separated for every fEe (X). For spaces of the type C (X), Problem (P2) has the following solution (see Theorem 14.24 in [47]). THEOREM 13.4. Let X be a completely regular (Hausdorff) topological space. Then the following are equivalent. (i) Every ring ideal in C (X) is an order ideal. (ii) Every finitely generated ring ideal in C (X) is a principal ring ideal (i.e., X is an F -space). (iii) C (X) is normal. Huijsmans and de Pagter considered (P2) in f-algebras. They only considered the unital case and their central result is the following generalization of Theorem 13.4 above (see [58, Theorem 6.6] for the proof). THEOREM 13.5. Consider the following conditions for an Archimedean f -algebra A with unit element.

(i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal.

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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

(iii) A is normal. (ii) (iii). If, in addition A is uniformly complete then (i) {:} (ii) {:} Then (i) (iii) .

'*

'*

Next we focus on the non unital case, which was considered by Triki in [90]. First we define the notion of a stable f-algebra. The f-algebra A is said to be stable if 1f (f) E (f) (where (f) is the principal ring ideal generated by f in A) for all 1f E Z (A). It is clear that A is stable if and only if 1f (1) c I for all ring ideals I in A and all 1f E Z (A). We now are in a position to present the result corresponding to Theorem 13.5 for semiprime f-algebras (see [90, Theorem 5.4]). THEOREM 13.6. Consider the following conditions for an Archimedean scmiprime f -algebra A. (i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal. (iii) A is stable and normal, and {f+} d or {f-} d is a modular ring ideal for all f E A. Then (i) (ii) (iii). Furthermore, if A is in addition uniformly complete then (i) ¢:> (ii) {:} (iii).

'*

'*

We proceed to the non semiprime case (see Theorem 5.5 in [90]). THEOREM 13.7. Let A be a non semiprime Archimedean f-algebra. Then the following are equivalent. (i) Every ring ideal is an orner zdeal. (ii) There exist a semiprime f-algebra B such that (a) every 7ing ideal in B is an order ideal, and (b) {b} d is a modular ring ideal in B for every b E H, so that A is isomorphic to the f -algebra B x JR endowed with the multiplication defined by (f, 0:) (g, 13) = (fg,O) for all i, g E B; 0:, 13 E JR.

Finally, note that the only i-algebras that we have considered in this section are i-algebras. Indeed, a Problem (PI) for more general i-algebras is futile since an i-algebra in which every order ideal is a ring ideal automatically is an f-algebra (see Page 144 in the classical book [42] or Proposition 1 in [12]). The situation for Problem (P2) is less clear. Indeed, in matrix algebras and algebras of formal power series in one variable, when ordered coordinatewise, every ring ideal is an order ideal. Thus we phrase Problem (P2) for lattice ordered algebras in general. PROBLEM

13.8. Study Problem

(P2)

for Archimedean i-algebras other than

i-algebras. 14. Representation of I-algebras

Let A be an f-algebra and recall that Ab denotes the i-subalgebra of all bounded elements in A (see Section 13 above). Several properties are satisfied by A if and only if they are satisfied by Ab and vice versa (see Sections 3,4 and 5 in [90]). Thus we can study some aspects of A via an investigation of Ab. For instance, if we assume that every ring ideal in A is an order ideal then A b has the same property and the converse holds if A in addition is uniformly complete (see [84] by Steinberg). In particular, if every ring ideal in A is an order ideal then order

LATTICE ORDERED ALGEBRAS

125

ideals and ring ideals coincide in Ab (see Theorem 13.2). It turns out that under the latter hypothesis, A b has a nice representation as a space of functions. This kind of a representation then is precisely the topic of this section. First assume that A has an identity element (Le., A is a
k (D) = n {M : M E D} (where it is understood that k (0)

= A). The

hull h (I) of an (-ideal I in A is

h(I) = {M E M (A): I

c M}. = h (k (D)).

The subset D of M (A) is said to be closed if D One thus defines the hull-kernel (or Stone) topology on M (A). It turns out that M (A) with respect to this hull-kernel topology is a compact Hausdorff space (see [53, Theorem 2.3]). Suppose at this point that A is in addition uniformly complete (instead of 'uniformly complete', Henriksen and Johnson in [53] use 'uniformly closed') and denote the identity element of A bye. Since A is unital, Ab is precisely the principal order ideal generated bye. Also recall that A and Orth (A) are isomorphic as f-algebra under the given condition. As usual, the f-algebra of all real-valued continuous functions on the compact Hausdorff space M (A) is denoted by C (M (A)). Henriksen and Johnson in [53, 3.2, p. 84] proved, using the Stone-Weierstrass theorem, the following variant of Stone's representation theorem. THEOREM 14.1. Let A be a uniformly complete f-algebra with identity element e. Then Ab and C (M (A)) are isomorphic as f-algebras. In particular, if e is a strong order unit in A then A and C (M (A)) are isomorphic as f-algebras.

The latter result can of course also be obtained via Kakutani's representation theorem. Indeed, being the principal order ideal in A generated bye, Ab is a uniformly complete Riesz space with e as a strong order unit. It follows that Ab is an M-space with respect to the M-norm 11.ll e defined by IIflle = inf {A > 0 : If I :'S Ae}

for all

f

E A

(see [72, Proposition 1.2.13]). Kakutani's representation theorem (see, for instance, [72, Theorem 2.1.3]) guarantees the existence of a compact Hausdorff space 0 so that Ab and C (0) are isomorphic as f-algebras and isometric as M-spaces. From an investigation of the cited proof of Kakutani's theorem, we see that 0 is the set of all algebra homomorphisms from Ab onto JR, or, equivalently, the set of all real valued Riesz homomorphisms that send e to 1. With respect to the weak topology a ((Abf ,Ab), where (Ab)'" is the order dual of Ab, the set 0 is a a ((Ab)'" ,Ab)-compact Hausdorff space. In view of Theorem 13.2, we observe that o is homeomorphic to M (Ab) and then to M (A) since M (Ab) and M (A) are also homeomorphic (see [53, Corollary 2.8]). We thus recover Theorem 13.1 above. The reader should also compare the above with Gelfand theory for Banach algebras

[94]. We proceed to representation theorems for non unital f-algebras. For the terminology and notations concerning the topological spaces under consideration, we refer the reader to the book [47] by Gillman and Jerison. The following two theorems deal with semiprime f-algebras (see Theorems 7.2 and 7.9 in [90]).

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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI

THEOREM 14,2, Let A be a uniformly complete semiprime f-algebra with a weak order unit, Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists an almost compact F -space 0 such that A b and Co (0) are isomorphic as f -algebras.

For almost compact. F-spaces, we refer to [47, 6J]. THEOREM 14.3. Let A be a semiprime Dedekind a-complete f -algebra. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exist a basically disconnected locally compact Hausdorff space 0 1 and a basically disconnected almost compact Hausdorff space O2 so that Ab is isomorphic as an f-algebra to one of the algebras

CK (Ot) ,Co (0 2 ) or CK (Od EB Co (0 2 )

.

Our next representation theorem represents non semiprime f-algebras (see [90, Theorem 7.10]). THEOREM 14.4. Let A be a uniformly complete non semiprime f-algebra A. Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists a locally compact F'-space 0 such that Ab is 'isomorphic as an f -algebra to the Cartesian product Co (0) x JR, where the multiplication in the f-algebra Co (0) x JR is defined by

(1,.x) (g, J.t)

= (1g,O)

for all f, g E Co (0) and.x, J.t E JR.

Note that the topological spaces 0 in Theorem 14.2 and in Theorem 14.4 are constructed in the same way as follows: for the uniformly complete f-algebra A (semiprime or not), the centre Z (A) is a uniformly complete f-algebra and its multiplicative identity IA is a strong order unit. According to Theorem 14.1, Z (A) is isomorphic as an f-algebra to the f-algebra C (M (Z (A))) of all real-valued continuous functions on the compact Hausdorff topological space M (Z (A)) (the set of all maximal f-ideals in Z (A)). It turns out that 0=

u {coz (1) : f

E C (M (Z (A)))).

In a similar way, the topological spaces 0 1 and O2 in Theorem 14.3 are constructed from a certain uniformly complete f-subalgebra of A (for more information, we refer to [90]). We end this section with two comments. Observe that representation theorems listed in this section apply to Ab and not to the whole f-algebra A. What one can say about A itself is the following. If A is unital then A can be embedded as an f-algebra into an algebra of extended functions on M (A) [53, Theorem 2.3], and if A is semiprime then A can be considered as an f-subalgebra of M (Orth (A)) (recall that if A is semiprime then A can be considered as an f-subalgebra of Orth (A)). In addition, the representation theorems that we presented in this section are not necessarily excessively restrictive due to the previously mentioned fact about transference of various properties of Ab to A. Our final comment about representing lattice ordered algebras deals with a matter of set theory. Zaanen started an ambitious program at around 1980, intending to prove all available material in vector lattices in as far as possible in an

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elementary way, i.e., without using representation theorems. A commonly quoted reason to adhere to Zaanen's program is the need to not use the Axiom of Choice unnecessarily. It is therefore interesting to know that one can avoid the Axiom of Choice and still use representation theorems, as long as the constructs that one has in mind depend on say countably many elements of a Riesz space. For f-algebras one of the most useful theorems in that direction is the following (combine Theorem 2.2 in [33] and Corollary 2.7 in [36]). THEOREM 14.5. Let A be a semiprime f-algebra. If D is a countable subset of A then the f -subalgebra generated by D in A can be r-epresented within ZermaloF'raenkel set theory as an f -subalgebra of the space of continuous functions on a metric space.

It should be noted at that the connection between the Axiom of Choice and representations of f-rings in terms of sub direct products of totally ordered algebras was discussed in the works [42] by Feldman and Henriksen, [67] by Luxemburg, and [9] by Banaschewski. Contrary to the possibility of locally representing countably many elements of any semiprime f-algebra as continuous functions without any choice, the global representation of f-algebras as sub direct products of totally ordered algebras can not be obtained without appealing to some transcendent tool from set theory. The latter kind of representation theorem is important for more than historical reasons. First of all, many researchers define f-algebras as subdirect products of totally ordered algebras. Secondly, Birkhoff and Pierce in their seminal paper [22] observed that the Axiom of Choice seems to be involved if one wishes to obtain (with the definition for f-algebras as in this survey) a representation theorem for f-algebras as a sub direct product of totally ordered algebras, which for convenience we will now name the Birkhoff-Pierce Representation Theorem. The three papers [42], [67], and [9] independently prove that the Boolean Prime Ideal Theorem is both sufficient as well as needed for the Birkhoff-Pierce Representation Theorem. Thus Stone's Representation Theorem for Boolean algebras is constructively equivalent to the Birkhoff-Pierce Representation Theorem for f-algebras. In turn, each of the latter representation theorems is constructively equivalent to the Kakutani Representation Theorem for vector lattices with a strong order unit. In about that same direction, we observe that it is still unknown whether the Boolean Prime Ideal Theorem suffices for that other main representation tool for vector lattices as vector sublattices of extended real valued continuous functions, the so-called Maeda-Ogasawara Representation Theorem (see [32]).

15. Linear biseparating maps on I-algebras A linear map T between two algebras A and B is said to be separating if T (I) T (g) = 0 in B whenever f 9 = 0 in A. If in addition T is bijective and its inverse T- 1 is separating as well then T is said to be biseparating. Clearly, if T is one-to-one and onto then T is biseparating if and only if

f9=

0 in A <=> T (I) T (g) = 0 in B.

If A and B are assumed to be f-algebras with unit elements, then T is separating if and only if Tis disjointness preserving, that is, If I II Igl = 0 in A implies IT (1)1 II IT (g) I = 0 in B. This follows directly from the equivalence fg = 0 <=> If I II Igl = 0,

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which holds in any semiprime I-algebra. For the same reason, the linear map T between two I -algebras with multiplicative identity is biseparating if and only if T is ad-isomorphism (Le., T is bijective and both T and T- 1 are disjointness preserving) The reader is encouraged to consult the beautiful memoir [1 J by Abramovich and Kitover for the theory of disjointness preserving linear maps on Riesz spaces. The study of when linear biseparating maps on algebras of real or complex valued continuous functions are weighted isomorphisms started in 1990 with the paper [63J by Jarosz and culminated in the work [5J by Araujo, Beckenstein and Narici with the following result. Let C (X) and C (Y) be the algebras of real or complex valued continuous functions on completely regular topological spaces X and Y, respectively. If T is a linear biseparating map then there exist a nonvanishing wEe (Y) and an homeomorphism h from the realcompactification vX of X onto v Y, such that T (f) (y) = w (y) I (h (y))

for all lEe (X) and y E Y.

Henriksen and Smith in [55J explored the aforementioned result by Araujo, Beckenstein and Narici in the more general setting of unital I-algebras. They proved that every positive linear biseparating map T between two unital I-algebras A and B closed under inversion is a weighted isomorphism, that is, there exist an invertible wEB and a Riesz isomorphism S : A ---> B which is simultaneously an algebra isomorphism, such that

T (f) = wS (f)

for all lEA.

Very recently in [30J, Boulabiar, Buskes, and Henriksen extended the latter result to all order bounded linear biseparating maps on arbitrary (not necessarily closed under inversion) unital I-algebras over the reals as well as over the complex numbers (for the theory of complex I-algebras, we refer to [17]). The theorem they obtained is the following. THEOREM 15.1. Let A and B be (real or complex) I-algebras with unit elements, and let T : A ---> B be an order bounded linear biseparating map. Then T is a weighted isomorph'ism.

In the previously mentioned memoir by Abramovich and Kitover, we find the following theorem. A d-isomorphism between two uniformly complete Riesz spaces A and Jyf is automatically order bounded as soon as every universally a-complete projection band in A is essentially one-dimensional (see [1, Corollary 15.3]). This theorem is used in [30], under the same condit.ions on A, to show that every linear biseparating map between two uniformly complete I-algebras A and B is a weighted isomorphism. We point out that a complex I-algebra is by definition uniformly complete. Thus the phrase 'uniformly complete unital I-algebra' is understood to mean either a uniformly complete unital I -algebra over the reals, or simply a unital I-algebra over the complex numbers. For t.he proof of the next theorem, see Proposition 5.1 and Theorem 5.2 in [30J. THEOREM 15.2. Let A and B be unilormly complete unital I -algebras and assume that every universally a-complete projection band in A is essentially onedimensional. Then every linear biseparating map from A onto B is order bounded and then a weighted 'isomorphism.

To obtain the result by Araujo, Beckenstein and Narici cited above as a consequence of the preceding theorem, Boulabiar, Buskes and Henriksen proved t.hat

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every algebra of all scalar-valued continuous functions on a completely regular topological space has the property that every universally a-complete projection band is essentially one-dimensional (see [30, Theorem 5.5]). THEOREM 15.3. Let X be a completely regular topological space X. Then every universally a-complete projection band in the Riesz space C (X) is essentially onedimensional.

Combining Theorems 15.2 and 15.3, we arrive at the next result. COROLLARY 15.4. Let X and Y be completely regular topological spaces. Then every biseparating linear map T : C (X) -> C (Y) is a weighted isomorphism. In particular, C (X) and C (Y) are isomorphic as f -algebras if and only if there exists a linear biseparating map from C (X) onto C (Y).

It is well known that if X and Yare completely regular topological spaces and S is an isomorphism from C (X) into C (Y) then there exists an homeomorphism h from vY into vX such that S (f) = f 0 h, where vX and vY denote the realcompactifications of X and Y, respectively (see Section 10 in [47]). The latter fact, together with Corollary 15.4 above, directly leads to the next corollary, which was proved earlier in an alternative way by Araujo, Beckenstein and Narici in [5, Proposition 3]. COROLLARY 15.5. Let X and Y be completely regular topological. Then for every l'inear biseparating map T : C (X) -> C (Y) there exist a non-vanishing function wE C (Y) and an homeomorphism h : vY -> vX such that

T (f) (y) = w (y)

f

(h (y))

for all f E C(X) and y E Y.

It is shown in [47, Theorem 8.3] that two realcompact X and Yare homeomorphic if and only if C (X) and C (Y) are isomorphic as f-algebras. Another classical result of rings of continuous functions theory is that if X is a completely regular topological space then C (X) and C (vX) are isomorphic as f-algebras (see Remark 8 (a) in [47]). It follows immediately that if X and Y are two completely regular topological spaces, and C (X) and C (Y) are isomorphic as f-algebras, then vX and v Yare homeomorphic. The latter implies that, without further assumptions, the conclusion that vX is homeomorphic to vY in Corollary 15.5 is best possible. Under additional assumptions, however, X and Y may themselves be homeomorphic as is shown in the next Corollary. COROLLARY 15.6. Let X and Y be completely regular topological spaces and assume either (i) or (ii) below. (i) X and Yare realcompact. (ii) The points of X, as well as those ofY, are Go-points. If there exists a linear biseparating map from C (X) onto C (Y) then X and Yare homeomorphic.

Under the condition (i), the result in Corollary 15.6 above follows straightforwardly from Corollary 15.5 while under the condition (ii), it stems directly from [73] by Misra. Finally, it should be noted that the result of Theorem 15.2 above is false if the universally a-complete projection bands in A fail to be essentially one-dimensional. Indeed, Example 7.13 in [1] produces a linear biseparating map T on the Dedekind

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complete (and then uniformly complete) unital i-algebra Lo ([0, 1]) of all equivalence classes of measurable functions on [0,1)' which is not order bounded and thus cannot be a weighted isomorphism.

References [1] Abramovich, Y. A. and A. K. Kitover, Inverses of disjointness preserving operators, Memoirs Amer. Math. Soc., 143 (2000), no 679. [2] Aliprantis. C. D. and O. Burkinshaw. Positive Operators. Academic Press, Orlando, 1985. [3] Amemiya, I., A general spectral theory in semi-ordered linear spaces, J. Fac. Sci. Hokkaido Univ. Ser I, 12 (1953), 111-156. [4] Anderson, F. W., Lattice-ordered rings of quotients, Canad. J. Math., 17 (1965), 434-448. [5] Araujo, J., E. Beckenstein and L. Narici, Biseparating maps and homeomorphic realcompactifications, J. Math. Ana. Appl., 12 (1995), 258-265. [6] Arens. R, Operations induced in function classes. Monatshelte Math., 55 (1951), 1-19. [7] Arens, R, The adjoint of bilinear operation, Proc. Amer. Math. Soc., 2 (1951), 839-848. [8] Arens, R. and J. L. Kelley, Characterizations of the space of continuous functions over a compact Hausdorff space, Trans. Amer. Math. Soc., 62 (1947), 499-502. [9] Banaschewski, B., The Prime Ideal Theorem and representations of I-rings, Algebra Univ., 25 (1988), 384-387. [10] Basly, M. and A. Triki, F F-algebres Achimediennes, Preprint. [11] Basly, M. and A. Triki, I-algebras in which order ideals are ring ideals, Indag. Math. 50 (1988), 231-234. [12] Basly, M. and A. Triki, On uniformly closed ideals in I-algebras, 2nd Conlerence, Functions spaces, Marcel Dekker (1995), 29-33. [13] Bernau, S. and Huijsmans, C. B., Almost I-algebras and d-algebras, Math. Proc. Cambridge. Phil. Soc. 107 (1990), 287-308. [14] Bernau, S. J. and C. B. Huijsmans, The order bidual of Almost I-algebras and d-algebras, Trans. Amer. Math. Soc. 347 (1995), 4259-4274. [15] Bernau, S. J. and C. B. Huijsmans, The Schwarz inequality in Archimedean I-algebras, Indag. Math. 7 (1996), 137-148. [16] Beukers, F. and C. B. Huijsmans, Calculus in I-algebras, J. Austral. Math. Soc. (Ser. A) 37 (1984), 110-116 [17] Beukers, F., C. B. Huijsmans and B. de Pagter, Unital embedding and complexification of I-algebras, Math, Z., 183 (1983), 131-144. [18] Bigard, A. and K. Keimel, Sur les endomorphismes conservant les polaires d'un groupe reticule archimedien, Bull. Soc. Math. France, 97 (1969), 381-398. [19] Bigard, A., K Keimel and S. Wolfenstein, Groupes et Anneaux Reticules. Lecture Notes in Math. vo!' 608, Springer Verlag, 1977. [20] Birkhoff, G., Lattice ordered groups, Annals of Math., 43 (1942), 298-331 [21] Birkhoff, G., Lattice Theory, 3rd Edition. Amer. Math. Soc. Colloq. Pub!. no. 25, American Mathematical Society 1967. [22] Birkhoff, G. and R S. Pierce. Lattice-ordered rings, An. Acad. Brasil. Cienc. 28 (1956), 41-69 [23] Boulabiar, K., A relationship between two almost I-algebra products, Algebra Univ., 43 (2000), 347-367 [24] Boulabiar, K, Products in almost I-algebras, Comment. Math. Univ. Carolinae, 41 (2000), 747-759. [25] Boulabiar, K, A representation theorem for d-rings, Submitted. [26] Boulabiar, K, Positive derivations on almost I-rings, To appear in Order. [27] Boulabiar, K, On positive orthosymmetric bilinear maps, Submitted. [28] Boulabiar, K and E. Chi!, On the structure of almost I-algebras, Demonstratio Math., 34 (2001), 749-760. [29] Boulabiar, K and M. A. Toumi, Lattice bimorphisms on I-algebras, Algebra Univ. 48 (2002), 103-116. [30] Boulabiar, K., Buskes, G. and Henriksen, M., A Generalization of a Theorem on Biseparating Maps, To appear in J. Math. Ana. Appl.

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[61] Huijsmans, C. B. and B. de Pagter, Averaging operators and positive contractive projections, J. Math. Ana. Appl., 113 (1986), 163-184. [62] Ionesco Tulcea, A. and C. Ionesco Tulcea, On the lifting property I, J. Math. Anal. Appl., 3 (1961), 537-546. [63] Jarosz, K., Automatic continuity of separating linear isomorphisms, Bull. Canadian Math. Soc., 33 (1990), 139-144. [64] Johnson, D. G., The completion of an archimedean I-ring, J. London Math. Soc, 40 (1965), 493-493. [65] Kudhicek, V., On some types of R-rings. Sb. Vysoke. Ucen{ Tech. Bmo 1-2 (1962), 179-181. [66] Lambek, J. Lectures on Rings and Modules, Blaisdell, 1966. [67] Luxemburg, W. A. J., Some Aspects 01 the Theory 01 Riesz Spaces, Univ. Arkansas Lecture Notes Math. 4, Fayetteville, 1979 [68] Luxemburg, W. A. J., A remark on a paper by D. Feldman and M. Henriksen concerning the definition of I-rings, Nederl. Akad. Wetensch. Indag. Math., 50 (1998), 127-130. [69] Luxemburg, W. A. J. and A. C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, 1971. [70] Lozanovsky, G., The functions of elements of vector lattices, Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1973), 45-54. [71] Martinez, J. The maximal ring of quotient I-ring. Algebra Univ .• 33 (1995), 355-369. [72] Meyer-Nieberg, P., Banach Lattices, Springer Verlag, Berlin Heidelberg New York, 1991. [73] Misra, P. R., On isomorphism theorems for C(X), Acta. Math. Acad. Sci. Hungar., 39 (1982), 179-180. [74] Nagasawa, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Semin. Rep., 11 (1959), 182-188. [75] Nakano, H., Modern Spectral Theory, Tokyo Math. Book Series II, Maruzen, Tokyo, 1950. [76] de Pagter, B., I-algebras and orthomorphisms (Thesis, Leiden, 1981). [77] Phelps, R. R., Extremal operators and homomorphisms, Trans. Amer. Math. Soc., 108 (1963), 265-274. [78] van Putten, B., Disjunctive linear operators and partial multiplication in Riesz spaces, (Thesis, Wageningen, 1980). [79] Quinn, J., Intermediate Riesz spaces, Pacific J. Math., 56 (1975), 225-263. [80] Scheffold, E., FF-Banachverbandsalgebren, Math. Z., 177 (1981),193-205. [81] Scheffold, E., Der Bidual von F-Banachverbandsalgebren, Acta Sci. Math., 55 (1991), 167179. [82] Scheffold, E., Uber Bimorphismen und das Arens-Produkt bei kommutativen DBanachverbandsalgebren, Preprint. [83] Scheffold, E., Uber den ordnungsstetigen Bidual von FF-Banachverbandsalgebren, Arch. Math., 60 (1993), 473-477. [84] Steinberg, S. A., Quotient rings of a class of lattice-ordered rings, Can. J. Math., 25 (1973), 627-645. [85] Steinberg, Stuart A. On the unitability of a class of partially ordered rings that have squares positive. J. Algebra 100 (1986), no. 2, 325-343. [86] Steinberg, Stuart A. On lattice-ordered algebras that satisfy polynomial identities. Ordered algebraic structures (Cincinnati, Ohio, 1982), 179-187, Lecture Notes in Pure and Appl. Math., 99, Dekker, New York, 1985. [87] Steinberg, Stuart A. Unital $l$-prime lattice-ordered rings with polynomial constraints are domains. Trans. Amer. Math. Soc. 276 (1983), no. 1, 145-164. [88] Steinberg, Stuart A. On lattice-ordered rings in which the square of every element is positive. J. Austral. Math. Soc. Ser. A 22 (1976), no. 3, 362-370. [89] Toumi, M. A., On some I-subalgebras of d-algebras, To appear in Math. Reports [90] Triki, A., Stable I-algebras, Algebra Univ., 44 (2000) 65-86. [91] Triki, A., On algebra homomorphisms in complex almost I-algebras, Comment. Math. Univ. Carolinae, 43 (2002), 23-31. [92] Zaanen, A. C., Riesz Spaces II, North Holland, Amsterdam-New York-Oxford, 1983. [93] Zaanen, A. C., Introduction to Operator Theory in Riesz Spaces, Springer Verlag, Berlin, 1997. [94] Zelazko, W., Banach algebras, Elsevier, Amsterdam-London-New York, 1973.

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DEPARTEMENT DU CYCLE AGREGATIF, INSTITUT PREPARATOIRE AUX ETUDES SCIENTIFIQUES ET TECHNIQUES, UNIVERSITE DU 7 NOVEMBRE

A CARTHAGE,

BP 51, 2070-LA MARSA, TUNISIA

E-mail address:karim.boulabiar«lipest.rnu.tn DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS-38677, USA

E-mail address: mmbuskes«lsunset. olemiss. edu DEPARTEMENT DE MATHEMATIQUES, FACULTE DES SCIENCES DE TUNIS, UNIVERSITE TUNIS EL MANAR, 1060-TUNIS, TUNISIA

E-mail address:abdelmajid.triki«lfst.rnu.tn

Contemporary Mathematics Volume 328, 2003

An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces Eggert Briem

ABSTRACT. It follows from a theorem of J. Wermer that if A is a uniformly closed algebra of continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space X, the property that b2 E Re A for all bE Re A implies A = Co(X). In other words, if the function h(t) = t 2 operates by composition on ReA then A = Co(X). This result was generalized by O. Hatori. He proved that in place of the function h(t) = t 2 one can put any real-valued function defined in a neighbourhood of 0, thus extending a similar result for the compact case. Here a simple alternative proof is given for the locally compact case.

1. Introduction

Let X be a compact Hausdorff space and B a uniformly closed subspace of CIR(X), the space of all continuous real-valued functions on X, which separates the points of X and contains the constant functions. A version of the Stone-Weierstrass theorem says that if b2 E B for all b E B then B = CIR(X). Clearly this result does not hold if, instead of assuming that B is uniformly closed, one assumes that B is a Banach space in some norm which dominates the sup-norm, as the example of any non-trivial real Banach function algebra shows. However, if B is the real part of a uniform algebra, a theorem of J. Wermer says that if b2 E B for all b E B then

B = CIR(X). Let us say that a real-valued function h, defined on an interval I of the real line, E B whenever b E Band b maps X into I. Thus b2 E B for all bE B means that h(t) = t 2 operates on B. The Stone-Weierstrass theorem W8.', generalized by K. de Leeuw and Y. Katznelson (see [4], theorem 4.21), they showed that h(t) = t 2 can be replaced by any continuous non-affine function (Le. a function not ofthe type h(t) = o:t+!3) defined on an interval, and the theorem of J. Wermer was similarily generalized by A. Bernard, S. Sidney and O. Hatori [1], [5] and [8]. Here one can even do without the continuity assumption, a function operating on the real part of a uniform algebra of infinite dimension is automatically continuous.

operates on B if hob

1991 Mathematics Subject Classification. Primary 46JlOj Secondary 46E15. Key words and phmses. uniform algebra, locally compact space, functional calculus. © 2003 American Mathematical Society

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In the case where X is locally compact, the functional calculus for a uniformly closed subspace of Co(X, JR), the space of all continuous real-valued functions vanishing at infinity on X, may be non-trivial (cf. [2]). What then about the real part of a uniformly closed subalgebra of Co(X), the space of all continuous complex-valued functions vanishing at infinity on X? Wermer's theorem, [10], clearly applies to this situation. It turns out that the functional calculus for the real part of a uniformly closed subalgebra, which is not a C* -algebra, of Co(X) is trivial. This result, which is due to O. Hatori, is to be found in [7]. To prove this one has to do more than just adapt the proofs for the compact case to the locally compact situation. The purpose of this paper is to give a simple alternative proof using antisymmetric decomposition of uniform algebras. 2. Proofs and results Let X be a locally compact Hausdorff space and let Xl denote the one-point compactification of X, with Xoo denoting the point at infinity. The functions in Co(X) have a natural extension to functions in C(Xl)' the space of all continuous complex-valued functions on Xl. If A is a uniformly closed subalgebra of Co(X), that separates the points of X and does not vanish identically at any point of X, then Al = AEI1C is a uniformly closed subalgebra of C(Xd that separates the points of Xl and contains the contant functions and A = {a E All a(x oo ) = O}. Thus, we can assume that there is a uniformly closed subalgebra Al of C(Xd containing the constant functions, where Xl is compact, and a point Xoo in Xl, such that A is obtained by restricting the functions in the set {a E Al Ia(x co ) = O} to the set X = Xl \ {xoo}. We need some results from the theory of uniform algebras. A subset E of Xl is a set of antisymmetry for Al if the only functions in Al that are real-valued on E are constant functions. Maximal sets of antisymmetry are closed generalized peak sets (intersections of peak sets), they form a partition of Xl and a continuous complexvalued function f on Xl is in Al if and only if its restriction to each maximal set of antisymmetry E is in the space AIlE of restrictions of functions in Al to E. Thus Al(Xd = C(Xd if and only if each maximal set of antisymmetry is a singleton. Also, AIlE is uniformly closed for each maximal set of antisymmetry, there is actually for each a in AIlE an element a' in Al with a' = a on E and II a' 1100=11 a llco,(E). (The latter norm is the supnorm w.r.t E). This material can f. ex. be found in [3]. If Al is the disc algebra on the closed unit disc in the complex plane and A is the algebra of functions vanishing at the origin, then any non-zero function in A takes real values of opposite signs in any neighbourhood of O. Thus any real-valued function, defined on an interval which is not a neighbourhood of 0, operates trivially on Re A. We shall therefore always assume that the interval on which an operating function is defined is a neighbourhood of O. We can now state the main result of this note. THEOREM 1. Let A be a uniformly closed subalgebra of Co(X) which separates the points of X and does not vanish identically at any point in X and let Re A be the space of real parts of functions in A. If Re A has a non-affine operating function h: I ........ JR where I, an interval, is a neighbourhood of 0, then Re A = Co(X, JR) and thus also A = Co(X).

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137

The first step in the proof of Theorem 1 is to show that operating functions are continuous. To prove this we need the following lemma. LEMMA 1. Let E be a maximal set of antisymmetry for A1 and let b E Re A. Then b(E) is either an interval or a singleton.

Proof. Suppose b(E) is not connected, and that b = Rea for some a in A 1. Then we can find two disjoint closed rectangles Ro and R1 in the complex plane such that a(E) is a subset of Ro u R1 but not a subset of anyone of the two rectangles. By Runge's theorem there is sequence of polynomials (Pn) converging uniformly to o on Ro and to 1 on R 1. Since AdE is uniformly closed, the sequence (Pn 0 a) converges on E to an element of A1IE. But the sequence converges on E to a function which takes only the values 0 and 1, contradicting the fact that E is a set of antisymmetry. 0 LEMMA 2. Let E be a maximal set of antisymmetry for A 1 such that E\ {xoo } =I- 0. Then there is an Xo E E such that for every to E I (resp. to in the interior of I), there exists bo E ReA with bo(xo) = to and bo(X) C I (resp. bo(X) is a subset of the interior of I).

Proof. If to = 0, the existence of the desired function is easily proved. We consider the case where to > O. (The proof for the case to < 0 is similar.) Let Xo be a generalized peak point in E, other than x oo , for A 1 • Simple calculations show that we can take 1 E A with I(xo) = 11/1100 = 1. Let c: be a positive number such that (-2c:, to] C I (resp. (-2c:, to] is a subset of the interior of I), and R the closed rectangle with corners -c: ± i and to ± i. Let

~

be an operating function for Re A. Then, unless

Proof. If A =I- Co(X,~), there is a maximal set of antisymmetry E for A1 which contains more than one point. Thus E \ {xoo} =I- 0. Let t be an arbitrary point in I. Then by Lemma 2 there exists and x E E and bEReA such that b(x) = t and b(X) C I. Since hob is in ReA, hob is continuous. It follows that h is continuous at t. We conclude that h is continuous on its domain of definition. If A = Co(X), h need only be continuous at 0, f. ex., if A = Co. 0

In the compact case, where A contains the constant functions, Re A is dense in CR(X) (cf. Proposition 4 below). Here we obtain a similar result. PROPOSITION 2. Let h : I --+ IR be an operating function for ReA which is non-affine in every neighbourhood of an interior point to for I. Let Xo E X and suppose that there is a function bo E ReA which maps X into the interior of I and satisfies bo(xo) = to· Then there is an open neighbourhood Xo of Xo such that every f E Co (X, ~) which vanishes outside Xo can be approximated uniformly on X by elements of ReA.

EGGERT BRIEM

138

Proof. We are assuming that X is locally compact and not compact so that

h(O) = o. We may assume that h is continuous by Proposition 1. Let us first look at the case where h is a polynomial of degree at most three in some neighbourhood of 0 but not affine there. Then bed E Re A for all b, e, d E Re A and hence fb E cl(Re A) for all b E cl(Re A) and all f in the algebra generated by Re A. Here cl(Re A) denotes the uniform closure of Re A. It follows that

Co(X,JR)· cl(ReA)

~

cl(ReA)

and hence ReA is dense in Co(X,JR). Suppose first that h is affine in some neighbourhood ofO. Replacing h by h(t)-,t we may assume that h = 0 in a neighbourhood of o. Put e = tC; l bo. For bEReA, r,t E JR and '{) E CO'(JR) the function

J

h 0 (rb o + tb - se)'{)(s)ds

is in cl(Re A) if r is close to 1, t is close to 0 and the support of'{) is contained in a small neighbourhood of 0, so that the expression above is defined. We differentiate twice w.r.t. 0, put t = 0 to obtain

b2 e- 2

J

h 0 (rbo - se)'{)"(s)ds E cl(ReA),

for all bEReA. (If le(x)1 is small the expression is 0.) Put

d= We have

d(xo) =

J

h 0 (rb o - se)'{)"(s)ds.

J

h 0 (rto - s)'{)"(s)ds = (h

* '{))"(rto).

Since h is not affine in any neighbourhood of to we can choose '{) and r, where the support of'{) is contained an arbitrarily small neighbourhood of 0 and r is arbitrarily close to 1 such that d(xo) '" O. Let

Xo

=

{x

E

Xld(x) '" O}.

Since b2 e- 2 d E cl(Re A) for all b E Re A it follows that

b1 ,b2 e- 2 d E cl(ReA) for all b1 , b2 E Re A. Put

M(ReA) = {J

E

Co(X,JR) If· cl(ReA)

~

cl(ReA)}.

be- 2 d

Above we have E M(ReA) for bEReA. Since these functions separate the points of Xo and since M(ReA) is an algebra we deduce that cl(ReA) contains every f E Co(X,JR) which vanishes outside Xo. Suppose now that h is not affine in any neighbourhood of o. For e E Re A let

B(e) = {u E cl(ReA) 13>' E JR+ s.t. lui::::; >'Iel}. If b ERe A and b + ie E A then be and be(b2 - e2 ) belong to B(e). Let also

M(e) = {g

E

Co(X, JR) Ig. cl(B(e))

~

cl(B(e))},

a subalgebra of Co(X, JR). We note that since h is continuous h also operates on

cl(ReA).

ALGEBRAS ON LOCALLY COMPACT SPACES

139

For bl ERe A, for u E B(e), r, t E lR and

J

(1)

h 0 (rb l

+ tu -

se)
is in cl(Re A), if rand t are sufficiently small and the support of

uie- i

J

ho (rb l - se)
Put

J

d= With i

= 2 and
h 0 (rb l - se)
instead of

u 2 e- 2 d E cl(Re A) for all u E B(e) and hence

(2) for all

Ul, U2

E B(e). With i

= 3 and u = be E B(e)

in (1) we get

b3 d E cl(ReA), and with i

=

1, u

= beW - e2 )

E B(e) and
b(b2

-

e2 )d E cl(Re A)

and thus

If we put

U2

be2 d E cl(ReA). = be2 d in (2) we find that ubd2 E cl(Re A)

for all u E B(e). Since u E B(e) implies ubd2 E B(e) we deduce that

bd 2 E M(e). Let us determine how well the functions bd2 separate the points of the set

Xo = {x

E

X Ib(x)e(x) i- a}.

Let x, y E Xo. Suppose that does not separate x and y for every d. Since A is an algebra we can choose bl E ReA such that bl(x) = 0 and bl(y) = 1. Then

bd 2

d2 (y)

= (b(x)/b(y))d 2 (x).

The right hand side is independant of r so that

d(y)

=

J

h 0 (r - se(y))
a differentiable function of r, is constant in for r in some neighbourhood of O. It follows that h 0 (r - se(y))
J

for all r in a neighbourhood of O. Since this holds for all

EGGERT BRlEM

140

dense in Co(X,JR). Otherwise, the functions bd2 separate the points of X o, and since these functions vanish outside X o, it follows that

{f E Co(X, JR) I I = 0 on Xo \ Xo}

~

M(c).

Thus by a simple calculation we have

{f

E

Co(X,JR) I I = 0 on Xo \ Xo}

~

cl(ReA).

o To proceed to the proof of Theorem 1 we need a local version of the so called Bernard's Lemma [1]. Let A be an infinite set and let C be a Banach space with norm I . II. Put

eOO(C) = {{c.d ICA E C and sup

AEA

I CA 11< oo},

where {cA} denotes a function from A into C such that {cA}(a) = a E A. Then eOO(C) is a Banach space in the norm

I

{cA }

Co

E C for each

11= sup II CA I . AEA

The space Re A is a Banach space in the norm

lib 11= inf{11 a 1100 Ia E A and clearly lib 1100::;11 b II. We thus have eOO(ReA)

~

and b = Rea},

eOO(Co(X,JR)).

Bernard's lemma (see [1]) says that if eOO(ReA) is dense in eOO(Co(X,JR)) then ReA = Co(X,JR). To prove that eOO(ReA) is dense in eOO(Co(X,JR)), we represent eOO(Co(X,JR)) as a space of functions on a compact Hausdorff space. We identify Co(X, JR) with a subspace of CIR(Xd in the obvious manner. Let A have the discrete topology and let (3(A x Xd) be the Stone-Cech compactification of Ax Xl' The space eOO(C(xd) can be identified with C({3(A x Xd) in a natural way

{fA}

FUll} where F{!lI}b,x) = 1'Y(x) for each b,x) in A x Xl' subspace of CIR ({3(A x Xl)) and we have the inclusions

eOO(ReA)

~

--+

eOO(Co(X,JR))

~

Thus eOO(Co(X,JR)) is a

CIR({3(A x Xd).

The subsets of the Stone-Cech compactification may be complicated but the subsets we are interested in have a simple description. For each I in C(Xl ) let {f} denote the net {fA} where IA = I for each A Let now Po be in {3(A x Xd. The map

I

--+

{f }(Po)

is a homomorphism of C(X l ) and is thus given by point-evaluation at some point in Xl' Put

Xo

5: 0

= {p E (3(A x Xd I {f}(p)

=

I(xo)

VI E CIR(Xl )}.

ALGEBRAS ON LOCALLY COMPACT SPACES

141

The sets x form a partition of {3(A x Xt} into closed sets. Their significance stems from the following local version of Bernard's lemma, similar to the one given by O. Hatori [6].

Bernard's lemma (local version) Let A denote the unit ball of Co(X,JR.), let x be in X and suppose that the restriction space lOO(ReA)lx is dense in CIR(x). Then there is a compact neighbourhood Kx of x such that ReAIKx = CIR(Kx )' Proof. For each A in A let 1>.. = A. By assumption there is an element {b A } in lOO(ReA) such that we have the inequality

I{fA} - {bA}1 < 1/2 on x and hence also on some open subset U of {3(A x Xt} containing X. By a simple calculation we have

Xo = {p E (3(A x Xl) I I{f}(p) - f(xo)1

:=; ~

Vf E CIR(Xl )}.

Thus there is a function fo in CIR(Xt} such that the set

{p

E

{3(A x Xt}

I I{fo}(p) -

fo(x)1

:=; 1/2}

is a subset of U. Put

Kx = {y E Xd Ifo(Y) - fo(x)1 :=; 1/2}, a compact neighbourhood of x. Then A x Kx is contained in U so that IfA - bA I :=; 1/2 on Kx for all A in A. Let M = SUPA II bA II, a finite quantity because (b A ) is in lOO(Re A). Take any f in Co(X, JR.) with II f lloo:=; 1. By induction we construct a sequence (b n ) of elements from ReA, with II bn II:=; M for all n, such that n-l

12n( f - L

2- i

bi ) - bnl < ~

i=O

on Kx. The function b =

E:o 2- b is in Band b = f on Kx. 0 i i

PROPOSITION 3. Let Xo E X and suppose that there is an open neighbourhood Xo of Xo such that every function in Co(X,JR.), which vanishes outside Xo can be unifromly approximated on X by elements from ReA. Then lOO(ReA) sepamtes the points of xo.

Proof. Suppose p,q are in xo. There is an element {fA} in lOO(Co(X,JR.», where each 1>.. vanishes outside X o, such that {fA}(P) = 0 and {fA}(q) = 1. By assumption we can for each A E A find a function bA in ReA with II fA - bA lloo:=; ~. Thus we have {bA}(p) :=; ~ and {bA}(q) ~ ~ since II{fA} - {bA}lIoo,{3(Axxt} :=; ~. Let a A = bA + iC A be in A for each A. Now, the A-net {e a ... - 1} is in lOO(A) and it separates p from q since {e an } does. Since

I{e a ... }(p)1 =

{e b... }(p)

:=; e l / 4

and I{e a ... }(q)1 = {e b... }(q) ~ e3 / 4

,

we see that lOO(A) separates p from q and thus lOO(ReA) does so as well. 0

EGGERT BRIEM

142

We also need the aforementioned extension of the Stone-Weierstrass theorem due to de Leeuw and Katznelson (see [4], Theorem 4.21). We can prove a version of this result, Proposition 4, in the same way as the original one, the proof is omitted. PROPOSITION 4. Let Y be a compact Hausdorff space and B a subspace ofCR(Y) which separates the points of Y and contains the constant functions. Suppose that B is also a normed space with the norm II· liB which dominates the uniform norm. Let h be a continuous function defined on an interval I. Suppose that h is nonaffine in every neighbourhood of an interior point to in 1. Suppose that there exists a positive real number 8 > 0 with (to - 8, to + 8) ~ I such that h 0 (to + u) E B for every u E B with Ilull < 8. Then B is uniformly dense in CIR(Y).

Proof of Theorem 1. We are going to use the local version of Bernard's Lemma. We thus have to show that the conditions stated there are satisfied. For this we use the result of de Leeuw and Katznelson above. The main obstacle is that we can not conclude that if {b.>.} is in [00 (Re A) then h e {b.>.} is in [00 (Re A) although the composite function is defined, i.e. we can not conclude that composition with h maps bounded nets to bounded nets. To overcome this difficulty we use a method of Sidney, [8], to obtain local boundedness for composition with h. Suppose A =I- Co(X) and thus also Al =I- C(X 1 ). Let E be a maximal set of antisymmetry for Al containing more than one point. Let further to be an interior point of I such that h is not affine in any neighbourhood of to and let bo be a function in Re A which maps X into the interior I for which to is an interior point of bo(E). By Lemma 2 such a function exists. We now choose E > 0 such that if b is in the E-ball, (ReA)., of ReA then ho(bo+b) is defined and to is an interior point of (b o + b)(E). We write (ReA).

= U{b E (ReA). I I h 0 (bo + b) II::; n}. n

The Baire Category Theorem shows that the closure of one of the sets on the right hand side has an interior point and thus there is a function b1 E Re A and positive numbers 8, lvI, with I b1 I +8 < E, such that

I he (bo +b 1 +c) II::; lvI for c in a dense subset of the 8-ball of Re A. Let b = bo + b1 , and let [00 (Re A) be as in the local version of Bernard's Lemma. We then have

h 0 {b + c.>.} E cl([OO(Re A)), the uniform closure of [OO(ReA), if {c.>.} E [OO(ReA) and such that b(xo) = to, an interior point of b(E). We restrict to Xo and deduce that

h 0 {to

I

{c.>.}

11< 8.

Take Xo E E

+ c.>.} E cl([OO(ReA)lxo),

for every {c.>.} E [OO(ReA)lxo whose quotient norm satisfies II {c.>.} 11< 8. Since {c} is constant on xo for any c E ReA, the space [OO(ReA)lxo contains the constant functions, it also separates the points of Xo by Proposition 2 and 3. Then Proposition 4 shows that [OO(ReA)lxo is dense in CIR(xo) and thus, by the local version of Bernard's Lemma, ReAIK = CIR(K) for some compact neighbourhood K of Xo. The theorem of Sidney and Stout, [9], then shows that AIK = C(K). By Proposition 2, there exists an open neighbourhood Xo of Xo such that every f E Co(X, 1R) which vanishes outside Xo can be approximated uniformly on X by

ALGEBRAS ON LOCALLY COMPACT SPACES

143

elements of ReA. We may assume K c Xo. Let Kl be a compact neighbourhood of Xo such that K 1 is in the interior of K. Since E is antisymmetric and contains more than one point, EnK l contains more than one point. On the other hand, we will show that {f E Co(X, IR) II = 0 outside Kd CAl. It will follow that En Kl = {xo}, which will be a contradiction proving A = Co(X, IR). Let 1 E Co(X, IR) with 1 = 0 outside K 1 . A function u E Co(X, IR) with lui::; 1 on X, u = 1 on K l , and u = 0 outside K can be uniformly approximated by functions in ReA. Thus, for every positive integer n, there exists bn E ReA such that 1 - ,& ::; bn ::; 1 on K 1 and bn ::; ,& outside K. Without loss of generality we may assume that Ibnl ::; 1 on X. Take Cn E A with Recn = bn and put an = (eCn-l)n. Then an E AI, e-1. ::; lanl ::; Ion Kl and lanl ::; e- n+1. outside K. Since AllK = C(K) and thus AllKl = C(Kd, there exists a positive real number M such that for every positive integer n there exists gn E Al such that gnan = 1 on Kl with IIgnlloo ::; M. Since AllK = C(K), there is a function af E Al with af = 1 on K. Then afgnan E Al and by a simple calculation lIafgnan - 11100 --+ 0 as n --+ 00, that is, 1 E AI' 0 References [lJ A. Bernard, Espaces des parties relies des Iments d'une algebre de Banach de fonctions, J. Funct. Anal. 10 (1972), 387-409. [2J E. Briem, Approximations from Subspaces of Co(X), J. Approx. Theory 112 (2001), 279-294. [3J A. Browder, Introduction to Function Algebras, W. A. Benjamin, Inc. !969. [4J R.B. Burckel, Characterizations of C(X) among its subalgebras (Lecture Notes in Pure and Appl. Math. 6). Marcel Dekker, New York 1972. [5J O. Hatori, Functions which operate on the real part of a uniform algebra, Proc. Amer. Math. Soc. 83 (1981), 565-568. [6J O. Hatori, Separation properties and operating functions on a space of continuous functions, lntemat. J. Math. 4 (1993), 551-600. [7J O. hatori, Range transformations on a Banach function algebra. IV, Proc. Amer. Math. Soc. 116 (1992), 149-156. [8J S. J. Sidney, Functions which operate on the real part of a uniform algebra, Pac. J. Math. 80 (1979), 265-272. [9J S. J. Sidney and E. L. Stout, A note on interpolation, Proc. Amer. Math. Soc. 19 (1968), 380-382. [10J J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 1423-1426. SCIENCE INSTITUTE, UNIVERSITY OF ICELAND. REYKJAVIK, ICELAND

E-mail address:briemillhi.is

Contemporary Mathematics Volume 328, 2003

Some mapping properties of p-summing operators with Hilbertian domain Qingying Bu

ABSTRACT. We prove that if H is a Hilbert space, Y is a Banach space and u : H ---+ Y is absolutely p-summing for some p :::: 1, then for any 1 < q < 00, u takes absolutely q-summable sequences in H into members of iq®Y, the projective tensor product of lq and Y.

Given a real or complex Banach space X and 1 ::; p < 00, we denote by and f;eak(X) the Banach spaces of sequences in X with norms II(xn)nlle;trong(X) = 11(llxnll)nllep and II(xn)nlle;eak(X) = sUP",*eB x * II(x*xn)nllep, respectively (cf. [4, pp. 32-36]). For 1 < p < 00, let fp(X) denote the space of all (strongly p-summable) sequences in X such that E~=l Ix~(xn)1 < 00 for each (X~)n E f't},eak(x*), normed by f~trong(x)

II(xn)nlltp(X) = sup

{I ~ x~(xn)1

:

II(x~)nlle~~eak(X.) ::; 1} ,

where pI is the conjugate of p, i.e., lip + 11pl = 1. With this norm fp(X) is a Banach space (cf. [1, 3]). Note: In [2] it was shown that fp(X) is exactly fp®X, the projective tensor product of fp and X. In this note we use this identification of fp(X) with fp®X to deduce a surprising mapping property of absolutely p-summing operators that have a Hilbert space domain. While the main result of this note can be derived from some by-now famous results of Kwapien, it was discovered because of the identification of fp(X) with fp®X, moreover, this identification leads itself to a proof that is a clean and clear application of Khinchin's inequality, Kahane's inequality, and Pietsch's Domination theorem - all fundamental aspects of the theory of p-summing operators. From the definitions, we have for 1 < p < 00, fp(X) ~ f;trong(x) ~ f;eak(x), and 11·lIe~..ak(X) ::; 1I'lIl~trong(X) ::; 11·llep(x)· Moreover, in case dimX = 00, all the containments are proper. For Banach spaces X and Y and a continuous linear operator 'U : X ---+ Y, define ft. : XN ---+ yN by (xn)n 1---+ (uxn)n. Then ft. is a linear operator. Thanks 2000 Mathematics Subject Classification. 46B28. © 145

2003 American Mathematical Society

146

QINGYING BU

to the Closed Graph Theorem, each of

it: f;eak(x)

---->

f;eak(y);

it:

f~trong(x) ----> f~trong(y);

it: fp(X)

---->

fp(Y)

is a continuous operator with

Ilitll£;:,eak(x)_e;:,eak(Y) = Ilitll£~trong(x)_e~trong(y) =

Ilitllep(x)-£p(Y) = Ilull·

We should mention here Khinchin's inequality (cf. [4, p. 10]) and Kahane's inequality (cf. [4, p. 211]) each of which plays a critical role in this paper. Let rn(t) denote the Rademacher functions (cf. [4, p. 10]), namely, rn : [O,IJ ----> ~, n E Pi! defined by rn(t) := sign(sin2n7rt).

< p < 00, there are positive constants Ap, Bp such that for any scalars a1, a2, ... ,an, we have

Khinchin's Inequality. For any 0

Kahane's Inequality. If 0 < p, q < which

([II t, r,('lx, I ,d')

1/, ,;

00,

then there is a constant Kp,q > 0 for

K",' ([II

t,

r,('lx,lI' dt )

1/,

regardless of the choice of a Banach space X and of finitely m.any vectors Xl, X2, ... , Xn from.X.

In 1967, A. Pietsch [5J introduced p-summing operators between Banach spaces, namely, a Banach space operator u : X ----> Y is called p-summing operator, 1 :::; p < 00, if it takes f~eak(x) into f~trong(y). Let IIp(X, Y) denote the space of all psumming operators from a Banach space X to a Banach space Y. If u : X ----> Y is a p-summing operator then, thanks to the Closed Graph Theorem, it : f~eak(x) ----> f~trong(y) is continuous. So we define the p-summing norm 7rp(u) on IIp (X, Y) to be 7rp (u) = Ilitll£;:,eak(x)_£~trong(y). With this norm IIp (X, Y) is a Banach space. There is an equivalent definition of p-summing operators, namely, a Banach space operator u : X ----> Y is p-summing if and only if there is a constant c > 0 such that for any XI,X2,'" ,Xn E X, (1)

In this case, 7rp (u)

= inf{c

> 0: for all possible c in (I)}.

In 1973, J. S. Cohen [3J introduced strongly p-summing operators between Banach spaces, namely, a Banach space operator u : X ----> Y is called strongly p-summing operator, 1 < p < 00, if it takes f~trong(x) into fp(Y). Let Dp(X, Y) denote the space of all strongly p-summing operators from a Banach space X to a Banach space Y. If u : X ----> Y is a strongly p-summing operator then, thanks to the Closed Graph Theorem, it : f~trong(x) ----> fp(Y) is continuous. So we define a

PROPERTIES OF p-SUMMING OPERATORS

147

strongly p-summing norm Dp(u) on Dp(X, Y) to be Dp(u) = lIulll~trong(X)-+lp(Y)' With this norm Dp(X, Y) is a Banach space. There is an equivalent definition of strongly p-summing operators, namely, a Banach space operator U ': X ~ Y is strongly p-summing if and only if there is a constant c > 0 such that for any Xll X2,'" , Xn E X, and any yi, Y2,'" , y~ E Y*,

(2)

In this case, Dp(u) = inf{c > 0: for all possible c in (2)}.

Note: Actually, U E Dp(X, Y) means that u takes absolutely p-summable sequences in X into members of ip®Y.

Main Theorem. Let 1 < p, q < 00; and let H be a Hilbert space and Y be a Banach space. Then IIp(H, Y) ~ Dq(H, Y), i.e., if u : H ~ Y is absolutely p-summing, then u takes absolutely q-summable sequences in H into members of

lq®Y.

PROOF. First consider H = i2 for n E N. Let u E IIp(l2' Y). By Pietsch's Domination Theorem [4, p. 44], there is a regular probability measure J.L on Bl'.J: such that for any x E i 2, lip (

Iluxll ::; 1Tp (U)' Lt'.J: I(x, z)IP dJ.L(z) Now for

Xl,X2,'"

,Xm

Ei

)

(3)

2, and Yi'Y2"" ,Y;" E Y*, we have n

Xk =

~Xk ·e· L.-J ,'&"',

k = 1,2,··· ,m.

i=l

Then m

m

L

I(UXk,

yZ)1

n

L I(Lxk,iuei, yZ)1 k=l

k=l

< <

i=1

t. (t, lx,.,1 (t,1 t.IlX,II' 1" ' (1' 1t, 2) ,/, ,

(He;, y;)I' ) II'

r,(t)(ue" Yk)

I" II"~ dt)

148

QINGYING BU

by Khinchin's inequality,

L· (t." (t.[1 t, I"
<

x."') II, .

(x.);"

<

r,(t)(ue, , yk)

",(t)"""

(X) • ( [

(x,);"

~

11,·

dt)

r,(t)"e,

r,(t)ue,

dt)

(4) Now by Kahane's inequality and (3),

([II t, r,(t)"" II" dt) 'I,' "K,.,.· ([II t, r,(t)ue,II'
(L'r I(t,
q ([

u) .

<

~

r,(t)e;, z)I' dp(z) )

r,(t)e;, z) dt) dP(Z)),,'

z)

u) .

B, .

dp(z) )'/'

r

dp(z)

B, .

(5) Combining (4) and (5),

So

U

E

Dq(H, Y) with 1

Dq(u) :::; -A . Bp· Kp,ql· 7rp(u). ql Now consider a general Hilbert space H. Let

Xl, X2,··· , Xm E

H and

Yi,y2,··· ,y:n, E Y*. Then there is an n E N such that span{xk}r is isometrically isomorphic to i 2. Let P be the orthogonal projection from H onto span{xk}r.

PROPERTIES OF p-SUMMING OPERATORS

149

Then by (6), m

m

L I(UPXk, Yk)1

k=1

k=1

<

:

ql

. Bp' Kp,ql' ll"p(U) . II (PXk)i"lI egtron 9 (X)

. Bp' Kp,ql' ll"p(U) ·11(Xk)i"lIe~tr.on9(X) ql This shows us that U E Dq(H, Y) and again 1 Dq(u) :::; A . Bp' Kp,ql' ll"p(u). ql The proof is complete. :

·11(Yk)i"lle~,.ak(X.)

·11(Yk)i"lle~,eak(X.).

o

ACKNOWLEDGEMENT. I am grateful to my advisor, Professor Joe Diestel, for his good suggestions for this paper.

References 1. H. Apiola, Duality between spaces of p-summable sequences, (p, q)-summing operators and characterization of nuclearity, Math. Ann. 219 (1976), 53-64. 2. Q. Bu and J. Diestel, Observations about the projective tensor product of Banach spaces, I - ip®X, 1 < p < 00, Quaestiones Math. 24 (2001),519-533. 3. J. S. Cohen, Absolutely p-summing, p-nuclear operators, and their conjugates, Math. Ann. 201 (1973), 177-200. 4. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. 5. A. Pietsch, Absolut p-summierende abbildungen in normierten riiumen, Studia Math. 28 (1967), 333-353. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS

E-mail address: qbu(llolemiss. edu

38677

Contemporary Ma.thema.tics Volume 328, 2003

The Unique Decomposition Property and the Banach-Stone Theorem Audrey Curnock, John Howroyd, and Ngai-Ching Wong* ABSTRACT. We show an affine version of the Banach-Stone theorem. Given compact convex sets K and S with unique decomposition property, we show that every surjective linear isometry T between the affine function spaces A(K) and A(S) induces an affine homeomorphism between K and S. Furthermore, T can be written as a weighted composition operator in this case.

1. Introduction

The celebrated Banach-Stone Theorem states that two compact Hausdorff spaces X and Y are homeomorphic if and only if the corresponding real continuous function spaces C(X) and C(Y) are linearly isometric (see, for example, [3, Chapter 7]). As is well known, see for example [2, Theorem 1.4.9], C(X) can be identified with the Banach space A(K) of real continuous affine functions on K = {'P E C(X)* : II'PII = 1 = 'P(l)}, the state space of C(X), where the norm in A(K) is the usual supremum norm. Here K is a Bauer simplex and consequently there is a natural reformulation of the Banach-Stone theorem in the context of affine geometry as follows. Two Bauer simplexes K and S are affinely homeomorphic if and only if their corresponding affine function spaces A(K) and A(S) are linearly isometric. Clearly an affine homeomorphism between any two compact convex sets K and S induces a linear isometry between A(K) and A(S). However, an example of J.T. Chan given in [9] shows that the converse of this cannot hold for arbitrary compact convex sets in locally convex (Hausdorff) spaces, even in finite dimensions. Thus it is natural to ask what conditions on K and S imply that this converse does hold. Lazar [11] showed that the converse holds if both K and S are Choquet simplexes. Ellis and So [9] extended this to the case when both K and S have the property that every pair of complementary closed faces is split, which applies to the state spaces of function algebras. In this paper we use another geometric condition on K and S, namely the unique decomposition property of Ellis [6, 7, 8], under which K and S are affinely homeomorphic whenever there is a linear isometry T between A(K) and A(S). 2000 Mathematics Subject Classification. Primary 46A55; Secondary 46B04. Key words and phrases. Affine function space, isometry, unique decomposition property. *Partially supported by Taiwan National Science Council Grants: 89-2115-Mll0-009,37128F. 151

152

AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG

Moreover, we show that in this case T can be written as a weighted composition operator. This includes Lazar's result as a special case and, more generally, the state spaces of C*-algebras. Further results by the same authors relating to skew-symmetries are to be found in [4]. We would like to express our deepest gratitude to Professor Cho-Ho Chu for many useful discussions. 2. Preliminaries We note that every compact convex set K (in a locally convex space) can be embedded into A(K)* as the state space {cp E A(K)* : Ilcpll = 1 = cp(l)} of A(K) with the weak* topology, where x E K is identified with the linear functional f

1-+

(j, x) = f(x)

for all f in A(K). If Y is a subset of K then we denote the convex hull of Y by co (Y). A point x in K is called an extreme point of K if K \ {x} is a convex set. The set of extreme points of K is denoted by 8K. In the above setting the closed unit ball BA(K)' of A(K)* is the convex hull of K and -K; namely co (K U -K) where -K denotes {-k: k E K}. Thus 8B A(K)' is contained in 8KU8(-K). The reverse inclusion follows easily from the fact that for cp E BA(K)" \ (K U -K) we have -1 < cp(l) < 1. Therefore, 8B A(K)" is equal to 8KU8(-K). A convex subset F of K is called a face of K if whenever x E F with x = >..y + (1 - >..)z for some y, z E K and >.. E (0,1), then both y and z are in F. A pair of faces (FI, F2) is said to be complementary whenever FI n F2 = 0 and K = co (FI U F 2 ). Thus, in this case, each x in K has a decomposition relative to (FI' F2); namely x = >..y + (1 - >..)z for some y E F I , Z E F2 and>" E [0,1]. If >.. is unique, for each x E K \ (FI U F 2 ) but independent of y and z, then FI and F2 are said to be parallel faces; parallel faces are automatically norm closed. If in addition y and z are unique then FI and F2 are called split faces. See, for example, [1, 2] for the general theory of compact convex sets and related topics. Let K be a compact convex set. Recall that the facial topology on 8K is given by defining

{F n 8K : F is a closed split face of K} to be the family of all closed sets. The facial topology is weaker than the relative topology on 8K. The centre Z(A(K)) of A(K) is the set of all those functions in A(K) whose restriction to 8K is facially continuous. The central functions h E Z(A(K)) are characterised by the following property (see, for example, [1, Corollary 11.7.4] or [2, Theorem 3.1.4]): for all f E A(K), there exists 9 E A(K) such that g(x) = h(x)f(x) for all x in 8K. The uniqueness of the continuous affine function 9 is clear, since a continuous affine function on a compact convex set is completely determined by its values on the extreme boundary, and consequently we may write 9 = h . f. In this way it is useful to think of the central functions as the multipliers of A(K). A compact convex set K is a Choquet simplex whenever for all bounded (real) linear functionals cp in A(K)* and 0 > 0 the set K n (cp + oK) is either empty or of the form 'IjJ + (3K for some '¢ in A(K)* and {3 ~ 0 (see, for example, [12]); note that {3 = 0 allows K n (cp + oK) to be a singleton. In a Choquet simplex every closed face is split (see [2, Theorem 2.7.2]).

THE UNIQUE DECOMPOSITION PROPERTY AND BANACH-STONE THEOREM

153

In [6, 7, 8], Ellis defined the unique decomposition property of K by the condition that for every

0 the set Kn( O. In particular, every Choquet simplex has the unique decomposition property. A result of Grothendieck [10], see also [5, p. 272], shows that the state space of a (unital) C*-algebra has the unique decomposition property. Thus the results of this paper apply, giving an affine homeomorphism between the state spaces K and S of two C*-algebras whenever the associated (real) affine function spaces A(K) and A(S) are linearly isometric. We give some examples below, the second and third of which show that the unique decomposition property and the condition of Ellis and So are independent geometric properties of a compact convex set. EXAMPLE 2.1. Let K be the state space of the C*-algebra M 2 , of all 2 x 2 matrices over C. Then, by Grothendieck's result, K has the unique decomposition property. Also, K is affinely homeomorphic to a closed ball in 1R3 (see [2, p. 241]) and satisfies the condition of Ellis and So since it has no proper complementary faces. EXAMPLE 2.2. Let K be a triangular bi-simplex in 3-dimensional space as in the figure below, Figure 1. Then no proper face is (geometrically) parallel to any other and hence K n (

o. It follows, by the geometric characterisation of Ellis, that K has the unique decomposition property. However (F, F') is a pair of complementary faces of K, but not split, and hence K does not satisfy the condition of Ellis and So. EXAMPLE 2.3. Let K be an icosahedron in 3-dimensional space. Then K satisfies the condition of Ellis and So because it has no proper complementary faces. However it does not have t.he unique decomposition property because it has

F

FIGURE 1. The bi-simplex of Example 2.2

154

AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG

faces (geometrically) parallel to each other and hence K n (cp + aK) can have empty interior for some cp E A(K)* and a > O. 3. Results

Throughout this section K and 8 will denote compact convex sets of locally convex (Hausdorff) spaces. We will also let T: A(K) ~ A(8) denote a surjective linear isometry. Notice that if T1 = 1 then for cp E A(8)* with Ilcpli = 1 = cp(1), we have IIT*cpll = licp 0 Til = licpli = 1 = cp(1) = cp(T1) = T*cp(1); here T*: A(8)* ~ A(K)* denotes the dual map of T. Consequently T*(8) = K and hence T* induces an affine homeomorphism a: 8 ~ K such that Tf(s) = T*s(f) = f(a(s)) for all s E 8; that is, T is a composition operator f f-+ f 0 a. In Proposition 3.3 we see that T is a weighted composition operator if and only if T1 is central. To do this we 'decompose' 8 by defining (3.1)

81

= {s E 8: (Tl)(s) = I}

and

82 = {s E 8 : (T1)(s)

= -I}.

It is clear that 8 1 and 8 2 are closed faces of 8. LEMMA 3.1. Let 8 1 and 8 2 be as in (3.1). Then 88 are closed parallel faces of 8.

~

8 1 U 8 2, and 8 1 and 8 2

PROOF. Observe that the dual map T* is a linear isometry from A(8)* onto A(K)*. Hence T* maps the extreme points of the closed unit ball of A(8)* onto the extreme points of the closed unit ball of A(K)*. Thus

T*(88 U 8( -8)) = 8K U 8( -K). Consequently, for each s E 88 we have T* s is in 8K or 8( - K) and hence

Tl(s) = T*s(l) = ±1. Therefore 88 ~ 8 1 U 8 2 and thus by the Krein-Milman Theorem 8 = co (81 U 8 2), since 8, 8 1 and 82 are all (weak*) compact. Thus 8 1 and 8 2 are complementary faces since they are clearly disjoint. For each s in 8 with s = >.x + (1 - >')y, where x E 8 1 , Y E 8 2, and>' E (0,1), we have T* s = >'T*x + (1 - >')T*y. Thus,

= T*s(1) = >'T*x(1) + (1- >')T*y(1) = >. - (1- >.) = 2>' - 1. This establishes the uniqueness of>. = ((Tl)(s) + 1)/2 and the result follows. Tl(s)

0

We now specialise to the case when K and 8 have the unique decomposition property. LEMMA 3.2. Let 8 1 and 8 2 be as in (3.1). 8uppose that K has the unique decomposition property. Then 8 1 and 8 2 are complementary split faces of 8. PROOF. By Lemma 3.1, for each s E 8\(81 U 8 2) we may write s = >.x + (1 - >.)y where x E 8 1 , Y E 8 2 and 0 < >. < 1, and>' is unique. We consider the decomposition T* s = >'T*x - (>. -1)T*y. Since x E 8 1 we have T*x E K and hence >'T*x is positive. Similarly, T*y E -K and hence (>. - I)T*y is positive. Also

IIT*sll = 1 = >. + (1- >.) = Ii>'T*xli + 11(>' -

I)T*yli· Thus, by the unique decomposition property, >'T*x and (>. -1)T*y are unique. By the uniqueness of >., we have T*x and T*y are unique. Since T is surjective, T* is injective and the result follows. 0

THE UNIQUE DECOMPOSITION PROPERTY AND BANACH-STONE THEOREM

155

By replacing T by T- 1 in (3.1) we may 'decompose' K by defining (3.2) K1 = {k E K : T- 11(k) = I} and K2 = {k E K : T- 11(k) = -I}. Applying Lemmas 3.1 and 3.2 to T- 1 we see that K1 and K2 are complementary split faces of K whenever S has the unique decomposition property. We say that T is a weighted composition operator whenever there exists a central function h in A(S) and a continuous affine mapping (1: S -+ K such that Tf = h· f 0 (1 for all f E A(K); that is, Tf(s) = h(s)f((1(s)) for all sEaS. The following proposition asserts that the linear isometry T is a weighted composition operator, with Tl(s) = ±1 on as, if and only if Tl is central. PROPOSITION 3.3. Let T: A(K) -+ A(S) be a linear mapping. Then the following are equivalent: a) T is an isometry and Tl is central; b) T is a weighted composition operator of the form T f = h . f 0 (1 for all f E A(K) where (1 is an affine homeomorphism and h(s) = ±1 for all sEas. PROOF. See [4, Theorem 3.3] or [13].

o

We now apply the above decompositions of K and S to prove our main theorem. THEOREM 3.4. Suppose that K and S have the unique decomposition property. Then the real affine function spaces A(K) and A(S) are linearly isometric if and only if K and S are affinely homeomorphic. Moreover, every linear isometry from A(K) onto A(S) may be written as a weighted composition operator. PROOF. It suffices to show necessity. Suppose that T is a linear isometry from A(K) onto A(S). We 'decompose'S into the complementary split faces Sl and S2 of (3.1) and, similarly, K into the complementary split faces K1 and K2 of (3.2). Since (T-1)* = (T*)-l, we have T*(Sd = K1 and T*(S2) = -K2' and hence we may define (1: S -+ K by (1(,xx + (1 - ,x)y) = ,xT*(x) - (1 - ,x)T*(y)

whenever x E Sl, Y E S2 and 0 :5 ,x :5 1. We see that (1 is an affine homeomorphism from S = co (Sl U S2) onto K = co (K1 U K 2). Moreover, to show that T is a weighted composition operator it suffices, by Proposition 3.3, to show that h = Tl is central. Let f E A(S) then, since (1: S -+ K is an affine homeomorphism, we may write f = gO(1 for some g E A(K). Note that for x E Sl we have Tg(x) = T*x(g) = g((1(x)) = h(x)f(x). Similarly for x E S2 we have Tg(x) = T*x(g) = -g((1(x)) = h(x)f(x). Therefore, for all x E as £;; Sl U S2 we have Tg(x) = h(x)f(x), and the result follows. 0 References [1] E.M. Alfsen, Compact convex sets and boundary Integmls, Ergebnisse der Mathematik, 57, (Springer-Verlag, Berlin-Heidelberg-New York, 1971). [21 L. Asimow and A.J. Ellis, Convexity theory and its applications in functional analysis, London Math. Soc. Monograph, 16 (Academic Press, London, 1980). [3] E. Behrends, M -Structure and the Banach-Stone Theorem, Lecture Notes in Mathematics 736, (Springer-Verlag, Berlin-Heidelberg-New York, 1979). [4] A. Curnock, J. Howroyd, and N.-C. Wong, Isometries of affine function spaces, preprint. [5] J. Dixmier, C*-Algebms (North-Holland Publishing Co., Amsterdem-New York-Oxford, 1982).

AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG

156

[6] A.J. Ellis, An intersection property for state spaces, J. London Math. Soc., 43 (1968), 173176. [7] A.J. Ellis; Minimal decompositions in partially ordered normed spaces, Proc. Camb. Phil. Soc., 64 (1968),989-1000. [8] A.J. Ellis, On partial orderings of normed spaces, Math. Scand., 23 (1968), 123-132. [9] A.J. Ellis and W.S. So, Isometries and the complex state spaces of uniform algebras, Math. Z., 195 (1987), 119-125. [10] A. Grothendieck, Un result at sur Ie dual d'une C*-algebre, J. Math. Pures Appl., 36 (1957), 97-108. [11] A.J. Lazar, Affine products of simplexes, Math. Scand., 22 (1968), 165-175. [12] R.R. Phelps, Lectures on Choquet's Theorem, Second Edition, Lecture notes in Mathematics 1757 (Springer-Verlag, Berlin, 2001). [13] T.S.R.K. Rao, Isometries of Ac(K), Proc. Amer. Math. Soc., 85 (1982), 544-546. SCHOOL OF COMPUTING, INFORMATION SYSTEMS AND MATHEMATICS, SOUTH BANK UNIVERSITY, LONDON SE1 OAA, ENGLAND. E-mail address:curnocaOsbu.ac.uk DEPARTMENT OF MATHEMATICAL SCIENCES, GOLDSMITHS COLLEGE, UNIVERSITY OF LONDON, LONDON SE14 6NW, ENGLAND. E-mail address:masOljdhOgold.ac.uk DEPARTMENT OF ApPLIED MATHEMATICS, NATIONAL SUN YAT-SEN UNIVERSITY, KAOHSJUNG 80424, TAIWAN, R.O.C.

E-mail

address:wong~ath.nsysu.edu.tw

Contemporary Mathematics Volume 328, 2003

A Survey of Algebraic Extensions of Commutative, Unital Normed Algebras Thomas Dawson ABSTRACT. We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.

Introduction Algebraic extensions have had striking applications in the theory of uniform algebras ever since Cole used them (in [5]) to construct a counterexample to the peak-point conjecture. Apart from this, their main use has been in (a) the construction of examples of general, normed algebras with special properties and (b) the Galois theory of Banach algebras. We shall not discuss (b) here; a summary of some of this work is included in [29]. In the first section of this article we shall introduce the types of extensions and relate their applications. The section ends by giving the exact relationship between the types of extensions. Section 2 contains a table summarising what is known about the extensions' properties. A theme lying behind all the work to be discussed is the following question:

(Q) Suppose the normed algebra B is related to a subalgebra A by some specific property or construction. (For example, B might be integral over A: every element b E B satisfies ao + ... +an_1bn-1 +bn = 0 for some ao, ... ,an -1 EA.) What properties of A (for example, completeness or semisimplicity) must be shared by B? This is a natural question, and interesting in its own right. Many special cases of it have been studied in the literature. We shall review the related body of work in which B is constructed from A by adjoining roots of monic polynomial equations. Throughout this article, A denotes a commutative, unital normed algebra, and A its completion. The fundamental construction of [1] applies to this class of 1991 Mathematics Subject Classification. Primary 46J05, 46J10. This research was supported by the EPSRC. © 2003 American Mathematical Society 157

THOMAS DAWSON

158

algebras. Algebraic extensions of more general types of topological algebras have received limited attention in the literature (see [19], [21]). If E is a subset of a ring then (E) will stand for the the ideal generated by E. 1. Types of Algebraic Extensions and their Applications 1.1. Arens-Hoffman Extensions. Let a(x) = ao + ... + an_lx n- l + xn be a monic polynomial over the algebra A. The basic construction arising from A and a(x) is the Arens-Hoffman extension, Ao. This was introduced in [1]. Most of the obvious questions of the type (Q) for Arens-Hoffman extensions were dealt with in this paper and in the subsequent work of Lindberg ([18]' [20], [13]). See columns two and three of Table 2.2. All the constructions we shall meet are built out of Arens-Hoffman extensions. DEFINITION 1.1.1. A mapping 0: A - B between algebras A and B is called unital if it sends the identity of A to the identity of B. An extension of A is a commutative, unital normed algebra, B, together with a unital, isometric monomorphism

O:A-B.

The Arens-Hoffman extension of A with respect to a(x) is the algebra Ao := A[x]/(a(x)) under a certain norm; the embedding is given by the map v: a t-+ (a(x)) + a.

To simplify notation, we shall let x denote the coset of x and often omit the indeterminate when using a polynomial as an index. It is a purely algebraic fact that each element of Ao has a unique representative of degree less than n, the degree of a(x). Arens and Hoffman proved that, provided the positive number t satisfies the inequality t n ~ ~~:~ lIakll tk, then

I~

bkX

k

=

~ IIbkll t

k

(bo, ... , bn -

l

E

A)

defines an algebra norm on Ao. The first proposition shows that Arens-Hoffman extensions satisfy a certain universal property which is very useful when investigating algebraic extensions. It is not specially stated anywhere in the literature; it seems to be taken as obvious. 1.1.2. Let A(l) be a normed algebra and let 0: A(l) _ B(2) be a unital homomorphism of normed algebras. Let al (x) = ao + ... + an_lXn - l + xn E A(l) [x] and B(1) = A~l}. Let y E B(2) be a root of the polynomial a2(x) := O(ad(x) := O(ao) + ... +O(an_l)X n - l +xn. Then there is a unique homomorphism 1>: B(l) _ B(2) such that PROPOSITION

B(1)

~

II

/9

r

B(2)

is commutative and 1>(x) = y.

A(l)

The map 1> is continuous if and only if 0 is continuous. PROOF.

This is elementary; see [7]

o

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

159

1.2. Incomplete Normed Algebras. A minor source of applications of Arens-Hoffman extensions fits in nicely with our thematic question (Q): these extensions are useful in constructing examples to show that taking the completion of A need not preserve certain properties of A. The method uses the fact that the actions of forming completions and ArensHoffman extensions commute in a natural sense. A special case of this is stated in [17]; the general case is proved in [7), Theorem 3.13, and follows easily from Proposition 1.1.2. It is convenient to introduce some more notation and terminology here. Let O(A) denote the space of continuous epimorphisms A --+ Cj when n appears on its own it will refer to A. As discussed in [1], this space, with the weak *-topology relative to the topological dual of A, generalises the notion of the maximal ideal space of a Banach algebra. In fact, it is easy to check that 0 is homeomorphic to 0(..4), the maximal ideal space of the completion of A. The Gelfand transform of an element a E A is defined by

a: n --+ C; W

1--+

w(a)

and the map sending a to a is a homomorphism, r, of A into the algebra, C(O), of all continuous, complex-valued functions on the compact, Hausdorff space O. We denote the image of r by A. A good reference for Gelfand theory is Chapter three of [24]. DEFINITION

1. 2.1 ([1)). The algebra A is called topologically semisimple if

r

is injective. If A is a Banach algebra then this condition is equivalent to the usual notion of semisimplicity. The precise conditions under which Aa is topologically semisimple if A is are determined in [1]. In [17] Lindberg shows that the completion of a topologically semisimple algebra need not be semisimple. In order to illustrate Lindberg's strategy we recall two standard properties of normed algebras.

1.2.2. The normed algebra A is called regular if for each closed subset E ~ 0 and wE O-E there exists a E A such that a(E) ~ {O} and a(w) = 1. The algebra is called local if A contains every complex function, f, on 0 such that every wE 0 has a neighbourhood, V, and an element a E A such that flv = alv. DEFINITION

It is a standard fact that regularity is stronger than localness; see Lemma 7.2.8 of [24). EXAMPLE 1.2.3. Let A be the algebra of all continuous, piecewise polynomial functions on the unit interval, I, and a(x) = x 2 - id/ E A[x]. Let A have the supremum norm. By the Stone-Weierstrass theorem, A = C(I) and hence n is identifiable with I. Clearly A is regular. We leave it as an exercise for the reader to find examples to show that Aa is not local. This is not hardj it may be helpful to know that in this example the space O(Aa) is homeomorphic to {(s, oX) E I xC: oX 2 = s}. This follows from facts in [1]. In the present example, neither localness nor regularity is preserved by (incomplete) Arens-Hoffman extensions.

THOMAS DAWSON

160

Finally we can explain the method for showing that some properties of normed algebras are not shared by their completions because, in the above, 'non-regularity' is not preserved by completion of Ao (nor is 'non-localness'). To see this, note that Ais clearly regular if A is and so by a theorem of Lindberg (see Table 2.2) the ArensHoffman extension (A)o is regular. But, by a result of [17] referred to above, this algebra is isometrically isomorphic to the completion of Aa. Of course Lindberg's original application was much more significant; there are simpler examples of the present result: for example the algebra of polynomials on I.

1.3. Uniform Algebras. It is curious that the application of Arens-Hoffman extensions to the construction of integrally closed extensions of normed algebras did not appear in the literature for some time after [1]. It was seventeen years later until a construction was given in [22]. Even then the author acknowledges that the constuction was prompted by the work of Cole, [5], in the theory of uniform algebras. Cole invented a method of adjoining square roots of elements to uniform algebras. He used it to extend uniform algebras to ones which contain square roots for all of their elements. Apart from feeding back into the general theory of commutative Banach algebras (mainly accomplished in [22] and [23]) his construction provided important examples in the theory of uniform algebras. We shall describe these after recalling some basic definitions. DEFINITION 1.3.1. A uniform algebra, A, is a subalgebra of C(X) for some compact, Hausdorff space X such that A is closed with respect to the supremum norm, separates the points of X, and contains the constant functions. We speak of 'the uniform algebra (A, X)'. The uniform algebra is natural if all of its homomorphisms wEn are given by evaluation at points of X, and it is called trivial if A = C(X).

Introductions to uniform algebras can be found in [4], [11], [26], and [16]. An important question in this area is which properties of (A, X) force A to be trivial. For example it is sufficient that A be self-adjoint, by the Stone-Weierstrass theorem. In [5] an example is given of a non-trivial uniform algebra, (B,X), which is natural and such that every point of X is a 'peak-point'. It had previously been conjectured that no such algebra existed. We shall describe the use of Cole's construction in the next section, but now we reveal some of the detail. PROPOSITION 1.3.2 ([5],[7]). Let U be a set of monic polynomials over the uniform algebra (A, X). There exists a uniform algebra (AU, XU) and a continuous, open surjection 7r: XU ---> X such that (i) the adjoint map 7r*: C(X) ---> C(XU) induces an isometric, unital monomorphism A ---> AU, and (ii) for every a E U the polynomial 7r*(a)(x) E AU[x] has a root Po E AU. PROOF. We let XU be the subset of X x such that for all a E U f(a)(K)

JO

cU consisting of the elements (K, >.)

+ ... + /(0) (K)>.n(0)-1 + >.n(o) n(a)-1 a 0

= 0

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORM ED ALGEBRAS

161

where o:(x) = f~Ol) + ... + f~(l)_lXn(Ol)-l + xn(Ol) E U. The reader can easily check that XU is a compact, Hausdorff space in the relative product topology and so the following functions are continuous: 11":

XU -+ X; (~, >.) ...... ~

POI: XU -+ C; (~, >.) ...... >'01

(0:

E U).

The extension AU is defined to be the closed subalgebra of C(XU) generated by 1I"*(A) U {POI: 0: E U} where 11"* is the adjoint map C(X) -+ C(XU) ; g ...... go 11". It is not hard to check that AU is a uniform algebra on XU with the required properties. 0 We shall call AU the Cole extension of A by U. Cole gave the construction for the case in which every element of U is of the form x 2 - f for some f E A. It is remarked in [22] that similar methods can be used for the general case; these were independently, explicitly given in [7]. By repeating this construction, using transfinite induction, one can generate uniform algebras which are integrally closed extensions of A. Full details of this, including references and the required facts on ordinal numbers and direct limits of normed algebras, can be found in [7]. Again this closely follows [5]. Informally the construction is as follows. Let v be a non-zero ordinal number. Set (Ao,Xo) = (A,X). For ordinal numbers T with 0 < T ~ V we define (A~" , X!:" )

(A.,-, X T )

if T = a

+1

and

= { l~ . (A u,Xu)u<.,-, (* 1I"p,u, 1I"p,u ) p~U
if T is a limit ordinal.

The construction requires sets of monic polynomials, Uu ~ Au [x], to be chosen inductively. The notation (Au, Xu )u<.,-, (1I";,u, 1I"p,u )P~U 0 be an ordinal number. Then (A.,-, X T)T
(A, X).

-

Thus (A1' Xl) is just a Cole extension of (Ao, Xo). When U1 is a singleton we call Al a simple extension of Ao; the same adjective can be applied to ArensHoffman extensions. An integrally closed extension, (Av, Xv), is obtained by taking v to be the first uncountable ordinal. At the successor ordinals the whole set of monic polynomials is frequently used to extend the algebra, but this set is larger than necessary. The same procedure is used to obtain the integrally closed extensions in other categories (to be discussed in Section 1.6).

THOMAS DAWSON

162

1.4. Some Applications of Cole's Construction. Cole's method has been developed by others, including Karahanjan and Feinstein, to produce examples of non-trivial uniform algebras with interesting combinations of properties. We cite the following example of Karahanjan. THEOREM 1.4.1 (from [15], Theorem 4). There is a non-trivial, antisymmetric uniform algebra, A, such that (1) A is integrally closed, (2) A is regular, (3) n is hereditarily unicoherent, (4) G(A) is dense in A, and (5) the set of peak-points of A is equal to n. In the above, G(A) is our notation for the invertible group of A. We refer the reader to [15] and the literature on uniform algebras for the definitions of other terms we have not defined here. A further example in [15] also strengthens Cole's original counter-example. Both examples (of non-trivial, natural uniform algebras on compact, metriseable spaces, every point of which is a 'peak-point') are regular. Feinstein has varied the construction to obtain such an example which is not regular in [10]. The same author also used Cole extensions in [9] to answer a question of Wilken by constructing a non-trivial, 'strongly regular', uniform algebra on a compact, metriseable space. Returning to the sample theorem quoted above, note that some of these properties (for example the topological property of 'hereditary unicoherence') are consequences of the combination of other properties of the final algebra. By contrast, (2) and (4) hold because they are true for the base algebra on which the example is constructed. It is therefore very useful to know exactly when specific properties of a uniform algebra are transferred to those in a system of Cole extensions of it. The known results on this problem are summarised in the first column of Table 2.2. Determining if an algebra's property is shared by its algebraic extensions has led to some interesting devices. We shall elaborate on this topic in the next section. We remark in passing that the methods used in [15] to show that the final algebra has a dense invertible group have been simplified in [8]; in particular there is no need to develop the theory of 'dense thin systems' in [15].

1.5. A Further Remark on Cole Extensions. The reader will notice from Table 2.2 that virtually all properties of uniform algebras are preserved by Cole extensions. The key to obtaining most of these results is the following result, originating with Cole. PROPOSITION 1.5.1 ([5]'[23]). Let (Ar,Xrk:;v be a system of Cole extensions of (A, X). There exists a family of unital contractions (T.,.,r: C(Xr) -+ C(X.,.)).,.::;r::;v such that for all a ~ T ~ v (i) T.,.,r(A r ) ~ A.,., and (ii) T.,.,r 07r;,r = idc(X u )'

o

PROOF. See [23]. For example, it is easy to see from the existence of T: C(Xu) Cole extension AU is non-trivial if A i= C(X).

-+

C(X) that the

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

163

The operator T was constructed in [5] for extensions by square-roots. In the case of a simple Cole extension, (A{o}, X{o}), there are at most two points y±(x:) in the fibre 7r- l (x:) for each x: E X and they correspond to the roots of the equation x 2 - f(x:) = 0 where o(x) = x 2 - f. The operator is then defined by

(g E C(X{o}), x: EX). For other sorts of monic polynomials it was not so obvious how to construct T. The basic techniques appeared in [22] (see the proof of Theorem 3.5) for simple extensions, and were further developed in the proof of Theorem 4 of [15], but it was not until [23] that a comprehensive construction was given. We must also mention the role of E. A. Gorin: he appears to have paved the way for [15] and [23].

1.6. Algebraic Extensions of Normed and Banach Algebras. As we have seen, algebraic extensions have had striking applications in the theory of uniform algebras. They have long been used as auxiliary constructions in the general theory of Banach algebras. Notable examples of this are in [14] and [25]; the latter explicitly uses Arens-Hoffman extensions. However algebraic extensions for Hormed algebras were apparently only studied in their own right in order to generalise the work of Cole and Karahanjan. We now turn to these generalisations. The basic extension generalising Arens-Hoffman extensions is called a standard normed extension. It is defined in the following theorem of Lindberg. THEOREM 1.6.1 ([22]). Let A be a normed algebra and U a set of monic polynomials over A. Let ~ be a well-ordering on U with least element 00' Then there exists a normed algebra, B u , with a family of subalgebras, (Bo)oEU, such that: (i) for all 0, (3 E U, Bo ~ Bf3 if 0 ~ (3, and, (ii) for all (3 E U, Bf3 is isometrically isomorphic to an Arens-HoffmBll extension of B
PROOF. See [22].

o

Lindberg shows how this leads to the construction of Banach algebras with interesting combinations of properties, one of which is integral closedness. Let the isometric isomorphism B
THOMAS DAWSON

164

Let ~Ot be the standard root of a E U, witll associated norm parameter tOt, and suppose (T/Ot)OtEU ~ B(2) is such that £I(a)(T/Ot) = 0 for all a E U. Tllen there is a unique, unital homomorphism ¢: B(l) --+ B(2) such that the following diagram is commutative B(1)

r~

~

B(2)

/9

A(1)

(Note added in proof: The map ¢ is continuous if and only if £I is continuous and

L OtEU

(n(a) -l)log+

(11~OtII) < +00 Ot

where log+ denotes the positive part of the logarithm, max(log,O).) PROOF. A simple application of transfinite methods and Proposition 1.1.2. 0 Purely algebraic standard extensions are defined in [22] and the main content of Lemma 1.6.2 is a statement about these. Narmania gives ([23]) an alternative construction for integrally closed extensions of a commutative, unital Banach algebra, A. His method is rather more conventional than the one used to define standard extensions. If U is a set of monic polynomials over A then the Narrnania extension of A by U is equal to the Banach-algebra direct limit of (As: 8 is a finite subset of U) where each As is isometrically isomorphic to A extended finitely many times by the Arens-Hoffman construction. As this paper is not readily available in English and we shall refer to the explicit construction of Narmania's extensions in the next result, we stop to report the precise details of this. If E is a set, the set of all finite subsets of E will be written E<wo. Let 8 = {ai, ... , am} ~ U and let tOt (a E U) be a valid choice of Arens-Hoffman normparameters (see Section 1.1). It is important to insist that distinct elements a, /3 E U are associated with distinct indeterminates XOt , x{3. Thus 8 is an abbreviation for {a1(xoJ, ... ,am(xo,J}. It is proved carefully in [23] that for q = Ls qsx~ll ... x~;;, E A[x U1 ' · •• ,xo: m ], the algebra of polynomials in m commuting indeterminates over A (s is a mult.iindex in No where No = {O}UN), then (8) +q has a unique representative whose degree in 3:O: j is less than than n(aj), the degree of aj(:ro: j ) (j = 1, ... ,m). For convenience we shall call such representatives minimal. Then if q is the minimal representative of (8) +q, 11(8) + qll := Ls IIqsll t~ll ••• t~: defines an algebra norm on As. The index set, U<wo is a directed set, directed by ~. The connecting homomorphisms VS.T (for 8 ~ T E U<wo) are the natural maps; they are isometries. Thus (sec [24] Section 1.3) Au is the completion of the normed direct limit, D := UsEU<wo As / "', where '" is an equivalence relation given by (8) + q '" (T) + r if and only if q - r E (8 UT) for 8, T E U<wo. Furthermore, the canonical map, Vs, which sends an element of As to its equivalence class in D, is an isometry. Note that A0 is defined to be A. We can now show how the types of extensions we have been considering are related. Many of the idea.'3 behind Proposition 1.6.3 are due to Narmania but we take the step of linking them to Cole and standard extensions.

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

165

1.6.3. Let A be a commutative, unital Banach algebra and U a set of monic polynomials over A. Then, up to isometric isomorphism, Au = Bu. If A is a uniform algebra then we have PROPOSITION

A

u

--

--

= (Au)" = (Bur,

where the closures are taken with respect to the supremum norm. PROOF.

It is easily checked that if B is a normed algebra then the homeomor-

phism 12(B) ---t O(B) induces an isometric isomorphism B" ---t (B)". It is therefore sufficient to prove that Au = Bu and that AU = (Bu)". The last equality follows very quickly from the universal property of standard extensions mentioned above and the simplicity of the definition of AU. We shall only prove the first identification; the second can be proved by a similar approach. Although what follows is routine, we hope that it will help to clarify the details of standard and Narmania extensions. As before let to (0 E U) be a valid choice of Arens-Hoffman norm-parameters for the respective extensions Ao. We shall show that there is then an isometric isomorphism between Bu and D (when defined by these parameters); the result then follows from the uniqueness of completions. For each a E U let Yo be the equivalence cla..<;s [({o(:c o )}) + xo] E D. Since Yo is a root of V0(0)(X) in D there exists, by the universal property of standard extensions, a (unique) homomorphism ¢: Bu ---t D such that ¢I A = V0 and for all a E U, ¢(~o) = Yo' Here, ~o is the the element of Bu associated with x by the isometric isomorphism 1/)0 : B
L,j!!j-111¢(bj)11 tb.

tb

Since the algebras vs(As) are directed there exists 8 E U<W(I such that ¢(bj ) E vs(As) (j = 0, ... ,n(,6) -1). We can assume that 8 = {al,'" ,am} and ol(X) = ,6(x). Let qo,·.· ,qn((3)-IE A[XQ1"" ,XOm] be the minimal representatives such that ¢(bj ) = [(8) + qj] (j = 0, ... ,n(,6) -1). So Ilbjll = Ilqjll (j = 0, ... ,n(,6) - 1). A routine exercise in the transfinite induction theorem shows that for all 'Y E U, ¢(B"() ~ UTE[O,"(]<w" vT(AT). It follows that the

THOMAS DAWSON

166

degree of qj in

XO
is zero. Hence n({3)-l

11¢(b)11 =

L

reS) + qjX~l]

=

j=O n(tJ)-l

(S) +

L

qj.T~l

j=O n({3)-l

=

L

IIqj

II t~l

=

Ilbll ,

j=O

from above. The penultimate equality above follows from noting that the representative of the coset is minimal and then expanding and collecting terms. By the transfinite induction theorem, .:J = U as required. 0

2. A Survey of Properties Preserved by Algebraic Extensions 2.1. Introduction. We summarise in Table 2.2 what is currently known about the behaviour of certain properties of normed algebras with respect to the types of extensions we have been considering. Some preliminary explanation of the entries is in order first. Extra information about the polynomial(s) generating an algebraic extension can help to determine whether certain properties are preserved or not. For example if a(x) has degree nand factorises completely over A with distinct roots AI, ... ,An E A such that for all W E fl, .x:(w) =I- :X;(w) if i =I- j then n(Ao<) decomposes into n disjoint homeomorphs of fl in which case very many properties of A, for example localness, are shared by Ao<' This property, referred to as 'complete solvability', is investigated in [12]. The condition on a(x) most frequently encountered in the literature is that it should be 'separable'. This means that its 'discriminant', which is a certain polynomial in the coefficients of a(x), is invertible in A. It is interesting to compare columns two and three. Of course one can make additional assumptions on the algebra (for example that A be regular and semisimple) but the resulting table would become too large and we have restricted it to three popular categories. References to the results follow the table. We should mention that some of the entries have trivial explanations. For example Sheinberg's theorem, that a uniform algebra is amenable if and only if it is trivial, explains the entries for amenability in column one. Also, applying the Arens-Hoffman construction to a uniform algebra need not result in a uniform algebra so not all the entries make sense. We have already met most of the properties listed in the table. We end this section by discussing the ones which have not yet been specially mentioned. 1. Denseness of the invertible group. Although this property is self-explanatory it might not be obvious why it is listed. However, the condition G(A) = A appears in the literature in various contexts; see for example [8].

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

167

2. The Banach algebra, A, is called sup-norm closed if A is uniformly closed in C(f2) (and therefore a uniform algebra). It is called symmetric if A is selfadjoint. 3. For the definitions of 'amenability' and 'weak amenability' we refer the reader to section 2.8 of [6]. All the properties in the table are preserved by forming the standard unitisation of a normed algebra. Most of these results are standard facts or ea'iy exercises; some are true by definition. However this question does not fit into our scheme because the embedding is not unital in this case.

2.2. Table. Cole extensions have only been defined for uniform algebra'l; the algebra is therefore assumed to be a uniform algebra throughout column one. Colulllns two and three, a'l mentioned above, refer to Arens-Hoffman extensions of a normed algebra, A, by a monic polynomial a(x); in column three it is given that a(:1:) is separable. Type of Extension: Property:

Cole

A.-H.

Ac.

A.-H. standard Narmania a sep.

for normed algebras

complete topologically semisimple non-local local regular

1.

2. 3. 4. 5.

• • • ?

• 0

•

• • •

•

0

? '?

?

'?

'!

• • • • •

• •

• • • •

0

0

•

0

0

•

•

0

•

'?

?

for Banach algebras

6. 7. S. 9. 10.

local regular dense invertible group sup-norm closed symmetric 11. amenable 12. weakly amenable

?

0 0 0 0

for uniform algebms

13. non-trivial 14. trivial 15. natural

• • •

Key • property is always preserved o property is sometimes, but not always preserved

? ?

? ?

?

•

• •

0

0

0

0

0

0

0

0

168

THOMAS DAWSON

? not yet determined - it doesn't always make sense to consider this property here References for the Entries. If we do not mention an entry here, it can be taken that the result is an immediate consequence of the definition or was proved in the same paper in which the relevant extension was introduced (that is in [5], [1], [22], or [23]). The results of row three are not hard to obtain, using appropriate versions of Proposition 1.5.1. Localness and regularity were discussed in Section 1.2. The main result about this is due to Lindberg in [18]; the same section of his paper also deals with the results on the symmetry of Arens-Hoffman extensions. That regularity passes to direct limits of such extensions has been widely noted by many authors, for example in [15]. Results of row eight follow from [8]; the case of Cole extensions was partially covered in [15], but the reasoning is not clear. The property of being sup-norm closed was investigated in [13]; this work was generalised in [28]. Finally, examples of amenable Banach algebras which do not have even weakly amenable Arens-Hoffman extensions have been known for a long time. For example, the algebra C E9 C under the multiplication (a, b) (e, d) = (ae, be + ad) is realisable as an Arens-Hoffman extension of C. Examples with both A and Ao semisimple have been found by the author. However the entries marked "?' in rows eleven and twelve represent intriguing open problems. 3. Conclusion The table in Section 2.2 still has gaps, and there are many more rows which could be added. For example it would be interesting to be able to estimate various types of 'stable ranks' (see [2]) of the extensions in terms of the stable ranks of the original algebras. (The condition G(A) = A is equivalent to the 'topological stable rank' of A not exceeding 1.) Remember too that there are many more questions which can be asked, of the form: 'if n has the topological property P, does n(Ao) have property P?' By way of a conclusion we repeat that algebraic extensions have proved immensely useful in the construction of examples of uniform algebras. There is therefore great scope for and potential usefulness in augmenting Table 2.2. It might also be valuable to reexamine the techniques used to obtain the entries to produce more general results (of the kind in [28] for example) in the context of question (Q).

References 1. Arens, R. and Hoffman, K., Algebraic Extension of Normed Algebras., Proc. Am. Math. Soc. 7 (1956), 203-210. 2. Badea, C., The Stable Rank of Topological Algebras and a Problem of R. G. Swan., J. Funet. Anal. 160 (1998), 42-78. 3. Batikyan, B. T., Point Derivations on Algebraic Extension of Banach Algebra., Lobachevskii J. Math. 6 (2000), 3-37. 4. Browder, A., Introduction to Function Algebras., W. A. Benjamin, Inc., New York, 1969.

ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS

169

5. Cole, B. J., One-Point Parts and the Peak-Point Conjecture., Ph.D. Thesis, Yale University, 1968. 6. Dales, H. G., Banach Algebms and Automatic Continuity., Oxford University Press Inc., New York,2000. 7. Dawson, T. W .. Algebmic Extensions of Normed Algebms., M.Math. Dissertation, accessible from the web at: http://xxx.lanl.gov/abs/math.FA/0102131, University of Nottingham, 2000. 8. Dawson, T. W., and Feinstein, J. F., On the Denseness of the Invertible Group in Banach Algebms., Proc. Am. Math. Soc. (to appear). 9. Feinstein, J. F., A Non-Trivial, Strongly Regular Unif01m Algebm., J. Lond. Math. Soc. 45 (1992), 288-300. 10. Feinstein, J. F., Trivial Jensen Measures Without Regularity., Studia Math. 148 (2001).6774. 11. Gamelin, T. W., Uniform Algebms., Prentice-Hall Inc., Engelwood Cliffs, N. J., 1969. 12. Gorin, E. A., and Lin, V ..J., Algebmic Equations with Cont'inuous Coefficients and Some Problems of the Algebmic Theory of Bmids., Math. USSR Sb. 7 (1969), 569-596. 13. Heuer, G. A., and Lindberg, J. A., Algebmic Extensions of Continuous Function Algebms., Proc. Am. Math. Soc. 14 (1963),337-342. 14. Johnson, B. E., Norming C(O) and Related Algebms., Trans. Am. Math. Soc. 220 (1976), 37-58. 15. Karahanjan, M. I., Some Algebmic Chamcterizations of the Algebm of All Continuous Functions on a Locally Connected Compactum., Math. USSR Sb. 35 (1979),681-696. 16. Leibowitz, G. M., Lectures on Complex Function Algebm,~., Scott, Foresman and Company, Glenview, Illinois, 1970. 17. Lindberg, J. A .• On the Completion of Tractable Normed Algebms., Proc. Am. Math. Soc. 14 (1963),319-321. 18. Lindberg, J. A., Algebmic Extensions of Commutative Banach Algebms., Pacif. J. Math. 14 (1964), 559-583. 19. Lindberg, J. A., On Singly Genemted Topological Algebras, Function Algebras (ed. Birtel, F. T.), Scott-Foresman, Chicago, 1966, pp. 334-340. 20. Lindberg, J. A., A Class of Commutative Banach Algebms with Unique Complete Norm Topology and Continuous Derivations., Proc. Am. Math. Soc. 29 (1971), 516-520. 21. Lindberg, J. A .. Polynomials over Complete l.m.-c. Algebms and Simple Integml Extensions., Rev. Roumaine Math. Pures Appl. 17 (1972), 47-63. 22. Lindberg, J. A., lntegml Extensions of Commutative Banach Algebms., Can. ,/. Math. 25 (1973), 673-686. 23. Narmaniya, V. G., The Construction of Algebmically Closed Exten,~ions of Commutative Banach Algebms., Trudy Tbiliss. Mat. Inst. Razmadze Akad. 69 (1982), 154-162. 24. Palmer, T. W., Banach Algebras and the Geneml Theory of *-Algebms. Vol. 1, Cambridge University Press, Cambridge, 1994. 25. Read, C. J., Commutative, Radical Amenable Banach Algebms., Studia Math. 140 (2000), 199-212. 26. Stout, E. L., The Theory of Uniform Algebms., Bogden and Quigley Inc., Tarrytown-onHudson, New York, 1973. 27. Taylor, J. L., Banach Algebms and Topology, Algebras in Analysis. (ed. Williamson, J. H.), Academic Press Inc. (London) Ltd., Norwich:, 1975, pp. 118-186. 28. Verdera, J., On Finitely Genemted and Projective Extensions of Banach Algebms., Proc. Am. Math. Soc. 80 (1980), 614-620. 29. Zame, W. R., Covering Spaces and the Galois Theory of Commutative Banach Algebms., J. Funet. Anal. 27 (1984), 151-171.

Acknowledgements The author would like to thank the Division of Pure Mathematics and the Graduate School at the University of Nottingham for paying for his expenses in

170

THOMAS DAWSON

order to attend the 4th Conference on FUnction Spaces (2002) at the Southern Illinois University at Edwardsville. The author is grateful to Mr. Brian Lockett who provided him with a translation of the paper [23]. Special thanks are due to Dr. J. F. Feinstein who offered much valuable advice and encouragement and also proofread the article. DIVISION OF PURE MATHEMATICS, SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF NOTTINGHAM, UNIVERSITY PARK, NOTTINGHAM, NC7 2RD, UK.

E-mail address:pmxtwdlDnottingham.ac.uk

Contemporary Mat.hematics Volume 328, 200a

Some more examples of subsets of Co and Ll[O, 1] failing the fixed point property P.N. Dowling, C.J. Lennard, and B. Thrett We give examples of closed, bounded. convex, non-weakly compact subsets of Co on which the right shift is expansive, and we construct two nonexpansive self-mappings (one affine and one non-affine) on these sets which fail to have a fixed point. We also prove that every closed, bounded, convex subset of L1 [0,1] with a non-empty interior fails the fixed point property for nonexpansive mappings. Finally, we extend this result by showing that every closed, bounded, convex subset of L1 [0,1] that contains a non-trivial order interval must fail the fixed point property. ABSTRACT.

1. Introduction

In [5], Llorens-Fuster and Sims construct examples of closed, bounded, convex subsets of Co that are not weakly compact but are compact in a topology that is slightly coarser than the weak topology, and they nonetheless fail the fixed point property for nonexpansive mappings. These examples led Llorens-Fuster and Sims to conjecture that closed, bounded, convex non-empty subsets of Co have the fixed point property if and only if they are weakly compact. This conjecture has been recently settled in the affirmative [2, 3]. All the examples constructed in [5] had a common feature - they all support a nonexpansive right shift. In the first part of this short note we produce a collection of sets of the type considered by Llorens-Fuster and Sims, but which do not support a nonexpansive right shift and yet they fail the fixed point property for nonexpansive mappings. In fact, we will produce two nonexpansive fixed point free mappings: one affine and the other non-affine. Variations on the themes of these examples are important in the papers [2, 3]. Llorens-Fuster and Sims [5] also proved that a closed, bounded, convex subset of Co with non-empty interior fails the fixed point property for nonexpansive mappings. In the second part of tllis note we prove an analogous statement in the setting of L1 [0, 1]. We also generalize this result to show that every closed, bounded, convex 2000 Mathematics Subject Classification. Primary 47HI0, 47H09, 46E30. The authors wish to thank Professor Kaz Goebel for his helpful suggestions concerning the proof of Theorem 3.2. The second author thanks the Department of Mathematics and Statistics at Miami University for their hospitality during part of the preparation of this paper. He also acknowledges the financial support of Miami University.

© 171

2003 American l\1athematical Society

P.N. DOWLING, C.J. LENNARD, AND B. TURETT

172

subset of L1 [0, 1J that contains a non-trivial order interval must fail the fixed point property. We refer the reader to the text of Goebel and Kirk [4J for any unexplained terminology. 2. The fixed point property in Co We begin this section with the Llorens-F\lster and Sims examples. We have slightly modified their examples to simplify the computations. Let r denote the set of sequences 'Y = ("(n)n in the interval (0,1). For all 'Y E r, define K., by

K...,

:=

{x

=

("(ntn)n E Co 112: h 2: t2 2: t3 2: ... 2: O} .

We define the right shift Ton K..., by

T( ("(1 t 1 , 'Y2t2, 'Y3 t3, 'Y4t4, ... ))

:=

("(1, 'Y2t 1, 'Y3 t2, 'Y4t3, ... ).

Note that if x = ("(ntn)n and y = ("(nsn)n are elements of K..." then Ilx - yll sUPn 'Ynlt" - 8 n l and IIT(x) - T(y)11 = SUPn 'Yn+1Itn - 8 n l· Clearly, if the sequence ("(n)n is decreasing, then T is a nonexpansive mapping on K..., - this is the case considered by Llorens-Fuster and Sims [5J. However, it is equally obvious that if the sequence ("(n)n is strictly increasing, then T is an expansive mapping on K...,; that is, IIT(x) - T(y)11 > IIx - yll whenever x i= y. We will show that even though some of these sets do not support a nonexpansive right shift, they do support nonexpansive fixed point free mappings. To simplify our computations we will only consider the sets K..., where the sequence 'Y = ("(n)n is in (0,1), is strictly increasing and satisfies 1 - 'Yn < 4- n for all n E N. EXAMPLE 2.1. Let I be the identity mapping on K..." let T be the right shift defined above, T2 = ToT, T3 = ToT 0 T, and so on. Define R: K..., --+ K..., by

R

:= ~I

+ -b T + -1a T2 + 21

4

T3

+... .

A simple calculation shows that if x = ("(1 t1, 'Y2t2, 'Y3t3, ... ) E K..." then

R(x) = ('Y1(~t1

+ ~),'Y2(~t2 + it1 + i),'Y3(~t3 + it2 + kt1 + ~), ... ). that if R(x) = x, then tn = 1 for all n E N, and thus

It is easily seen x is not an element of K...,; that is, R is fixed point free on K...,. To see that R is nonexpansive on K..." let x = ("(ntn)n and y = ("(nsn)n be elements of K...,. Then, 12 + - 4 1 + ... + 2,,1...,1 ) < 1 for each n E N. since 1 - 4- n < 'Yn < 1, we have 'Yn( -...,,, "",,-1 Consequently,

IIR(x) - R(y)11

sup hnl~(tn - sn) n

+ i(tn-1

- sn-d

+ ... + 2~ (t1

-

< sup hn(~ltn - snl + iltn-1 - 8 n -11 + ... + 2~' It 1 n

< sup {'Yn(_l+ _1_ + ... + +) max 'Yilti - Sil} n 2...,,, 4"",,-1 2""1 l::;'i::;n

< sUP'Ynltn n

8n

l

Ilx-yll· Thus R is a nonexpansive mapping on K...,.

st)l} Sl

J)}

173

EXAMPLES OF FIXED POINT FREE MAPS

The mapping R, given in example 2.1, is an affine mapping on K"I' Our next example is non-affine on K"I' EXAMPLE 2.2. For an element x = (xn)n in K"I' we denote by bl,X) the sequence bl,Xl,X2,X3, ... ). We define a mapping S: K"I -+ Co by S(x):= X, for each x E K"I' where x = (Xl, X2, X2, ... ) is the decreasing rearrangement of the sequence bl' x); that is, X = bl, x)*. Note that

X= (Xl,X2,X3, ... ) =

(11

(~~) ,12 (~:) ,13 (~:) , ... ).

Also Xl 2: X2 2: X3 2: ... 2: 0 and 0 < 11 < 12 < 13 < ... < 1. Therefore ~ > ~ > ~ > ... > O. Since ~ = max("("x) > 1 S(x) does not necessarily belong ~-n-n~ ~ -, to K"I' However, the mapping S is nonexpansive on K"I because the operation of decreasing rearrangement is nonexpansive on Co, so for all x and y in K"I' we have IIS(X) - S(Y)II = IIx -

yll

IIbl,X)* - bl,Y)*11 :::; Ilb1.x) - b1.y)11

=

=

Ilx - YII·

We now introduce a modification U of S that will be nonexpansive and fixed point free on K"I' Define U : K"I -+ Co by

Since

11j AXj - 1j AYj I :::; IXj - Yj I for all j E N, it follows that IIU(x) - U(Y)II :::;

Ilx - yll = IIS(x) - S(Y)II :::; IIx - YII·

Thus U is a nonexpansive mapping on K"I' Furthermore, since 1j AXj =

1j (1 A ~) for all j E Nand 1 2: 1 A ~ 2: 1 A ~ 2:

1 A ~ 2: ... 2: 0, U maps K"I into K"I' To finish, we will show is fixed point free on K"I' Suppose, to get a contradiction, that there exists x E K"I such that x = U(x). Thus, for all j EN, Xj = 1j AXj. A well-known fact about decreasing rearrangements that we will use is that for for each mEN, all W E

ct,

WI

+ ... + Wm

:::;

wi + ... + w;;'.

Since 1 = (1n)n is strictly increasing with limit 1, while X is decreasing with 1 > Xl 2: 11, there exists a unique kEN such that Xk+1 < 1k+l and Xk 2: 1k. Thus, for all mEN with m > k, we have, k + Xl

+ ... + Xm+l > (Xl + ... + Xk) + (Xl + ... + xm+d (Xl

+ ... + Xk) +

bl A Xl + ... + 1k A Xk + 1k+l A Xk+1 + ... + 1m+1 A Xm+1 = (Xl + ... + Xk) + bl + ... + 1k + Xk+l + ... + xm+d bl + ... + 1k) + (Xl + ... + Xk + Xk+1 + ... + Xm+1) bl + ... + 1k) + (Xl + ... + Xm+l) > bl+"'+1k)+bl+Xl+"'+Xm),

174

P.N. DOWLING, C.J. LENNARD, AND B. TURETT

It follows that for all

Tn

E N with Tn

> k, k

Xm

+l

>

bl+"'+'Yk)-k+'Yl='YI-L(1-'Yj) j=1

> 'Y1 -

=

L (1 -

'Yj) ~ 'Yl -

j=1

=.

L 4-

J

1

= 'Yl -

:3

1

>

:3'

j=1

This contradicts the fact that x E Co and so completes the proof that U is fixed point free on K-y.

3. The fixed point property in £1 [0, 1] One of the most notable works in metric fixed point theory is the construction of Alspach [1] of a non-empty weakly compact convex subset of Ll[O, 1] which fails the fixed point property. We begin this section by recalling some of the details of Alspach's construction. Let C:= {f E Ll[O, 1] : 0::; f(t) ::; 1, for all t E [0, I]}. Now define T: C -+ C by

Tf(t) := {min{2f (2t), I} max{2f(2t - 1) - 1, O}

for 0 ::; t ::; ~ for ~ < t ::; 1.

for all f E C. Alspach showed that the mapping T is an isometry on C which has two fixed points; namely 0 and X[O,lj' Alspach also showed that T is an isometric self map of the closed convex subset Co := {f E C : J~ f dm = 1/2} of C, such that T is fixed point free on Co. Here, m denotes Lebesgue measure. We now follow a modification of Alspach's example due to Sine [7]. Define S : C -+ C by S(f) := X[O,lj - f, for all f E C. The mapping S is clearly an isometry of C onto C. Thus the mapping ST is a nonexpansive mapping on C. Sine proved that ST is fixed point free on C. In [5], Llorens-FUster and Sims prove that a closed bounded convex subset of Co with non-empty interior fails the fixed point property. We will use the above construction of Alspach, and modification by Sine, to prove a result analogous to the Llorens-FUster and Sims result in the setting of Ll [0, 1]. Specifically we prove the following result. THEOREM 3.1. Let K be a closed, bounded, convex subset of Ll [0, 1] with nonempty interior. Then K fails the fixed point property for non expansive mappings. PROOF. By translating and scaling, we ca.n assume that K contains the unit ball of £1 [0, 1]. Consequently, the set C, constructed above, is a subset of K. Define the mapping R : K -+ K by

Rf(t) := min{lf(t)l, I}, for 0::;

f

E

t::; 1, for all f

E

K.

It is easily seen that R is a. nonexpansive mapping on K and R(f) E C for all K. Now define U: K -+ K by

U(f) := ST(R(f)), for all

f

E

K.

The mapping U is nonexpansive since all of the mappings, R, S, and Tare nonexpansive.

EXAMPLES OF FIXED POINT FREE MAPS

175

We now show that U is fixed point free. Suppose that f E K is a fixed point of U, that is, U(f) = f. Since f E K, R(f) E C, and since ST maps C into C, f = U(f) = ST(R(f)) E C. Note that the mapping R restricted to C is the identity on C. Therefore, f = ST(R(f)) = ST(f) and so f is a fixed point of ST in C. This contradicts Sine's result that ST has no fixed point in C [7], and thus the proof is complete. D THEOREM 3.2. Let K be a closed, bOllnded, convex sllbset of Ll[O, 1] that contains an order interval [h,g] := {f E L1[0, 1] : h ::; f ::; 9 a.e.}, for some h,g E Ll[O, 1] with h ::; 9 a.nd h ::f g. Then K fa'ils the fixed point property for nonexpansive mappings,

a a

a

PROOF. By translating by -h, we may assume that h = and 9 ~ a.e. with 9 non-trivial. Next, note that there exists a real number c > and a measurable set E with Lebesgue measure rn(E) > 0, such that 9 ~ CXE. By rescaling K by lie, we may assume without loss that c = 1. Now, define the mapping R : K ~ [0, xel ~ K by R(f) := ifi /\ XE. Note that R is nonexpansive and R equals the identity on [0, xel. At this stage, consider E. There exists to in the interval [0,1] such that rn(En [0, to]) = ~ rn(E), Let E1.1 := En [0, to] and E1.2 := En (to, 1]. Clearly E is the disjoint union of E1.1 and E 1,2 and rn(E 1,1) = rn(El,2) = ~ rn(E). Proceed iteratively from here. Similarly to above, there exist pairwise disjoint measurable subsets E2,1, E 2,2, E2,3 and E2,4 of [0, 1] such that El,l = E 2,1 U E2,2, E 1 ,2 = E 2,3 U E 2.4 , and each rn(E2,k) = ~ rn(E). Repeating this construction inductively, we produce a family of measurable subsets (EO,1 := E, En,k : n E N, k E {I, ... , 2n}) of [0,1] such that (XE",k)n.k is a dyadic tree in Ll [0,1]. Moreover, letting the measure v be defined on the measurable subsets of E by v = (1/rn(E))rn, it follows that the Banach space L 1 ( E, v) is isometrically isomorphic to L 1 ( [0, 1], m) = L 1 [0, 1] via the mapping Z defined as follows: Z(XEn, k):= X[k-l k), 2l'r'2"'t"

for each XE",k' Then Z is extended to L := the linear span of the functions XE",k in the usual way. Of course, Z is an isometry on L. Finally, since L is dense in Ll(E, v), with dense range in Ll [0,1], Z extends to a linear isometry from L1(E, v) onto Ll[O, 1]. Let W := ST be Sine's variation on Alspach's example, as described above, and note that W maps the order interval C := [0, X[O,I]] into C. Let's use W to define E : [0, xel ~ [0, XE], by E := Z-1 W Z. We have that E is a fixed point free Ll [0, 1]-isometry on [0, xel. Finally, we define U : K ~ [0, XE] ~ K via U := E R. In a manner analogous to the argument in the proof of Theorem 3.1 above, we see that U is a fixed point free Ll[O, 1]-nonexpansive mapping on K. D REMARK 3.3. In [6], MatlI'ey proved that closed, bounded, convex, non-empty subsets of reflexive subspaces of Ll [0, 1] have the fixed point property for nonexpansive mappings. Consequently, Maurey's result, in tandem with Theorem 3.2, shows that reflexive subspaces of U [0,1] cannot contain a non-trivial order interval. In fact, as pointed out by the referee, the argument in the proof of Theorem 3.2 shows

176

P.N. DOWLING, C.J. LENNARD, AND B. TURETT

that infinite-dimensional subspaces of £1[0,1] which contain non-trivial order intervals actually contain isometric copies of £1 [0, 1] and thus are nonreflexive. The authors thank the referee for his/her comments. References 1. D. Alspach, A fixed point free nonexpansive mapping, Proc. Amer. Math. Soc., 82 (1981), 423-424. 2. P.N. Dowling, C.J. Lennard and B. Thrett, Characterizations of weakly compact sets and new fixed point free maps in co, to appear in Studia Math. 3. P.N. Dowling, C.J. Lennard and B. Thrett, Weak compactness is equivalent to the fixed point property in co, preprint 4. Kazimierz Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990 5. Enrique Llorens-Fuster and Brailey Sims, The fixed point property in co, Canad. Math. Bull. 41 (1998), no. 2, 413-422. 6. B. Maurey, Points fixes des contractions de certains faiblement compacts de Ll, Seminaire d'Analyse Fonctionelle, 1980-1981, Centre de Mathematiques, Ecole Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19. 7. R. Sine, Remarks on an example of Alspach, Nonlinear Anal. and Appl., Marcel Dekker, (1981), 237-241. DEPARTMENT OF MATHEMATICS AND STATISTICS, MIAMI UNIVERSITY, OXFORD, OH

45056

E-mail address: dowlinpnGmuohio. edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PITTSBURGH, PITTSBURGH, PA

15260

E-mail address: lennard+lDpi tt. edu DEPARTMENT OF MATHEMATICS AND STATISTICS, OAKLAND UNIVERSITY, ROCHESTER,

48309 E-mail address: turettlDoakland. edu

MI

Contemporary Mathelnatics Volume 328. 2003

Homotopic composition operators on Hoo (Bn) Pamela Gorkin, Raymond Mortini, and Daniel Suarez We characterize the path components of composition operators on Hoo(B n ), where Bn is the unit ball of en. We give a geometrical equivalence for the compactness of the difference of two of such operators. For n = 1, we give a characterization of the path components of the algebra endomorphisms. ABSTRACT.

1. Introduction

Consider the Hardy space H2 on the unit disk D. Littlewood's subordination principle tells us that for an analytic self-map ¢ of D and a function f in H2, the function f 0 ¢ is once again in H2. Thus one defines the composition operator C'" on H2 by C",(f) = f 0 ¢. The interplay of operator theory and function theory leads to several interesting results. One of these results is Berkson's theorem on isolation of composition operators (see [1] and [14]): THEOREM 1 (Berkson). Let ¢ be an analytic self-map of D. If ¢ has mdial limits of modulus one on a set E of positive measure, then for every other analytic self-map 'l/J of D, the following estimate holds:

IIC", - C",II2:

Jmea;(E),

where C'" and C'" are the corresponding composition opemtors on H2.

Thus, Berkson's theorem tells us that every such operator is isolated in the set of composition operators in the operator norm topology. For example, the identity operator, C z , is at least a distance of ..[f72 from every other composition operator on H2 (as is C"', where ¢ is any inner function). However, not every composition operator is isolated. If ¢ is analytic and ¢ : D ~ sD for some s with 0 < s < 1, then it is easy to check that

Thus, C'" is not isolated. Inner functions induce highly noncompact operators, as well as isolated operators. The operators C'" for which ¢(D) is contained in sD for some s with 0 < s < 1 2000 Mathematics Subject Classification. Primary 47B33; Secondary 47B38. Key words and phrases. composition operator, path components, compact differences.

© 177

2003 American lvlathematical Society

178

PAMELA GORKIN. RAYMOND MORTINI. AND DANIEL SUAREZ

are compact. As Shapiro and Sundberg [14J indicate in their paper, "compact composition operators are dramatically nonisolated." They show that the set of compact composition operators is path connected, and therefore these operators are never isolated. It is interesting to ask which composition operators are, in fact, isolated. Shapiro and Sundberg studied this problem, and showed (among other things) that if ¢ is au analytic self-map of D that is not an extreme point of the algebra HOC(D), then C is not isolated; in their words, "isolated composition operators can only be induced by extreme points." This allowed them to exhibit an example of a non-compact non-isolated operator. They also raised several questions at the end of their paper: (1) Characterize the components in Comp(H2), the space of all composition operators on H2. (2) Which composition operators are isolated? (3) Characterize composition operators whose difference is compact. Before stating the final question, we remind the reader that the essential norm of an operator T defined on a Banach space H is the distance to the compact operators; that is, liT lie = inf{IIT - KII : K compact on H}. It is clear that IITII ~ IITlle, and therefore every essentially isolated operator is isolated. In fact, because of the abundance of weakly null sequences in H2, all the results on isolation appearing in Shapiro and Sundberg's paper hold true if we replace the norm with the essential norm (see [14], p. 148). Thus they raised the following question. (4) Is every isolated operator essentially isolated? Other papers of interest on this subject include [10J. Of course, one is not limited to the space H2, and the study of composition operators on various spaces has lead to a large body of literature. In this paper, we are interested in the same problems for composition operators on HOC (Bn), where Bn denotes the open unit ball in en. While the problem 011 H2 seems to be difficult, MacCluer, Ohno and Zhao [l1J were able to obtain partial results about operators on the algebra HOO(D). They showed that for two analytic self-maps of the disk, C and C.p are in the same path component in the space of composition operators, Comp(HOC(D)), if and only if IIC", - c.pll < 2. In particular, an operator C is isolated in Comp(HOC(D)) if and only if IIC", - C",II = 2 for any other analytic self-map 'IjJ of the disk. The authors show that this result can be rephra..<;ed in terms of the pseudohyperbolic metric, and they posed the question of whether or not every isolated composition operator on HOC is, in fact, essentially isolated. The answer to this la..<;t question was given by Hosokawa, Izuchi and Zheng [8J. Their technique was to develop something called asymptotic interpolating sequences, or a.i.s. for short. Essentially, this definition allowed them to interpolate sequences with a good bound on the norm (see [6J for more information about these sequences). They then used these sequences and Blaschke products to obtain HOC(D) functions that provide good estimates for the essential norm of the difference of two composition operators.

HOMOTOPIC COMPOSITION OPERATORS ON Hoo(Bn)

179

In this paper, we give simpler proofs of the results obtained by Hosokawa, Izuchi and Zheng, and combine them with the proofs of MacCluer, Ohno and Zhao. Because our proofs are significantly simpler and do not refer to asymptotic interpolating sequences or Blaschke products, we are able to obtain the results on the ball in en. While our results do not rely on interpolating sequence results, they do rely on a construction of Gamelin and Garnett relating interpolating sequences to peak sets [5]. The proof we will provide is simple enough to be applicable to other algebras. We conclude the paper with an example of such an application: Using these same techniques, we are able to "lift" these results to obtain a characterization of the path components of endomorphisms of HOO(D). After we completed this paper, we learned that some of the results on composition operators on Hoo(Bn) were also obtained by Carl Toews [15]. Two other papers directly related to the results described here are [3] and [9]. Finally, we mention that recent results on Shapiro and Sundberg's first question in the space H2 can be found in [12].

2. Preliminary results Our goal is to prove results about composition operators on Hoo(Bn) and endomorphisms of HOO(D). In this section, we present proofs of several lemmas that will be important in obtaining estimates on norms and essential norms of operators. Our discussion begins with functions of n variables and Z = (Zl' Z2, .•. , zn), where each Zj is a complex number. As usual, the associated norm of Z is given by

Izi =

(z, z)1/2,

and the unit ball Bn is the set of all Z E en for which Izi < 1. We let Hoo(Bn) denote the space of bounded holomorphic functions on Bn. If n = 1 our situation reduces to the familiar space of functions on the unit disk: HOO(D). We need some background on general uniform algebras, some information specific to Hoo(Bn) and some deeper results for HOO(D). Everything we need is presented in this paper. Let A be a uniform algebra. The maximal ideal space of A, denoted M(A), is the set of complex-valued, linear, multiplicative maps of A that map the identity of A to the value 1 E C. Since evaluations at points of Bn are linear multiplicative functionals on Hoo(Bn), we may think of Bn as a subset of M(Hoo(Bn)). It is well known that M(A) is a compact Hausdorff space when endowed with the weak-* topology induced by A * . We will always consider this topology for M(A). For an element a E A, the Gelfand transform of a is a complex-valued map defined on M(A) by a(x) = x(a). This map establishes an isometric isomorphism between A and a closed subalgebra of C(M(A)). It is usual to identify the function with its Gelfand transform, since the meaning is generally clear from the context. For x, y E M(A), the pseudohyperbolic and hyperbolic metrics are defined, respectively, by

p(x, y) = sup{l/(y)1 : I E A, 11/11 ::; 1, and I(x) = O} and y) h( x, y ) -- Iog 11 + p(x, ( ). -p x,y

180

PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ

It is well-known that P is a [0, 1]-valued metric and that h is a [0, +oo]-valued metric on M(A). The triangle inequality for h immediately implies that the condition p(x, y) < 1 (i.e. h(x, y) < 00) is an equivalence relation on M(A). If A is a uniform algebra and ¢ : M(A)-+M(A) is a continuous map such that a 0 ¢ E A for all a E A, then the map C'" defined by

C",a = a 0 ¢ is an endomorphism of A. We will write C'" E End (A). Conversely, if E E End (A), we may define ¢ : M(A)-+M(A) by ¢(x)(a) = x(E(a)), obtaining E = C"'. The Shilov boundary of A is the smallest closed subset of M(A) on which every function in A attains its maximum. We denote the Shilov boundary by 8A. LEMMA 2. Let A be a uniform algebra and let C"', C1/J E End (A). Let V M(A) be a set whose closure contains 8A. Then,

(2.1)

sup xEV

PROOF.

c

2p(¢(x),1/I(x)) J :::; IIC", - C1/J1I :::; 2 sup p(¢(x), 1/I(X)). 1 + 1 - p(¢(x), 1/I(x))2 xEV

First we show that if x, y

(2.2 )

1

E

M(A), then

II 2p(x, y) + Jl - p(x,y )2:::; X

II 2 ( ) Y A*:::; p x, y .

-

The proof uses the techniques of [4, p. 144]. Let I E A with Ilfll < 1. It is clear that 1 = U - f(y))/(1 - I(y)f) E A, l(y) = and 11111 :::; 1. By definition of p then lj(x)1 :::; p(x, y), and consequently

°

If(x) - f(y)1 :::;

11 -

f(y)f(x)1 p(x, y) :::; 2p(x, y).

Taking the supremum over Ilfll < 1 we get the upper inequality. Now we turn to the lower inequality. For simplicity, we write p = p(x, y). Choose fn E A with I!fnl! < 1, In(x) = and fn(y) > p-l/n. Let Pn = fn(Y) and Ln(z) = (tn - z)/(I- tnz), where tn = (1- Jl- P~)/Pn' Therefore Ln 0 fn E A, IILn 0 Inll :::; 1 and

°

IIx -

yl!A* ~ I(Ln

0

fn)(x) - (Ln

0

fn)(Y) I = Itn - L .. (Pn)l·

A simple computation shows that

Itn - Ln(Pn)1 = 1Pn(1 - t;) 1-+ 2p . I-tnPn 1+~ The lemma will follow immediately from (2.2) and the following chain of identities: IIC",-C1/J1I

sup sup IU 0 ¢)(x) - U 0 1/I)(x) I 11/11=1 xE8A sup sup IU 0 ¢)(x) - U 0 1/I)(x) I 1I/1I=lxEV

sup sup 11(¢(x)) - f(1/I(x))1 11/11=1 sup 11¢(x) -1/I(x)IIA*.

xEV

xEV

o LEMMA

IIC", - C1/J1I

3. Let A be a uniform algebra and let C"', C,,}

< 2 is an equivalence relation.

E

End (A). The condition

HOMOTOPIC COMPOSITION OPERATORS ON Hoo(Bn)

181

PROOF. It is obvious that the relation is reflexive and symmetric. So, suppose that ¢,'l/J and t.p define endomorphisms of A such that IICeI> - Cvlli < 2 and IICob C
=

sup p(¢(x),'l/J(x)) < 1 and

0'2

=

xEM(A)

sup p('l/J(x),t.p(x)) < 1. xEM(A)

Using the hyperbolic metric h on M(A) we obtain

1 + 0'1 h(¢(x), t.p(x)) :::; log - XEM(A) 1 - 0'1 sup

1 + 0'2 + log - = /3, 1-

0'2

and consequently sUPXEM(A) p(¢(x), t.p(x)) :::; (ei3 -1)/(ei3 + 1) tion of (2.1) yields the desired result.

< 1. A new applica0

LEMMA 4. Let A be a uniform algebra. If {fn} is a sequence of functions in the unit ball of A tending pointwise to zero on 8A, then {fn} tends to zero weakly in A.

PROOF. As indicated in the introduction, using the Gelfand transform, we

may think of A ~ C(8A). Let x be any element of the dual space of A. By the Hahn-Banach theorem, x has a continuous norm preserving extension to the space of continuous functions on the Shilov boundary. Therefore, there exists a finite measure ILx on 8A such that x(f) =

r

loA

f dJtx •

But Ilfnll :::; 1 for all n, and fn ...... 0 pointwise on 8A, so we may apply the Lebesgue dominated convergence theorem to conclude that 3;(fn) ...... O. Therefore, the sequence {fn} converges to zero weakly. 0 We will need another estimate, but this will depend on the pseudohyperbolic metric particular to the ball, Bn. For a, z E Bn, let Sa = lal 2 , and the

Jl -

((z' a)) a and Qa = 1- Pa. Relevant a,a computations can be found in [13, p. 25]. On Bn, the pseudohyperbolic metric induced by Hoo(Bn) is given by projections Pa and Qa be given by Pa(z)

(

p a, z

)=

=

la - Pa(z) - saQa(z) I 1 _ (z, a) ,

In what follows, for points a and z in the ball and numbers sand t in the closed interval [0,1]' we let as = a + s(z - a) and asH = a + (s + t)(z - a). LEMMA

(2.3)

5. Let a, z

E

Bn. For s, t

E

[0,1] .satisfying t :::; 1 - s we have

tp(a,z) p(a + s(z - a), a + (s + t)(z - a)) :::; 1 _ (1 _ t)p(a, z)·

PROOF. We can assume that a =I z, because otherwise there is nothing to prove. We consider first the case s = O. Since Pa and Qa are linear operators satisfying Pa(a) = a and Qa(a) = 0, the nmnerator of p(a, a + t(z - a)) satisfies

a - Pa(a + t(z - a)) - saQa(a + t(z - a)) = ta - tPa(z) - tsaQa(z).

182

PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ

We have assumed that a =I- z, and therefore a - Pa(z) - saQa(z) =I- O. A simple computation gives

Ia -

p(a,a+t(z-a)) =

Pa{a + t(z - a)) - satQa(z) 1 - (a + t(z - a), a)

I

I

t(a - Pa(z) - saQa(z)) + (z, a) - (a, a) - t(z - a, a)

I

1 - (z, a)

< 11/p(Z,a) -1(1- t)(z - a,ta)I/la - Pa(z) - saQa(z)ll· But

I(z -

a,a)1 = I(a - Pa(z) - saQa(z),a)1 ::; la - Pa(z) - saQa(z)llal. Therefore t tp(a,z) p(a. at) < = . . - (1/ p(a, z)) - (1 - t) 1 - (1 - t)p(a, z)

(2.4)

This proves (2.3) for s = O. For the general case we can assume s =I- 1 since otherwise t = 0 and there is nothing to prove. From (2.4) we obtain

<

p(a s, as + t(z - a)) p(as,a s + (t/(I- s))(z - as)) t/(1 - s) (l/p(a s,z)) - (1- (t/(I- s)))'

But

p(a + s(z - a), z) p(z + (1 - s)(a - z), z) p(z, z + (1 - s)(a - z)). So we may apply (2.4) again to conclude that

I-s p(as,z)::; (l/p(a,z)) -s Combining this with our estimate on p(a s , as+t) above, we see that

t p(a s , as+t) ::; (1/ p(a, z)) - (1 - t)

o

Simplifying, we obtain the desired conclusion. 3. Composition operators on Hoc (Bn)

In this section we study composition operators on Hoc(Bn); that is, given an analytic self-map ¢> of the unit ball, we look at maps C'" : Hoc(Bn) ~ Hoc(Bn) defined by C",(f) = f 0 ¢>. These maps are all endomorphisms of the algebra HOC (Bn). For the special case of n = 1 we will say more in the final section of the paper. We are interested here in estimates on the essential norm of the difference of two composition operators. If T is a bounded operator, we denote its essential norm by IITlle. THEOREM

6. Let ¢> and '!/J be holomorphic self-maps of Bn such that

max{II¢>II, II'!/JII} =

1.

HOMOTOPIC COMPOSITION OPERATORS ON H"""(Bn)

183

Let

e=

max {lim sup p(¢(Z), 'lj!(Z)) , lim sup p(¢(Z), '¢(Z))} . 1'I/J(z)l-l

1",(z)l-l

Then (3.1) PROOF. By hypothesis there is a sequence of points {Zj} in Bn, such that p(¢(Zj),-¢(Zj)) ~ e, and one of them, say {¢(Zj)}, converges to a point ( on the boundary of Bn. Without loss of generality we may assume that p( ¢( Zj ), 'lj!( Zj)) > e-1fj. Let kj E Hoo(Bn) be a function of norm one, whose existence is guaranteed by the pseudohyperbolic distance definition, satisfying kj(¢(zj)) > e - 1fj and kj('lj!(zj)) = O. Consider the functions

f(z) = (1 + (z, () )/2 and g(z) = (1 - (z, () )/2. Then f,g E HOO(B n ), f(() = 1, If(7])1 < 1 for all 7] E aBn satisfying 7] -I- (, and g(() = O. We will now produce a sequence of functions, {hj}, tending to zero weakly for which Ihj(¢(zj)) - hj (1/I(Zj)) I ~ e. We proceed as follows. Let j E N. Since f(¢(zj)) ~ 1, we may choose Zmj so that If(¢(zmj))lj > 1 - 1 fj. Now since 9 -I- 0 on Bn, there exists an integer Ij such that

J

Ig(¢(zmj))1 1/ 1j 2: J1- 1fj. Consider the functions hj = (gl/1 j )(J3)kmr Then Ilhjll ~ 1, hj (1/I(zmj)) = 0, and Ihj(¢(zmJ)1 > (1 - 1fj)(e - 1/mj). We note that hj ~ 0 on the Shilov boundary and, by Lemma 4, hj ~ 0 weakly. Thus for any compact operator K we have IIC", - C,p

+ KII > 2:

IIC",hj - C,phj + Khjll Ihj(¢(zmj)) - hj('lj!(zmj))

+ (Khj)(zmJI ~ e·

= Ihj(¢(zmJ) + (Khj)(zmj)1

This proves the lower inequality in (3.1). For the upper inequality, let I: > 0 and choose 8 with 0 to 1 so that

p(¢(Z), 'lj!(z)) ~

e + I:

on the set {1¢(z)1

< 8 < 1 close enough

> 8} U {1'lj!(z)1 > 8}.

Now choose a = a(I:,8) E (0,1) close enough to one so that p(¢(z),a¢(z)) < I: and p('lj!(Z) , ml'{z)) < I: on the set {1¢(z)1 ~ 8}n{I1/I(z)1 ~ 8}. Since max{lla¢lI, Ila'lj!ll} ~ a < 1, the operator K ~f Co", - Co,p is compact. If Z and ware any two points of B'\ we may view Z and w as elements of the dual space of Hoo(Bn) in the obvious way. Therefore, for any function f in the unit ball of Hoo(Bn) we may apply (2.2) to conclude that If(z) - f(w)1 ~ 2p(z,w). Henceforth, applying (2.2) to the functions f and fo(z) = f(az), we have

I(C",f)(z) - (C,pf)(z) - (Kf)(z)1

~

If(¢(z)) - f(a¢(z))1 + If('lj!(z)) - f(a'¢(z))1 < 2p(¢(z), a¢(z)) + 2p('lj!(z), a'lj!(z)) < 4c

when z E {I¢I ~ 8} n Hl/)I ~ 8}, while

I(C",f)(z) - (C,pf)(z) - (Kf)(z)1

< If(¢(z)) - f('lj!(z)) I + If(a¢(z)) - f(a1/l(z))1 < 2p(¢(z), 'lj!(z)) + 2p(¢(z), 'lj!(z)) < 4e + 41:

184 when

PAMELA GORKIN. RAYMOND MORTINI, AND DANIEL SUAREZ Z E

{1cf>1

> 8} U {I'!/JI > 8}. Since the function f is arbitrary, IIG - G", - KII

4e + 4c, and since c is arbitrary we obtain (3.1).

~

D

COROLLARY 7. Let cf> and '!/J be two holomorphic self-maps of the unit ball. Then G - G", is compact if and only if either max {11<;b11, II'!/JII} < 1, or lim sup p(cf>(z), '!/J(z))

=

leI>(z)I~1

lim sup p(cf>(z), '!/J(z))

= O.

1"'(z)I~1

PROOF. It is clear that G and G", are compact, if max {11cf>11, 11'!/J11l < 1. On the other hand, if max {11cf>11, II'!/JII} = 1 and e is the parameter of Theorem 6, then (3.1) says that Gel> - G", is compact if and only if e = o. D Our next goal is to characterize the path components of composition operators on Hoo(Bn). We write G '" G", to indicate that there is a norm-continuous homotopy of composition operators joining Gel> with G",. Also, if K denotes the ideal of compact operators, we write G "'e G", to indicate that there is an essential norm-continuous homotopy of classes {G", + K: cp: Bn -4 B n holomorphic} joining Gel> + K with G", + K. Let cf> be a holomorphic self-map of Bn. For x E M (HOO (Bn)) we can define

cf>(x) E M(Hoo(Bn)) by the rule <;b(x)(f) ~f x(f 0 cf». Thus we can extend cf> : B n -4Bn to a self-map of M(Hoo(Bn)), which we also denote by cf>. The continuity of this extension is immediate. We now have everything we need to prove the main theorem of this paper. As indicated in the introduction, this theorem unifies and extends many of the results appearing in [11], as well as [8]. THEOREM 8. Let cf> and '!/J be holomorphic self-maps of the unit ball in en. Then the following are equivalent.

(a) (b) (c) (d)

G '" G",. G "'e G",. IIGeI> - G",II < 2. SUPzEBn p(cf>(z), '!/J(z)) < 1.

PROOF. (a) => (b) is obvious. (c) ¢:} (d). A boundary for HOCJ (Bn) is a closed set F c M (HOCJ (Bn)) such that Ilfll = SUPxEF If(x)1 for all f E HOCJ(Bn). It is clear that the closure B n of B n in M(HOCJ(Bn)) is a boundary for HOCJ(B n ), and since oHOCJ(Bn) is the intersection of all the boundaries [4, p. 10], then oHOCJ(Bn) c F. The equivalence then follows from (2.1). (b) => (c). By hypothesis there is a family {cf>t}, with t E [0, 1], of holomorphic self-maps of Bn such that cf>o = cf>, cf>1 = '!/J and for every c > 0 there is some 8 > 0 satisfying

< c: if It - sl < 8. Then we can take finitely many points ty = 0 < ... < tm = 1 in [0,1] such that IIGt. - GeI>tHl lie < 1/2 for every i = 1, . .. , m - 1. We claim that IIGt - Gel>. lie

(3.2)

sup p(cf>t. (z), cf>t'+l (z)) < 1

zEBn

for every i. In fact, if r = max{llcf>d,lIcf>tHlll} < 1, then both functions map B n into the closure of r Bn, and since the pseudohyperbolic diameter of this ball

HOMOTOPIC COMPOSITION OPERATORS ON

Hoo(Bn)

185

is smaller than 1, we are done. If some of the maps have norm 1, then the first inequality of (3.1) tells us that there is some 0 < 8 < 1 close enough to 1 such that sup p(¢t;{Z), ¢tHl (Z)) < 3/4, {I., 1~t5}U{I"+11~6} while the set {I¢t, I < 8} n {I¢t'+ll < 8} is mapped by both functions into the ball 8Bn, whose pseudohyperbolic diameter is smaller than 1. Our claim follows. Since the closure of Bn in M(Hoo(Bn)) contains the Shilov boundary, (3.2) and (2.1) imply that IIC" - C'<+1 11 < 2 for i = 1, ... , Tn - 1. Lemma 3 now says that (c) holds. (d) =} (a). By (d) there exists 0: < 1 such that sup p(¢(z), 'l/J(z)) -:; 0:. zEBn

We define a map ¢t = ¢

+ t('l/J -

IIC, - C. II

¢) for t E [0,1]. Now, if t < s < 1

< 2 sup p(¢t(z), ¢s(z)) zEBn

< <

2(s - t)p(¢(z), 'l/J(z)) 1 - (1 - (s - t)) p(¢(z), 'l/J(z)) 20: (s - t) 1 _ 0:'

where the first inequality holds by (2.1), since 8Hoo(Bn) C 13", and the second inequality from (2.3). From this, we see that t 1--+ Ct is a continuous mapping. 0 Therefore C'" lies in the same path component as C. As a corollary, we obtain the following generalization of the work on isolated points in [11]. COROLLARY 9. Let ¢ be a holomorphic self-map of the ball. Then C is isolated in the set of composition operators if and only if C is essentially isolated. PROOF. Since IIC - C",II ~ IIC - C",lIe, it is clear that if C is essentially isolated, then it is isolated. If C is isolated and 'l/J =I- ¢, Theorem 8 implies that SUPzEBn p(¢(z),'¢(z)) = 1. This can only happen if there are points z E Bn such that 1¢(z)I--+1 or I'l/J(z) 1--+1, and p(¢(z), 'l/J(z))--+1. Hence, Theorem 6 says that IIC - C'" lie ~ 1, and C is essentially isolated. 0

4. Examples So what are some examples of isolated operators? If ¢ : Bn --+ Bn has radial limits of Euclidean norm 1 on a set of positive measure, we claim that C is isolated. If 'l/J =I- ¢ there must exist a set of positive measure in 8Bn on which ¢ has radial limits of norm 1 and 'l/J does not equal ¢ (see [13, Ch. 5]). Thus, there exists a sequence {zd c Bn for which ¢(Zk) --+ ( E 8Bn and 'l/J(Zk) --+ 'Tf, with 'Tf =I- (. Therefore p(¢(Zk), 'l/J(Zk)) --+ 1 and Lemma 2 tells us that IIC - C",II = 2. In particular, the automorphisms of B n induce isolated composition operators. It is clear from Theorem 8 that if ¢, 'l/J are holomorphic self-maps of Bn and O - C'" is compact, then C '" C"'. It is not completely clear, though, that the converse fails. In [8] Hosokawa, Izuchi and Zheng constructed an example that shows this for n = 1. By eliminating variables, every example that works for n = 1 can be made to work for general n. Here we construct a simpler example of

186

PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ

holomorphic self-maps of the ball, 4> and '1/), such that C'" '" C'" but C - C,,) is not compact. Let n = {UJ ED: ~1-=-!:1 > ~} be a nontangential region in D at the point Z = 1. We want to estimate p(w, (w + 1)/2) for wEn. We recall that p(z, w) = Iz - wi/II - zwl for z, wED. By straightforward calculation,

( ( + 1)/2) -1

p w, w

=

11 -lwl 2+ 1- wi Il-w I

(1 -lwl)(1 + Iwl) 1 3 Il-w I + ~

~

and

- Iwl21 11+ 1l-w

p(w, (w + 1)/2)-1

> ~ (1 + 1- Iw12) 1- w

1-lwl 2

1+ 11_wI2(1-~w)

1+ 1+ Iwl > ~

>

2

- 2

when wEn. That is,

1

2

3 ~ p(w, (w + 1)/2) ~ 3

(4.1)

for all wEn. Let c.p : D---+n be a one-to-one and onto holomorphic function and define 4>, 'I/J : Bn---+Bn by

= (c.p(Z1), 0, ... ,0) and that 114>11 = 11'1/)11 = 1. For z

4>(Z1,"" zn) It is clear that

'I/J(Z1, ... , zn) E

= ((c.p(zt) + 1)/2,0, ... ,0).

Bn, a straightforward calculation shows

p(¢(z), 'I/J(z)) = p(c.p(zd, (c.p(zd

+ 1)/2).

Since c.p(zd E n, the inequalities in (4.1) show that p(4)(z), 'I/J(z)) E [1/3,2/3]. Therefore Theorem 8 says that C'" '" C"', while Corollary 7 says that C - C'" is not compact..

5. Endomorphisms of HOO(D) In this section we investigate the path components of elldomorphisms of H OO (D). For x E M(HOO(D)), the Gleason part of x is P(x) = {y E M(HOO(D)) : p(x,y) < I}. Since the condition p(:r, y) < 1 is an equivalence relation, the Gleason parts form a partition of M(HOO(D)). In [7] Hoffman produced a continuous and onto map Lx: D---+P(x) such that Lx(O) = x and foLx E HOO(D) for every x E M(HOO(D)) and f E HOO(D). There are two possibilities: either Lx(z) = x for all zED (so P(x) = {x}) or Lx is one-to-one. We write G = {x E M(HOO) : Lx is one-to-one} and

r

= {x E M(HOO) : Lx = {x}}.

It is well-known that every endomorphism T of HOO(D) can be factored as T = C",CL", , where 4> is a holomorphic self-map of D and x E M(HOO(D)). Although it is clear that this factorization is not unique, two different factorizations of the same endomorphism are related in the following way (see [2]): if p(x, y) < 1, then there is a biholomorphic map r of D (depending on x and y) such that Ly(z) = Lx(r(z))

HOMOTOPIC COMPOSITION OPERATORS ON

Hoo(B n )

for every zED. This means that every endomorphism of the form T also be factored as

187

= Cq,CLy can

T = Cq,CLy = Cq,CTCL", = CToq,CLx ' Of course, if x E rand p(x, y) < 1, then x = y and T = C Lr . LEMMA 10. Let x E G and A = Ej=1 )..,jCq,j' where composition operators on H'XJ(D). Then IIACLJI = IIAII.

)..,j

E C and Cq,j are

PROOF. Since IIACLxll ::; IICLxllliAIl ::; IIAII, one direction is easy. For the other direction, if 0 < f < 1, there exists a function f in the ball of HOO(D) such that IIA(f)11 > (1 - f)IIAIi. By the definition of the norm, there exists r with o < r < 1 such that n

L )..,jf(cPj(z)) ZETD j=l

sup IA(f)(z)1 = sup

ZETD

> (1 - f)2I1AII·

By a result of Hoffman [7, p. 91]' there exist Blaschke products bk such that (bk 0 Lx)(z) ~ z uniformly on compact subsets of D. But rD is a precompact subset of D, and therefore cPj (r D) is precompact for each j. That is, there is 0 < 0: < 1 such that Uj=l cP j(rD) C o:D. Fix {J with 0: < {J < 1. Since f is analytic, there is 8> 0 such that for z, wE (JD with Iz - wi < 8 we have If(z) - f(w)1 < f. Clearly we can also require 8 < (J - 0:. Therefore we may choose k sufficiently large so that I(b k 0 Lx)(cPj(z)) - cPj(z)1 < 8 for all z E rD. Thus, for k that large, z E rD and f as above, (b k o Lx)(cPj(z)) E (3D and consequently If(bk(Lx(cPj(z))) - f(cPj(z))1 < f. Therefore there exists a constant At depending only on nand )..,l"",)..,n such that II ACLx II

~

sup I(ACLx(f

0

bk))(z)1

zErD

n

sup

IL

zErD j=l

)..,j(f 0 bk 0 Lx)(cPj(z))1

n

>

sup

IL

zErD

> Letting

f ~

)..,jf(cPj(z))I- Mf

j=l

(1 - f)2I1AII- Mf.

0 yields the desired result.

o

THEOREM 11. Let T 1, T2 E End(HOO(D)). Then the following ar'e equivalent.

(a) T1 rv T2 in End(HOO(D)). (b) IIT1 - T211 < 2. (c) There exist x E M(HOO(D)) and holomorphic self-maps cP. 'ljJ of D such that T1 = Cq,C Lx • T2 = CtfJCLx and IICq, - CtfJlI < 2. PROOF. Suppose that (a) holds. Then there is a homotopy

G: [0, 1] ~End(HOO(D)) with G(O) = T1 and G(I) = T 2. We can find finitely many points 0 = it < ... < tn = 1 such that IIG(tj) - G(tj+d II < 2 for j = 1, ... , n - 1. Lemma 3 then says that IIG(O) - G(I)1I < 2. Suppose that (b) holds and write T1 = CLxoq, and T2 = CLyo"" where x,y E M(HOO(D)) and
PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ

188

p(Lx(¢(O)), Ly( - CLyo'!'lI = 2. Thus (b) implies that p(x,y) < 1. If x E r, then Tl = T2 = C Lx ' If x E G and we write T2 = CLxo.p, where 1/J is a holomorphic self-map of D. Lemma 10 says that 2> IITI - T211 = IICt/> - C,pll, so (c) holds. If (c) holds Theorem 8 says that there is a homotopy of composition operators F(t), with t E [0,1] such that F(O) = Ct/> and F(l) = C,p. By Lemma 10, G(t) ~f F(t)CLx is a homotopy of endomorphisms connecting Tl with T 2, which proves (a). 0 Acknowledgement. The last author thanks Bucknell University for its hospitality and peaceful environment during the preparation of this paper. References [1] E. Berkson, Composition opemtors isolated in the uniform opemtor topology, Proc. Amer. Math. Soc. 81 (1981),230-232. [2] P. Budde, Support sets and Gleason parts, Michigan Math. J. 37 (1990), 367-383. [3] P. Galindo and M. Lindstrom, Factorization of homomorphisms through HOO(D), preprint. [4] T. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. [5] T. Gamelin and J. Garnett, Distinguished homomorphisms and fiber algebras Amer. J. Math. 92 (1970), 455-474. [6] P. Gorkin and R. Mortini, Asymptotically interpolating sequences in uniform algebms, J. London Math. Soc., to appear. [7] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74-111 . [8] T. Hosokawa, K. Izuchi, and D. Zheng, Isolated points and essential components of composition opemtors on HOC, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1765-1773. [9] U. Klein, Kompakte multiplikative Opemtoren auf uniformen Algebren, Mitt. Math. Sem. Giessen 232 (1997), 1-120. [10] B. MacCluer, Components in the space of composition opemtors, Integr. Equ. Oper. Theory 12 (1989), 725-738. [11] B. MacCluer, S. Ohno, and R. Zhao, Topological structure of the space of composition oper'ators on HOC, Integr. Equ. Oper. Theory 40 (2001), 481-494. [12] J. Moorhouse and C. Toews, Differences of composition opemtors, preprint. [13] W. Rudin, Function theory in the unit ball of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241. Springer-Verlag, New York-Berlin, 1980. [14] J. Shapiro and C. Sundberg, Isolation amongst the composition opemtors, Pacific J. Math., 145 (1990), 117-151. [15] C. Toews, Topological components of the set of composition opemtors on HOC (B N ), preprint.

en,

DEPARTMENT OF MATHEMATICS, BUCKNELL UNIVERSITY, LEWISBURG, PENNSYLVANIA

17837

E-mail address: pgorkinlDbucknell. edu DEPARTEMENT DE MATHEl\fATIQUES, UNIVERSITE DE METZ, ILE DU SAULCY,

F-57045 METZ,

FRANCE

E-mail address:mortinilDponcelet.univ-metz.fr DEPARTA~IENTO DE MATEMATICA, UNIVERSIDAD DE BUENOS AIRES, PAB. I, CIUDAD UNIVER-

(1428) NUNEZ, CAPITAL FEDERAL, ARGENTINA E-mail address:dsuarezlDdm.uba.ar

SITARIA,

Contemporary Mathematics Volume 328, 2003

Characterization of conditional expectation in terms of positive projections J.J. Grobler and M. De Kock ABSTRACT. A description of positive, order continuous projections in ideals of measurable functions is given in terms of conditional expectation-type operators. The dual of such an operator can also be represented as a conditional expectation operator. We use this result to characterize conditional expectation in terms of a positive, order continuous projection that preserves one, and such that an extension of the dual also preserves one.

1. Introduction

Dodds, Huijsmans and De Pagter (see [2]) give a complete description of positive projections in ideals of measurable functions in terms of conditional expectationtype operators. Let E be an ideal of measurable functions such that Loo(O, E, Jl) C E C L1 (0, E, Jl). They characterize a positive, order continuous projection T : E -+ E with the property that T and the operator dual T' preserves one, in terms of conditional expectation. We extend this result to an ideal of measurable functions L with the property that Loo(O, E, Jl) C L, but we omit the assumption L C Ll (0, E, Jl). In this case we prove that a positive, order continuous projection S : L -+ L, which is onepreserving and with the property that an extension of its dual is also one-preserving, can be characterized in terms of the conditional expectation. In order to prove this extension, we need the fact that if an operator can be characterized in terms of conditional expectation, then its dual can also be characterized in terms of conditional expectation.

2. Preliminaries Let (0, E, Jl) be a a-finite measure space. The vector lattice of all Jl-a.e. finite E-measurable functions on 0, with the usual identification of Jl-a.e. equal functions, is denoted by Lo(O, E, Jl). M+(O, E, Jl) denotes the set of all (equivalence classes of) real, positive, Jl-measurable functions into [0,00]. We define x Vy := sup{x, y}. A linear subspace G of a vector lattice E is called a sublattice if x Vy belongs to G for all x, y E G. A linear subspace A in which Ixl :::; IYI, with yEA implies x E A, is called an ideal. A subset AcE is order bounded if A is contained in an order interval, i.e. A is bounded from above and below. We denote the set of order bounded linear operators © lRO

2003 American Mathematical Society

190

J.J. GROBLER AND M. DE KOCK

from the vector lattice E into itself by Cb(E). Cb(E) is a Dedekind complete vector lattice (see [3), p3). All vector lattices considered will be Dedekind complete. A net (xQ)QEf in E is called order convergent if there exists a net (YQ)QEr satisfying (YQ) 1 0 and Ix - x", I ~ y", for all Q E r, where r is an index set. We write X ---> x in order. If Eo is a sub-a-algebra of E, then we denote the restriction of p, to Eo by IL again. Et denotes the collection of subsets of Eo of positive measure. The characte1"istic function of a set A E E is denoted by lA. We write 1 rather than In. For any measurable function Jon 0, the support of f is denoted by supp(J) = {w EO: f(w) =1= o}. Our attention will, to a great extent, be focused on ideals of measurable functions on (0, E, I.L), i.e., on ideals L in the vector lattice Lo(O, E, p,). The set Z E E is called an L-zero set if every .f E L vanishes p,-a.e. on Z. There exists (modulo IL-llUll sets) a maximal L-zero set ZI in E and the set 0 1 = 0 \ ZI is called the carrier of the ideal L. There exists a sequence An i 0 1 in E such that 1.L(An) < 00 and IAn E L for all n E N, (see [5], p143). Clearly, the carrier of L is equal to n, if and only if L is order dense in Lo(n, E, p,). Let L c Lo(O, E, p,) be an order dense ideal with order continuous dual L~. We identi(y L~ with an ideal L' of functions in Lo(n,E,p,), and we will assume that L' is again an order dense ideal (which is always the case if L is a Banach function space; (see [5), Theorem 112.1). Equivalent to this assumption is that L ~ separates f gd,.L the points of L. The duality relation between Land L' is given by (J, g) = for f ELand gEL' (see [5), Section 86). Let 8 E Cb(L) with L an ideal of functions in Lo(O, E, p,). We define its order continuous adjoint 8' : L' ---> L' by (g, 8' J) = (8g, J) for all f E L' and gEL (see [5], Section 97). Then 8' E Cb(L'). If there is no reason for confusion, we will denote (n, E, p,) by (E) only. Q

In

DEFINITION 1. Let E and F be vector lattices and let T : E ---> F be a linear operator. Then (i) T is positive (denoted by T ~ 0) whenever Tx ~ 0 for all x ~ 0; T is called strictly positive (denoted by T »0) if Tx > 0 for all x > O. (ii) T is order continuous whenever Txc< -+ 0 in order for every net (x satisfying X -+ 0 in order. (iii) T is order bounded if it maps order bounded subsets into order bounded subsets. Q )

Q

For a Banach function space (E, II . liE) defined on some finite measure space ~ E ~ Ll (E, I.L), we define the following (see [2), p69).

(n, E, p,) for which Loo(E, p,)

DEFINITION 2. (i) The linear map T : E -+ E is called averaging if for all f E Loo(E) and all gEE we have that T(JTg) = Tf· Tg. (ii) T: E -+ E is called contractive if II T II ~ 1. DEFINITION 3. Let (n, E, p,) be a probability space (i.e. 1.L(n) = 1) and let Eo be a sub-a-algebra of E. For fELl (E), we denote by lFP(J I Eo) the IL-a.e. unique Eo-measurable function with the property that

i

lFf'(J I Eo)dl.L

=

i

fdp,

for all A E Eo. The function lFP(J I Eo) is called the conditional expectation of f with respect to Eo. If there is no reason for confusion, we will denote the p,-a.e. Eo-measurable function lElL('IE o) by lE('IE o) only. The existence of lE(J I Eo) is a consequence of

CHARACTERIZATION OF CONDITIONAL EXPECTATION.

191

the Radon-Nikodym theorem. The conditional expectationlE('IEo) can be extended from a mapping from Ll (E) into itself, to a mapping from M+(E) into itself. If f E M+(E,), then 1E(f I Eo) E .l\J+(E) is defined by 1E(f I Eo) = suplE(fn I Eo), where 0 :::; fn E Ll (E) (71. = 1,2, ... ) satisfy 0 :::; fn i f J,L-a.e. The conditional expectation operator has the following properties. For a proof of properties (i) to (vi) we refer to [4], p7; for property (vii) we refer to [3], p7. 1. (i) lE(o:f + /1g I Eo) = 0:1E(f I Eo) + /11E(g I Eo) for all f,g E M+(E) and for all 0:::; 0:,/1 E R (ii) 0 :::; f :::; 9 in M+(E) implies that 0 :::; 1E(f I Eo) :::; lE(g I Eo) and if 1E(lfll Eo) = 0, then it follows that f = O. By virtue of positivity we have 11E(f I Eo)1 :::; 1E(lfll Eo). (iii) 0:::; fn i f IJ,-a.e. implies that 0:::; lE(fn I Eo) i 1E(f I Eo) J,L-a.e. (iv) lE(gf I Eo) = glE(f I Eo) for all f E M+(E) and all 9 E M+(Eo). (v) If 9 E M+(Eo) and f E M+(E), then fA gdJ,L = J~ fdJ,L for all A E Eo if and only if 9 = 1E(f I Eo) IJ,-a.e. (vi) If Eo c Ao are sub-a-algebras of E, then 1E(f I Eo) = 1E(1E(f I Ao) I Eo) for all 0:::; f E M+(E). (vii) If f E M+(E) is such that 1E(f I Eo) E Lo(E), then we also have that f E Lo(E). PROPOSITION

DEFINITION

4. The domain domlE('IE o) of 1E('IEo) is defined by

domlE('IE o) := {f E Lo(E) : 1E(lfll Eo) E Lo(Eo)}. Clearly, domlE(·IE o) is an ideal in Lo(E) which contains L 1 (E). For f E dom 1E(·1 Eo), we define: 1E(f I Eo) := 1E(f+ I Eo) -1E(r I Eo). This defines a positive linear operator 1E('IEo) : domlE('IE o)

--+

Lo(Eo) C Lo(E).

Let (n, E, l.l) be a probability space and let L carrier n. Set

c

M(L) = {m E Lo(E) : 1E(lmfll Eo) E L

Lo(n, E, J,L) be an ideal with

V f E L}.

Since L C Lo(E), we have that mf E domlE('IE o) for all m E M(L) and f E L. For m E M(L) we define Smf : L --+ L by

Smf := lE(mf I Eo)

V f E L.

Sm is order bounded and ISml :::; Simi' Sm is also order continuous. The following proposition will be applied in the sequel. A proof can be found in [3], (p8). PROPOSITION

2. Let (n, E, J,L) be a probability space and Eo C E a sub-a-

algebra. (i) If f E domlE('IE o) and 9 E Lo(Eo), then it follows that gf E domlE('IE o) and lE(gf I Eo) = glE(f I Eo). (ii) If f E Lo(E), then f E domlE('IE o) if and only if there exists a sequence {A n }:'=1 in Eo such that An in and .

{

JAn

IfldJ,L <

00

V

71.

= 1,2, ....

J.J. GROBLER AND M. DE KOCK

192

Moreover, if f E domlE(·IE o), then, for all A E Eo with

i

lEU I Eo)dft =

i

fA Ifldft < 00,

fdft·

The following lemma will be applied in t.he sequel. LEMMA

3. For a linear subspace N of Loo(E) the following statements are

equivalent. (i) There exists a sub-a-algebra Eo such that N = Loo(Eo). (ii) N is a subalgebra of Loo(E) containing the constants such that fn N, Ifni::; u E Loo(E) (11. = 1,2, ... ) and fn - t f a.e. imply that fEN.

E

The proofs of the following propositions and corollaries rely mainly on t.he proofs by Dodds, Huijsmans and De Pagter (see [2]). Let L c Lo(E) be an ideal of measurable functions such that Loo C L. We then have the following. PROPOSITION

4. Let S : L

-t

L be an or'der continuous, positive linear opemtor

for which (i) Sf E Loo(E) whenever f E Loo(E),

(ii) SUSg) = Sf· Sg for' all f E Loo(E) and all gEL. Then there exists a sub-a-algebra Eo of E and there exists a 0 ::; m E M(L) such that Sf = lE(mf I Eo) for all f E L. We use the following proposition in the proof of the main result. PROPOSITION

5. For a linear operator S : L

-t

L, the following statements are

equivalent.

(i) S is positive and or'der continuous, S2 = S, SI = 1 and the range R(S) of S is a sublattice.

(ii) There exist a sub-a-algebra Eo of E and a function 0 ::; m lE(mIEo) = 1 such that Sf = lE(mf I Eo) for all f E L.

E

M(L) with

Because the range of a strictly positive projection is a sublattice, we obtain the following result. COROLLARY

6. For a linear operator S : L

-t

L the following statements are

equivalent.

(i) S is a strictly positive, order continuous project'ion with SI = 1. (ii) There exists a sub-a-algebra Eo of E and a strictly positive function m M(L) with lE(ml Eo) = 1 such that Sf = lE(mfl Eo) for all f E L.

E

In the following proposition we consider the case where the operator no more preserves onc, but where the image of the indicator function is strictly positive. We derive a similar result as in Corollary 6 for S strictly positive. PROPOSITION 7. Let S : L ments are equivalent.

-t

L be a linear operator, then the following state-

(i) S is a positive or'der continuous projection onto a sublattice such that SI is strictly posit'ive. (ii) There exist a s'nb-a-algebra Eo ofE, 0 ::; m E Lo(E) and a strictly positive function k E Lo(E) with lE(mk I Eo) = 1, such that Sf = klE(mf I Eo) for all f E L.

CHARACTERIZATION OF CONDITIONAL EXPECTATION.

193

We have the following basic characterization of conditional expectation on L1 (E). A proof can be found in [2], p71. PROPOSITION 8. (Douglas R.G. and Seever) If T is a continuous linear map on L 1(E), then the following statements are equivalent.

(i) There exists a sub-a-algebra Eo of E such that for all f E L1 (E) we have that Tf = lEU I Eo). (ii) T is a contractive projection which preserves 1. 3. Main characterization of conditional expectation. We prove that if an operator can be characterized in terms of the conditional expectation, then its dual can also be characterized in terms of the conditional expectation. As before, we let L c Lo(E) be an ideal of measurable functions which contains Loo(E). LEMMA 9. If S : L --+ L is a linear operator such that Sf = lEU IEo) for all f E L, then S' : L' --+ L' satisfies S' 09 = lE(g I Eo) for all E-measurable 09 E L'.

ProoF. Let f ELand gEL'. Then (j, S'g)

(Sf,g) l

SfgdJ.l

l l

lE iL U I Eo)gdJ.l

!l

n

lEiL(glEiLU I Eo) I Eo)dJ.l

llEiLU I Eo) ·1E1L(g I Eo)dJ.l l

f S'lE iL (g I Eo )dJ.l

Thus, we have proved that (3.1)

(j,S'g)

=

(j,S'IE(gIE o )).

For any Eo-measurable g, we have that (3.2) It follows from (3.2) that for Eo-measurable 9 we have that S'g = 9 and from (3.1) 0 and (3.2), for arbitrary gEL' that S'g = S'lE(g I Eo) = lE(g I Eo). Now we are able to prove the main result, where we characterize conditional expectation in terms of a positive, order continuous projection and an extension of its dual. PROPOSITION

10. If S is a linear map, then the following statements are equi-

valent.

(i) Ther'e exists a sub-a-algebra Eo of E such that Sf = lEU I Eo) for' all f E L. (ii) S: L --+ L is a positive order continuous projection such that SI = 1 and S' has an extention S' : L' + Loo --+ L' + Loo satisfying S'I = 1.

194

J.J. GROBLER AND M. DE KOCK

Proof. (i) =} (ii) It follows from Proposition 5 that S is a positive, order continuous projection such that SI = 1. Since the conditional expectation operator is defined on L1 (E) into L1 (E), it follows that

S : L n L 1(E)

-4

Denote the restriction of S to L n L1 (E) by

8' : L' + Loo(E) and 8' is an extension of S' : L' have (3.3)

-4

-4

L n L 1(E).

8.

Then

L' + Loo(E)

L'. Because for

f

E

L' and gEL n L1 (E), we

(g, 8' J) = (8g, J) = (Sg, J) = (g, s' J).

Since L n L1 (E) is dense in L, (3.3) holds for all gEL, and so 8' f = S' f for all f E L'. For all gEL' + Loo(E), it follows from Lemma 9 that 8'g = lE(g 1 Eo), so 8'1 = 1, by the properties of conditional expectation. (ii) =} (i) We first note that since Loo (E) c L we have L' C L1 (E) and also that L' is dense in L 1 (E). An argument of Ando (see [1], (p401)) shows that S' is contractive for the L1 (E)-norm. In fact, if gEL', it follows from the assumption SI = 1 that,

In IS'

gld{t

In In IgIIS(1

S' 9 sgn S' gd{t

<

sgnS'g)ld{t

< InlgIS(lsgnS'gl)dJL <

In In

IglSld{t Igld{t.

Since L' is dense in L1 (E) for the L1 (E)-norm, we can extend S' to a contraction on LdE). Since S is a projection, the same holds for S'. Thus, by Proposition 8, there exists a sub-a-algebra Eo of E such that S' 9 = lE(g 1 Eo) for every 9 E Ll (E) and therefore also for all gEL'. It follows from Lemma 9 that S" f = lEU 1Eo) for all f E L". By restricting S" to L, we therefore have that Sf = lEU 1 Eo) for all f E L. 0

References [1] ANDO, T., 1966, Contmctive projections in Lp-spaces, Pacific J. Math., 17,391-405. [2] DODDS, P.G., HUIJSMANS, C.B., and DE PAGTER, B., 1990, Chamcterizations of Conditional Expectation type-opemtors, Pacific J. Math., 141,55-76. [3] GROBLER, J.J. and DE PAGTER, B., 1999, Opemtors representable as Multiplication Conditional Expectation opemtors, To appear in J. of Operator Theory. [4] NEVEU, J., 1975, Discrete-pammeter martingales, North Holland/American Elsevier, Amsterdam Oxford New York. [5] ZAANEN, A.C., 1982, Riesz spaces II, North Holland, Amsterdam, New York. SCHOOL FOR BUSINESS MATHEMATICS, POTCHEFSTROOM UNIVERSITY FOR CHE, POTCHEFSTROOM 2520, SOUTH AFRICA" MATHEMATICS DEPARTMENT, KENT STATE UNIVERSITY, KENT, OH 44240 E-mail address:srsjjgClpuknet.puk.ac.za • mdekockClmath. kent. edu

Contemporary Mathematics Volume 328, 2003

The Krull nature of locally C* -algebras Marina Haralampidou ABSTRACT. Any complete locally m-convex algebra, whose normed factors in its Arens-Michael decomposition are Krull algebras is also Krull. In particular, any locally CO-algebra is a Krull algebra. Considering perfect projective systems, we give another proof of the fact that any Frechet locally CO-algebra is a Krull algebra. Furthermore, a proper complete locally m-convex H* -algebra with continuous involution and a normal unit is a locally C* -algebra, hence Krull. The class of Krull (topological) algebras is closed with respect to cartesian products, topological algebra isomorphic images, and perfect projective limits.

1. Introduction and preliminaries

Every closed (left) ideal of a CO-algebra E is the intersection of the (closed) maximal regular (left) ideals containing it (see, for instance, [2: p. 56, Theorem 2.9.5]. Thus, E is a Krull algebra in the sense of Definition 1.1. A natural question arises here whether, in general, any locally CO-algebra is Krull. In that direction, using the Arens-Michael decomposition, we get that a complete locally m-convex algebra (E,(Po)oEA) is a Krull algebra, if each factor Eo. = E/ker(po.)' Q E A is a Krull algebra (Proposition 2.1). As a consequence, we get that any locally CO-algebra is Krull (Corollary 2.2). Besides, a proper complete locally m-convex H*-algebra (E, (Po.)o.EA) with continuous involution and a unit element e, so that po.(e) = 1 for every Q E A, is a Krull algebra (Corollary 2.6). Based on the notion of a perfect projective system (Definition 2.7), we provide another proof of the fact that any Fn3chet locally C* -algebra is Krull (Theorem 2.10). By the term topological algebm we mean an algebra, which is a topological linear space such that the ring multiplication is separately continuous (see [11: p. 4, Definition 1.1] and/or [12: p. 6]). A topological algebra E is called a Q'-algebm, if every maximal regular left or right ideal in E is closed (see [5: p. 148, Definition

1.1]). A locally m-convex algebm is a topological algebra E whose topology is defined by a family (Po.)o.EA of submultiplicative seminorms, i.e. Po.(xy) :::; Po.(x)Po.(y) for 1991 Mathematics Subject Classification. Primary 46H05, 46H10, 46H20. Key words and phrases. Krull algebra, Q'-algebra, Arens-Michael decomposition, locally C*algebra, perfect projective system of topological algebras, perfect projective limit algebra, Frechet locally CO-algebra, proper algebra, locally m-convex H*-algebra. © 2003 American Mathematical Society 195

MARINA HARALAMPIDOU

196

all x, y E E, 0: E A (see for instance [11] and/or [12]). Such a topological algebra is denoted by (E, (PoJ"'EA)' A complete metrizable locally m-convex algebra E is called a Prechet locally m-convex algebra. In this case, the topology of E is defined by a countable family (Pn)nEN of submultiplicative seminorms. A C*-seminorm is a seminorm P on an involutive algebra E, satisfying the C*-condition, namely, p(x*x) = p(x)2 for every x E E [13: p. 1, Definition 1]. Such a seminorm is submultiplicative and *-preserving [ibid. p. 2, Theorem 2]. A locally pre-C* -algebra is an involutive locally (m-) convex algebra (E, (P",)",EA), such that each p"" 0: E A is a C* -seminorm, while a complete algebra, as before, is called a locally C* -algebra [8: p. 198, Definition 2.2]. A Frechet locally C* -algebra is an involutive Frechet locally (m-) convex algebra (E, (Pn)nEN) where each Pn is a C* -seminorm. A locally m-convex H* -algebra is an algebra E equipped with a family (P"')"'EA of Ambrose seminorms in the sense that P"" 0: E A arises from a positive pseudoinner product <, >"" such that the induced topology makes E into a locally mconvex topological algebra. Moreover, the following conditions are satisfied: For any x E E, there is an x* E E, such that

< xy,z >",=< y,x*z >", < yx, z >",=< y, zx* >", for any y, z E E and 0: E A. x* is not necessarily unique. In case, E is proper (viz. Ex = (0), implies x = 0), then x* is unique and * : E -+ E : x f-+ x* is an involution (see [4: p. 451, Definition 1.1 and p. 452, Theorem 1.3]). Throughout of this work the considered algebras are over the field of complexes. To fix notation we recall the following. Let (E, (P"')"'EA) be a complete locally m-convex algebra and

(1.1)

P'" : E

-+

E/ker(p",) == E", : x

f-+

p"'(x) :=x + ker(p",)

the respective quotient maps. Then Ilx",ll", := p"'(x), x E E, 0: E A defines on E", an algebra norm, so that E", is a normed algebra and the morphisms P"" 0: E A are continuous. E"" 0: E A denotes the completion of E", (with respect to II . II",). A is endowed with a partial order by putting 0: :::; /3 if and only if p"'(x) :::; P(3(x) for every x E E. Thus, ker(p(3) <;;;; ker(p",) and hence the continuous (onto) morphism

(1.2)

j",(3 : E(3

-+

E", : x(3

f-+

j",(3(x(3) := x""

0::::;

/3

is defined. Moreover, j ",(3 is extended to a continuous morphism !",(3 : E(3

-+

E""

Thus, (E"" j",(3), (E",'/",(3), 0:, /3 E A with (resp. Banach) algebras, so that

(1.3)

E

2:!

0: :::;

0::::;

/3.

/3 are projective systems of normed

- -

lim E", 2:! lim E", (Arens-Michael decomposition)

within topological algebra isomorphisms (cf., for instance, [11: p. 88, Theorem 3.1 and p. 90, Definition 3.1] and/or [12: p. 20, Theorem 5.1]). Concerning the following notion see [7]. DEFINITION 1.1. A topological algebra is called a Krull algebra, if every proper closed left (resp. right) ideal is contained in a closed maximal regular left (resp. right) ideal.

THE KRULL NATURE OF LOCALLY C'-ALGEBRAS

197

For the statements (i) and (iii) in the next proposition see [7: Lemma 3.8). PROPOSITION 1.2. Let E, F be topological algebms and ¢ : E ---7 F a continuous epimorphism. Then the following hold true: (i) If E is a Krull algebm and ¢ closed, then F is a Krull algebm. (ii) If E is a Krull algebm and F a Q'-algebm, then F is a Krull algebm. (iii) If F is a Krull algebm and ¢ closed with ker(¢) ~ I for every proper closed left or right ideal in E, then E is a Krull algebm. (iv) If F is a Krull algebm and ¢ a closed injection, then E is a Krull algebm. PROOF. (ii) For a proper closed left ideal J in F, ¢-l(J) is a proper closed left ideal in E with ker(¢) ~ ¢-l(J) and J = ¢(¢-l(J)) (see also [3: p. 316, Proposition B.5.4)). Thus, ¢-l(J) ~ M for some closed maximal regular left ideal Min E. Hence J ~ ¢(M), so that ¢(M) is a maximal regular left ideal in F (ibid.), closed by Q'. Similarly, for proper closed right ideals. (iv) Immediate from (iii). 0 COROLLARY 1.3. A topological algebm is a Krull algebm if and only if a topological algebm isomorphic image of it is so. PROPOSITION 104. Let (E"')"'EA be a family of topological algebms and F = II"'EA E", the respective cartesian product topological algebm. Then F is a Krull algebm, if each E"" Q E A is a Krull algebm. The converse is true in case the factors are Q' -algebms. PROOF. Consider the canonical continuous epimorphisms (projections)

(1.4)

11"", :

F

---7

E", : x = (X"')"'EA

1--+

1I"",(x)

:=

x""

Q

E A.

Let I be a proper closed left ideal in F. Since the multiplication is separately continuous, it follows that the closure 11"",(1) of the left ideal 11"", (I), is a closed left ideal in E",. Moreover, 11"",(1) i- E", for some Q E A. Otherwise, II"'EA 11"",(1) = F. It is easily seen that n"'EA 11";;1 (E",) = ILEA E",. Besides, I = I = n"'EA 11";;1(11"",(1)). Hence I = II"'EA E", = F, a contradiction. Now, since E", is a Krull algebra, 11"",(1) ~ M, for some closed maximal regular left ideal M. Hence I ~ 1I";;1(M) with 1I";;1(M) a closed maximal regular left ideal in F. Similarly, for proper closed right ideals. The above argument shows that F is a Krull algebra. 0 For the rest of the assertion apply (ii) of Proposition 1.2. 2. The Krull property for locally C*-algebras

We provide first the following result akin to that of Proposition 104. PROPOSITION 2.1. Let (E, (P"')"'EA) be a complete locally Tn-convex algebm, such that the norrned algebms E"" Q E A in its Arens-Michael decomposition, are Krull algebms. Then E is a Krull algebm, as well. On the other hand, if E is a Krull algebm, then a factor E", is a Krull algebm if it is also a Q' -algebm. PROOF. E ~ lim E", within a topological algebra isomorphism, say ¢ (see f-(1.3)). Consider the continuous epimorphic image p",(I) (see (1.1)) of a proper closed left ideal I in E. Claim that the closed left ideal p",(I) is proper in E", for

MARINA HARALAMPIDOU

198

some Q E A. Suppose the contrary. Then based on M. Exarchakos, concerning the first equality of the next rels, we get ¢(E)

= ll!!!Ea = ll!!!Pa(I) = ll!!!(fa(¢(I))) = ¢(I),

here fa denotes the restriction to ll!!! Eo of the projection map 7ra : TIaEA Eo -+ Eo, Q E A (cf. also [11: p. 87, Lemma 3.2 and p. 89, (3.24)]; we note that ¢(I) is a closed left ideal in lim Eo). Thus, E = I, which is a contradiction. So, since tEa, Q E A is a Krull algebra, it follows that Pa(I) ~ M for some closed maximal regular left ideal M, and hence I ~ p~l(Pa(I)) ~ p~l(M). Besides, p~l(M) is a maximal regular left ideal in E, closed by the continuity of Po. An analogous result holds for proper closed right ideals. The last part of the assertion follows from (ii) of Proposition 1.2. 0

By [1: p. 32, Theorem 2.4], the factor normed algebras, in the Arens-Michael decomposition of a locally C*-algebra, are C*-algebras and hence Krull (see, for instance [2: p. 56, Theorem 2.9.5]). Thus, Proposition 2.1 implies the next. COROLLARY 2.2. Every locally C*-algebm is a Krull algebm. By Proposition 1.4 and Corollary 2.2, we get the next. COROLLARY 2.3. The cartesian product of locally C* -algebras is a Krull (locally C* -) algebm. In view of Corollary 2.2, Theorem 4.7 in [6: p. 3732] is improved as follows: THEOREM 2.4. A locally C* -algebm is dual if and only if it is complemented. Let (E, (Pa)aEA) be a proper complete locally m-convex H* -algebra with continuous involution. Then E can be made into a locally pre-C* -algebra, via a family (qa)aEA of C*-seminorms given by (2.1)

so that, (2.2)

qa(X) ~ Po(x) for every x E E,

Q

E A.

(Namely, the respective topology on E is weaker than the given one). Moreover,

(2.3)

Po(xy) ~ qa(x)Pa(Y) for every x, y E E,

Q

E A.

(See [9: p. 265, Proposition 2.3]). In that framework we get the next two results. PROPOSITION 2.5. Let (E, (Po)oEA) be a proper complete locally m-convex H*algebm with continuous involution and a unit element e. Then the following are equivalent: 1) po(e) = 1 for every Q E A (: normal unit). 2) (E, (Pa)aEA) is a locally C*-algebm. PROOF. 1) ==> 2): Let qa, Q E A be the seminorms given by (2.1). By (2.3), p",(x) ~ qo(x) for every x E E, Q E A. Hence (see also (2.2)) Po = qa for every Q E A. Namely, (E, (Pa)aEA) is a locally C*-algebra. 2) ==> 1): C*-property implies Pa(e)(l-po(e)) = 0 for every Q E A. If po(e) = 0 for some Q E A, then Ileoli o = 0, where eo = e + ker(Pa) is the respective unit

THE KRULL NATURE OF LOCALLY C'-ALGEBRAS

199

element in the factor algebra EOl == EOl (see aslo [1: p. 32, Theorem 2.4] and [11: p. 91, Theorem 4.1]). Thus eOl = 0, which is a contradiction. Thus, pOl(e) =I- 0 for every Q E A, hence POl (e) = 1 for every Q E A. 0 As a consequence of Corollary 2.2 and Proposition 2.5 we have the next. COROLLARY 2.6. Every proper complete locally m-convex H* -algebm with continuous involution and a normal unit is a Krull algebm. Our next aim is to provide another proof to the fact that a F'rechet locally C*-algebra is Krull (see Corollary 2.2). To do this, we use the notion involved in the next. DEFINITION 2.7. A projective system {(EOl , fOl,8)}OlEA of topological algebras is called perfect, if the restrictions to the projective limit algebra

= ~EOl = {(x Ol ) E

II

EOl: fOl,8(X,8) = XOl , if Q:S; (3 in A} OlEA of the canonical projections 7r0l : I10lEA EOl -+ E Ol , Q E A, namely, the (continuous algebra) morphisms

(2.4)

(2.5)

E

fOl = 7rOl IE =limE", : E

-+

EOl ,

Q

E A,

t--

are onto maps. The resulted projective limit algebra E = lim EOl is called perfect (topological) algebm. ~

LEMMA 2.8. Every Frechet locally m-convex algebm (E, (Pn)nEN) gives a perfect projective system of normed algebms. PROOF. For any n

:s; m in N, the connecting maps

(2.6)

with

fnm(X + ker(Pm)) = x + ker(Pn) are onto algebra morphisms (see, for instance, [11: p. 86, (3.6) and (3.7)]). So, since {(En, fnm)}nEN is a denumerable projective system of normed algebras, it follows that fn, n E N (see (2.5)) are onto, as well (see [10: p. 229, Theorem 8]). 0 The proof in the next result is an adaptation of that in Proposition 2.1. PROPOSITION 2.9. Any perfect projective limit of Krull algebms is a Krull algebm. PROOF. Let {(EOl,fOl,B)}OlEA be a perfect projective system of Krull algebras. Consider the projective limit algebra E = limEOl (see (2.4)), which is a closed subalgebra of the cartesian product topological algebra I10l EA EOl (see, for instance, [11: p. 84, Lemma 2.1]). For a proper closed left ideal I in E, fOl(I) is a (closed) left ideal in E Ol , Q E A. If f Ol (I) = EOl for every Q E A, then ~

1= limfOl(I) ~

= limfOl(I) = limEOl = E, ~

~

(see also [ibid. p. 87, Lemma 3.2]), which is a contradiction. Thus, fOl(I) =I- EOl for some Q E A. Since E Ol , Q E A is a Krull algebra, there exists a closed maximal regular left ideal, say M, with fOl(I) ~ M and hence I ~ f;;l(1Ol(1)) ~ f;;l(M),

200

MARINA HARALAMPIDOU

where J;;l(M) is a closed maximal regular left ideal in E and this terminates the proof for closed left ideals. Similarly, for closed right ideals. D THEOREM

2.10. Any Prichet locally C*-algebm is a Krull algebm.

PROOF. Let (E, (Pn)nEN) be an algebra as in the statement. By [1: p. 32, Theorem 2.4], the respective normed algebras En, n E N in the Arens-Michael decomposition of E, are C* -algebras and hence Krull (see, for instance, [2: p. 56, Theorem 2.4.5]. In particular, {(En' Jnm)}nEN is a perfect system of normed algebras (see Lemma 2.8 and relation (2.6)). Proposition 2.9 assures that the projective limit algebra lim En is a Krull algebra and hence E is a Krull algebra, as it fol+-lows from Corollary 1.3 and the fact that E ~ lim En within a topological algebra +-isomorphism (see (1.3)). D References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

C. Apostol, b*-algebms and their representation, J. London Math. Soc. 3(1971), 30-38. MR 44:2040. J. Dixmier, C*-Algebms, North-Holland, Amsterdam, 1977. MR 56:16388. R.S. Doran and V.A. Belfi, Chamcterizations of CO-Algebras. The Gel'fand-Na'tmark Theorems, Marcel-Dekker, 1986. MR 87k:46115. M. Haralampidou, On locally convex H*-algebms, Math. Japon. 38(1993), 451-460. MR 94h:46088. M. Haralampidou, Annihilator topological algebras, Portug. Math. 51(1994), 147-162. MR 95f:46076. M. Haralampidou, On complementing topological algebms, J. Math. Sci. 96(1999), 3722-3734. MR 2000j:46085. M. Haralampidou, On the Krull property in topological algebms (to appear). A. Inoue, Locally C* -algebras, Mem. Faculty Sci. Kyushu Univ. (SerA) 25(1971), 197-235. MR 46:4219. A. EI Kinani, On locally pre-C*-algebm structures in locally m-convex H*-algebms, Thrk. J. Math. 26(2002), 263-271. G. Kothe, Topological Vector Spaces, I, Springer-Verlag, Berlin, 1969. MR 40:1750. A. Mallios, Topological Algebms. Selected Topics, North-Holland, Amsterdam, 1986. MR 87m:46099. E.A. Michael, Locally multiplicatively-convex topological algebms, Mem. Amer. Math. Soc. 11(1952). (Reprinted 1968). MR 14,482a. Z. Sebestyen, Every C*-seminorm is automatically 8ubmultiplicative, Period. Math. Hung. 10(1979), 1-8. MR 80c:46065.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ATHENS, PANEPISTIMIOPOLlS, ATHENS 15784, GREECE E-mail address:mharalamOcc.uoa.gr

Contemporary Mathematics Volume 328, 2003

Characterizations and automatic linearity for ring homomorphisms on algebras of functions Osamu Hatori, Takashi Ishii, Takeshi Miura, and Sin-Ei Takahasi ABSTRACT. Automatic linearity results for certain ring homomorphisms between two algebras, in particular, semi-simple commutative Banach algebras with units are proved. For this purpose a representation by using the induced continuous mapping between the maximal ideal spaces and ring homomorphisms on the field of complex numbers is given. Ring homomorphisms on certain non-complete metrizable algebras into the algebras of analytic functions are also considered. A characterization of the kernel of complex-valued ring homomorphism on a commutative algebra is given. As a corollary of the results a complete description of ring homomorphisms on the disk algebra into itself is given in terms of prime ideals.

Introduction A ring homomorphism between two algebras is a mapping which preserves addition and multiplication. If we assume that the mapping is linear, then it is an ordinary homomorphism. In the case where the two algebras are just the field C of complex numbers, the assumption cannot be avoided; there are ring homomorphisms of C into C which are not linear nor conjugate linear (cf. [9]). The history of ring homomorphisms on C probably dates back to the investigation of Segre [19] in the nineteenth century and that of Lebesgue [12]. A similar remark applies to finite-dimensional Banach algebras. But this is not the case for several infinitedimensional ones; for instance, Arnold [1] proved that a ring isomorphism between the two Banach algebras of all bounded operators on two infinite-dimensional Banach spaces is linear or conjugate linear (cf. [5]). Kaplansky [8] proved that if p is a ring isomorphism from one semi-simple Banach algebra A onto another, then A is a direct sum Al EB A2 EB A3 with A3 finite-dimensional, p linear on All and p conjugate linear on A 2 . It follows that a ring isomorphism from a semi-simple commutative Banach algebra onto another with infinite and connected maximal ideal space is linear or conjugate linear. 2000 Mathematics Subject Classification. Primary 46JlO, 46E25; Secondary 46J40. The first, the second, and the fourth author were partialy supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.

© 201

2003 AJnerican Mathematical Society

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It is interesting to study ring homomorphisms on Banach algebras which are not necessarily injective or surjective. We may expect that a number of ring homomorphisms on infinite-dimensional Banach algebras are automatically linear or conjugate linear. By a routine work we see that a a ring homomorphism is real-linear if it is continuous. On the other hand, we can also arrive at automatic linearity for several ring homomorphisms by results in [14, 15, 20, 21, 13]; they studied and characterized *-ring homomorphisms between commutative Banach algebras with involutions and ring homomorphisms on regular commutative Banach algebras with additional assumptions. The heart of this paper is automatic linearity results for certain ring homomorphisms of a much more general nature. Throughout the paper A and B denote semi-simple commutative Banach algebras with units eA and eB respectively. The maximal ideal space for A is denoted by MA. In this paper, we denote the Gelfand transform of a E A also by a. For a ring homomorphism of C into C, we simply say a ring homomorphism on C. Let T be a ring homomorphism on C and x E MA. Then the complex-valued mapping p on A defined by

p(a) = T(a(x)),

aEA

is a typical example of a ring homomorphism. Semrl [20, Example 5.4] showed that there exists a complex-valued ring homomorphism other than this type. In section 2 we show that if a ring homomorphism of A into B satisfies a certain condition, say (m), then it is represented by a modified version of the above. Many ring homomorphisms satisfy this weak and rather natural condition (m): *-ring homomorphisms on involutive algebras; p{A)(y) = C for every y E M B ; p(A) contains a subalgebra of B. Thus our result generalizes the previous ones in [14, 20, 21, 13]. In section 3, by using results in section 2, we deduce some automatic linearity results for ring homomorphisms: p with (m) is real-linear on a closed ideal of finite co dimension in A; if p(CeA) = CeB and p(A) contains an element with an infinite spectrum, then p is linear or conjugate linear. It is a natural question: under the two hypotheses (1) p(CeA) c CeB and (2) p(A) contains an element with an infinite spectrum, does it follow that p is linear or conjugate linear? We give an affirmative answer under stronger hypotheses: (1) and (2)' p(A) contains an element whose spectrum contains a non-empty open subset. Problems in the same vein are also considered not only for Banach algebras but also for algebras of analytic functions. Bers [3] proved that if U and V are plane domains and H(U), H(V) are the rings of analytic functions on U, V respectively, then any ring isomorphism of H(U) onto H(V) is induced by a conformal (or anti-conformal) equivalence of V with U, thus the ring isomorphism is linear (or conjugate linear). Nakai [17] and Rudin [18] have shown this also holds for open Riemann surfaces. Ring homomorphisms which are not necessarily injective or surjective are also considered by many mathematicians (cf. [7, 10]). Among them, Becker and Zame [2] have proved automatic continuity and linearity for ring homomorphisms from certain complete metrizable topological algebras into the algebra of analytic functions on connected, reduced analytic spaces. In section 4 we also consider ring homomorphisms into the algebras of analytic functions. In particular, we consider the case of a ring homomorphism p from the algebra Rs of rational functions on C with poles off a subset SeC into an algebra of analytic funtions. Here Rs is a metrizable topological algebra, but it cannot be

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a Banach algebra by the Baire category theorem. We show that, for certain subsets S, p is automatically linear or conjugate linear, if the range of p contains a 11011constant function. We also give an example of S such that a ring hommomorphism that is neither linear nor conjugate linear, and whose range contains non-constant functions, is possible. In the final section we study ring homomorphisms into C: ring homomorphisms whose ranges contain only constant functions. We characterize the kernels of ring homomorphisms from a unital commutative algebra into C, which is compared with the one to one correspondence between maximal ideals and complex (linear) homomorphisms on commutative Banach algebras. As a corollary we show that there exists an injective ring homomorphism from an algebra which consists of analytic functions into C. We also give a complete description of the ring homomorphisms on the disk algebra in terms of prime ideals. We say that a ring homomorphism 7 on C is trivial if 7 = 0 or 7(Z) = Z (resp. z) for every Z E C. Other ring homomorphisms on C are said to be non-trivial. We note some properties of non-trivial ring homomorphisms on C, which are used later in this paper. For a proof of the existence of non-trivial ring homomorphisms, historical comments, and further properties, see [9]. It is easy to see that every non-zero ring homomorphism 7 on C fixes rational real numbers and 7(i) = i or -i. If 7 is non-trivial, then 7 does not preserve complex conjugation. (This is a standard fact. Here is a proof. Suppose 7 does preserve complex conjugation: 7(Z) = 7(Z) for every Z E C. Then 7(JR) C JR, that is, 7 is a ring homomorphism on the set of all real numbers R If x> 0, then 7(X) = (7( JX))2 > O. It follows that 7 is order preserving on R Since 7(r) = r for every rational real number r, we have 7(X) = x for every real number x. Thus 7(Z) = Z (resp. 7(Z) = z) for every Z E C if 7(i) = i (resp. 7(i) = -i), which is a contradiction.) It is easy to see that T is non-trivial if and only if 7 is discontinuous at every (resp. one) point in C. Thus, if T is non-trivial, then it is unbounded on every neighborhood of zero. It follows that there exists a sequence {w n } of complex numbers which converges to 0 such that IT(Wn)1 tends to infinity as n -> 00 if 7 is non-trivial. If the ring homomorphism on C is onto, then it is said to be a ring automorphism on C. Note that there is a non-zero ring homomorphism on C which is not a ring automorphism. We also note that there is a non-trivial ring automorphism on C (cf. [9, 11]).

1. Partial representation If ¢ is a non-zero complex homomorphism on A, then there exists a unique x E MA such that ¢(a) = a(x) for every a E A. By this fact a well-known representation of a (linear) homomorphism VJ from A into B follows: There exists a continuous mapping defined on {y E MB : VJ(a)(y) :I 0 for some a E A} into MA such that

VJ(a)(y)

= a((y)),

a E A,

y E {y E MB : VJ(a)(y)

:I 0 for

some a E A}.

On the other hand, if is a continuous mapping of MB into AfA and Ty is a ring homomorphism on C for every y E AfB , then

p(a)(y) = Ty(a((y)),

a E A,

y E MB

defines a ring homomorphism from A into the algebra of all complex-valued functions on M B . Thus it defines a ring homomorphism from A into B under the condition that T. (a ( (. )) is in B for every a E A, and this is the case when M B

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is finite. A problem is the converse: Is every ring homomorphism represented as above? A negative answer is known even in the case where B = C by the example due to Semd [20, Example 5.4]. Nevertheless, we show that a partial representation is still possible in this section. DEFINITION 1.1. Let p be a ring homomorphism of A into Band y a point in M B . The induced ring homomorphism py of A into C is defined by

py(a) = p(a)(y), Let Ie : C

-+

A be defined by IdA)

= AeA

aE

A.

for every A E C. We denote Ty

= pyole.

For every y E M B , the induced mapping Ty is a ring homomorphism on C. DEFINITION 1.2. Let p be a ring homomorphism of A into B. We denote: Mo = {y E MB : Ty = a}; !vlt = {y E MB : Ty(Z) = Z for every Z E C}; ALl = {y E MB : Ty(Z) = Z for every Z E C}; Md,l = {y E MB : Ty is non-trivial and Ty(i) = i}; Md,-l = {y E !vIB : Ty is non-trivial and Ty(i) = -i}. LEMMA 1.3. Let p be a ring homomorphism of A into B. Then M o, Ml U Md,l and M-1 UMd,-l are clopen (closed and open) subsets of MB. The subsets M1 and M-l are closed in M B . PROOF. By the definitions it is easy to see that Mo = {y E MB : p(ieA)(y) = a}, M1 U Md,l = {y E MB : p(ieA)(y) = i}, and ALl U Md,-l = {y E MB : p(ieA)(y) = -i}, so they are clopen since p(ieA) is continuous on M B . Next we show that Jl,ft is a closed subset of MB. Let y E Md,l' Since Ty is non-trivial, there exists a complex number A such that Ty(A) =I- A. Put

Then G is an open neighborhood of y. We also see that G n M1 = 0. It follows that ]\,{1 is a closed subset of MB since M1 U M d,l is clopen. In the same way, we see that M-1 is a closed subset of M B . 0 Suppose that p is a ring homomorphism of A into B. If y E M 1 , then it is easy to see that Py is a non-zero complex homomorphism on A. Thus there exists a unique cp(y) in MA with

p(a)(y) = a(CP(y)),

a E

A.

In a way similar to the above we arrive at a partial representation as follows:

p(a)(y) =

a, { a(CP(y)), a(CP(y)),

yEMo, yE M 1 , y E M_ 1 .

If y E Md,l U Md.-1, then the situation is complicated, in particular, ring homomorphisms with large Md,l U Md,-l are possible (cf. [20, Examples 5.3 and 5.4]).

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2. Ring homomorphisms which satisfy the condition (m) In general the kernel of a non-zero ring homomorphism of A into C is a prime ideal and need not be a maximal ideal. (See section 5 in this paper.) In this section we consider ring homomorphisms P of A into B which satisfy the condition that the kernel of the induced ring homomorphism Py for each y E MB defined by

Py(f) = p(f)(y),

f EA

is a maximal ideal. DEFINITION 2.1. Let P be a ring homomorphism of A into B. We say that P satisfies the condition (m) if Py is zero or ker Py is a maximal ideal of A for every yEMB. By the following Lemma 2.2, if py(A) = C for every y E M B , in particular, if p(A) :J CeB, then (m) is satisfied. A *-ring homomorphism also satisfies the condition (m). (See the proof of Corollary 2.5.) LEMMA 2.2. Let Po be a non-zero ring homomorphism of A into C. Then the following are equivalent. (1) The kernel ker Po of Po is a maximal ideal of A. (2) The equation po(A) = PO(CeA) holds. (3) There exist a non-zero ring homomorphism 7 on C and an x E MA such that the equation po(a) = 7(a(x)) holds for every a E A. In this case 7 = Po 0 Ie. Such a 7 and x are unique. (4) The mnge Po(A) is a subfield ofC which contains a non-zero complex number. PROOF. First we show that (1) implies (2). Suppose that ker Po is a maximal ideal. Then there exists a non-zero complex homomorphism

p(a)(y) = {7y(a(
yEMo.

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In particular, p(a)(y) = a(q>(y)) for every y E Ml and p(a)(y) = a(q>(y)) for every y EM_I' The set q>(Md,l U Md,-d is a (possibly empty) finite subset of MA and q>-l(X) n (Md.l U Md,-d is an open subset of MB for each x E q>(Md,l U Md,-d. PROOF. Let y E MB \ Mo. Then by the condition (m) ker PY is a maximal ideal of A, so by Lemma 2.2 there exists a unique q>(y) E MA such that

py(a) = PY o Ic(a(q>(y))) holds for every a E A. By the definition Ty = py 0 Ie and since py(a) = p(a)(y) we see that p(a)(y) = Ty(a(q>(y))) holds for every a E A. If y E M l , then Ty(A) = A for every A E C, so p(a)(y) = a(q>(y)). If y EM_I, then Ty(A) =). for every A E C, so p(a)(y) = a(q>(y)). If y E M o, then p(a)(y) = O. Put Md = Md,l UMd,-l. We show that q>(Md) is a finite subset of MA. Suppose not. Then there is a countable subset {Xn}~=l of q>(Md). For each n choose a point Yn E Md with q>(Yn) = Xn · Since TYI is unbounded near zero, there exists an al E A such that lIalli < 2- 1 and hI (al(xd)1 > 2. By induction we can find, for every n, an E A such that an(xd = ... = an(xn-d = 0, lIanll < 2- n , and

ITYn(an(xn))1 > 2n + ITYn(al(Xn) + ... + an-l(xn))l· (Choose b2 E A with b2 (xd = 0 and b2(X2) = 1. Since TY2 is unbounded near zero, there is a non-zero complex number 02 such that 1021 < Ilb211- l 2- 2 and ITY2(0)1 > 22+ITy2 (al(x2))I· Then put a2 = 02b2. We have a2(xl) = 0, IIa211 < 2- 2, and ITy2 (a n (x2))1 > 22+ITy2 (al(x2))1. Suppose that al, .. ' ,an-l E A are choosenso that the conditions are satisfied. Choose bn E A with bn(xd = ... = bn(xn-d = 0 and bn(xn) = 1. Since TYn is unbounded near zero, there is a non-zero complex number On such that 10nl < Ilbn ll- 1 2- n and

ITYn (on)1 > 2n + ITYn (al(xn) + ... + an-l(xn))l· Then a2 = onbn is a desired function for n.) Then E::'=l an converges in A, say to a. Then a(xn) = al(x n ) + ... + an(xn) since the Banach norm on A dominates the uniform norm on AlA. On the other hand

so that p(a) is unbounded, which is a contradiction proving that q>(Md) is a finite set. Let q>(Md) = {Xl, ... , xn} and Yj = q>-l(Xj) n Md for each j = 1,2, ... , n. Choose an a E A such that a(xl) = 1, a(x2) = ... = a(xn) = O. Then p(a)(y) = 1 if y E Y l while p(a)(y) = 0 if y E Md \ Y l . Because p(a) is continuous, Y1 is clopen in Md; but Md is open in M B , so Y l is open in MB. In the same way we see that Yj is an open subset of MB for each j = 2,3, ... n. Finally we prove that q> is continuous. Since Yj is open and q>(Yj) = Xj, we only need to prove that q> is continuous at each point in Ml U M_ l . Let y E Ml and {y>.hEA be a net which converges to y. Without loss of generality we may assume that {y>.} c Ml U Md,l since Ml U Md,l is clopen. Suppose that {q>(y>.)} does not converge to q>(y), that is, there is an open neighborhood G of q>(y) such that for every A E A there exists a A' 2:: A with q>(YN) rf. G. There exist a finite number of points aI, ... , am in A and a positive real number e such that

{x

E

MA : laj(x) - aj(q>(y))1 < e,j = 1,2, ... , m}

C

G.

AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS

207

Since {p(aj)(y,x)} converges to p(aj)(Y) for each j, there exists a Ao E A such that

JTy>. (aj(iP(y,x))) - aj(iP(y))J < e holds for every A ~ AO and j = 1,2, ... , m. Suppose that A ~ Ao, then there exists a A' ~ A such that iP(y,x/) f/. G, so that Jajl(iP(y,x/)) - ajl(iP(y))J ~ e for some j'. lt follows that YN E Md,l. We also see that iP(y,x/) E {Xl, ... ,Xn } \ {iP(y)}. There exists an a E A such that a(iP(y)) = 1 and a = 0 on {Xl, ... ,Xn } \ {iP(y)}. We conclude that for every A with A ~ Ao there exists a A' ~ A such that p(a)(y,x/) = 0 and p(a)(y) = 1, which is a contradiction since p(a) is continuous on M B . Thus we have that {iP(y,x)} converges to iP(y), so iP is continuous at y. In the same way we see that iP is continuous at each point in M_ I . We have proved that iP is continuous on MB \Mo. 0 Note that the set Md,l U Md,-l need not be a finite set or even a closed subset of AlB (cf. [20, Example 5.3]). In [21] the authors proved the following corollary in the case where A is regular and satisfies a certain additional condition. Now we can remove these conditions. COROLLARY 2.4. Let p be a ring homomorphism from A into B. Suppose that py(A) = C for every y E M B . Then there exists a continuous mapping iP of MB into MA and a non-trivial ring automorphism Ty on C for every y E Md,l U Md,-l s1Lch that Y E MI , a(iP(Y))' { p(a)(y) = a(iP(y)), Y EM_I, Ty(a(iP(y))), Y E Md,l U Md,-l.

Moreover iP(Md,1 U Md,-d is a finite subset of MA. PROOF. By Lemma 2.2 we see that ker py is a maximal ideal, so the condition (m) is satisfied. The conclusion follows by Theorem 2.3. In particular, Ty = Py 0 Ie is onto, thus it is a non-trivial automorphism on C for y E Md,l U Md,-l. 0

Theorem 2.1 in [13] for the case of unital and semi-simple commutative Banach algebras is also deduced from Theorem 2.3 COROLLARY 2.5. Suppose that A is involutive and B is symmetrically involutive. Let p be a *-ring homomorphism. Then MB = Mo U MI U M-I and there exists a continuous function iP from MB \ Mo into AlA such that

a(iP(Y))' p(a)(y) = { 0, --:a('-=-iP..,...( y77")) ,

yE

MI ,

yE Mo,

y EM_I·

PROOF. Since p is a *-ring homomorphism, it it easy to see that Ty(Z) = Ty(Z) for every Z E C and for every y E MB \ Mo. It follows that Ty is 0 or linear or conjugate linear. Thus the conclusion follows. 0

3. Automatic linearity One of the reasons for ring homomorphisms between infinite-dimensional Banach algebras to be linear or conjugate linear is that the range contains an element with large spectrum. In this section we show evidence of this.

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O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKA HAS I

COROLLARY 3.1. Let p be a ring homomorphism of A onto B. If MB contains no isolated point, then p is real-linear. If MB is infinite and connected, then p is linear or conjugate linear. PROOF. Since p is a surjection, py(A) = C for every y E M B , so the condition (m) is satisfied by Lemma 2.2. We also have that the induced mapping is injective. Thus Md,l U Md,-l is a (possibly empty) finite set. Because Md,l U Md,-l is open (by Lemma 1.3), each point of Md,l U Md,-l is isolated in M B . If MB contains no isolated point, then Md,l U Md,-l = 0. Thus p is real-linear. If lvIB is infinite and connected, then MB contains no isolated point, so Md,l U Md,-l = 0. It follows by Lemma 1.3 that MB = lvh or MB = M_ l . Thus P is linear or conjugate linear. 0 COROLLARY 3.2. Let p be a ring homomorphism of A into B. Suppose that p satisfies the condition (m). Then there exists a (possibly empty) finite subset {Xl, ... , xn} of M A such that p is real-linear on the finite-codimensional closed ideal {a E A: a(xj) = O,j = 1,2, ... ,n} of A. PROOF. Put {Xl,""X n } = (Md,l UMd,-d. (The set is finite by Theorem 2.3.) Then for every a E {a E A: a(xj) = O,j = 1,2, ... ,n} p(a)(y)

=

{Ty(a((Y))), 0,

y E Ml U M_l' Y E Mo U Md,l U Md,-l.

Since Ty is real-linear for every y E Ml U M_l' the conclusion follows.

o

COROLLARY 3.3. Let p be a ring homomorphism from A into B such that p(CeA) = CeB. Then we have that Mo = 0, and there exists a continuous mapping from MB into MA such that one of the following three occurs. (1) P is linear: p(a)(y)

= a((y)),

a E A,

y E MB .

a E A,

y E MB .

(2) p is conjugate linear: p(a)(y)

= a((y)),

(3) There exists a non-trivial ring automorphism T on C such that p(a)(y)

= T(a((y))),

a E A,

y E M B.

In particular, if there exists an a E A such that the spectrum of p( a) is an infinite set, then p is linear or conjugate linear.

PROOF. For every y E MB, we have py(CeA) = C, so py(A) = py(CeA) = c. Thus ker Py is a maximal ideal of A by Lemma 2.2, so that the condition (m) is satisfied. Since p(ieA) E CeB, MB = Ml U Md,l or MB = M_l U Md,-l. Suppose that Md,l U Md,-l = 0. Then (1) or (2) occurs. Suppose that there exists some Yd E Md,l and some Yl E M l · Then there is a complex number>' with Tyd (>') =I=- >., so that p(>.eA) is not a constant function, which contradicts our hypothesis. Thus Md,l =I=- 0 implies that MB = Md,l. It is also easy to see that Ty is identical for every y E Md,l since p(CeA) consists of constant functions. Thus (3) follows. In the same way we see that (3) follows if Md,-l =I=- 0. Suppose that there exists an a E A such that p(a)(MB) is infinite, then (3) does not occur since ( M B) is a finite set in this case. It follows that p is linear or conjugate linear. 0

AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS

209

Corresponding results for ring homomorphisms on rings of analytic functions are proved by Kra [10, Theorem I]. Suppose that p is a ring homomorphism from A into B which satisfies two conditions: p(CeA) C CeB; there exists an a E A such that p(a)(MB) is infinite. Does it follow that p is linear or conjugate linear? Although the authors do not know the answer, we can provide a positive answer under a stronger condition. THEOREM 3.4. Let p be a ring homomorphism from A into B. Suppose that the following two conditions are satisfied: (i) p(CeA) C CeB; (ii) there exists an a E A such that p(a)(MB) contains a non-empty open subset ofC. Then p is linear or conjugate linear. PROOF. Since p(CeA) C CeB we may suppose that pole is a non-zero ring homomorphism on C. We have two possibilities: po Ie(i) = i; po Ie(i) = -i. We show that, in the first case, po Ie(z) = z for every complex number z, so it will follow that p is linear on A. (In the same way we see that p is conjugate linear if pole( i) = -i.) Suppose that pole( i) = i. We show that pole is continuous on C. For this it is enough to show that pole is continuous at O. Suppose not. Then there is a sequence {w n } of non-zero complex numbers which converges to 0 such that {poIe(w n )} does not converge to O. Without loss of generality we may assume that Ip 0 Ie(w n ) I ~ 00 as n ~ 00. Let a be in A such that p(a)(MB) contains a non-empty open subset G of the complex plane. Let s be a complex number in G such that the real part and the imaginary part of s are both rational numbers. Put Zn = S + 1/ po Ie(w n ). Then there is a positive integer mo such that Zm E G for every m ~ mo since Ip 0 Ie(w n )I ~ 00 as n ~ 00, so ZmeB - p(a) ~ B- 1 . Thus we have (8 + l/wm)eA - a ~ A-I. Then 8 + l/wm is in the spectrum of a for every m ~ mo, which is a contradiction since Is + l/wnl ~ 00 as n ~ 00. It follows that pole is continuous at 0, thus on C, so pole( w) = w for every complex number w since p(ieA) = i. Then we see that p is linear on A. D Note that either of the two conditions (i) and (ii) in the above theorem itself does not suffice for p to be linear or conjugate linear. Let T be a non-trivial ring automorphism on C. Suppose that x E MA and tp from A into C is defined by tp(a) = a(x) for every a E A. Put P = TO tp. Then p is a ring fomomorphism with (i) since p(CeA) = C, but P is neither linear nor conjugate linear; p is not even real-linear. Let D be the closed unit disk in the complex plane. Let D + 3 = {z E C: Iz - 31::; I} and X = D U (D + 3). Define p(f)(z) = {f(Z)' f(z - 3),

zED zED+3

for every f E C(D). Then p is a ring homomorphism from C(D) into C(X) with the condition (ii). But p is neither linear nor conjugate linear. Even more is true. There is a ring homomorphism with the condition (ii) which is not real-linear. Recall that the disk algebra A(D) is the algebra of all complex-valued continuous functions on D which are analytic on the interior D of D. Suppose that

K = {O} U {l/n : n is a positive integer}, X

= {2} U D

and Y

= K U {z E C : Iz - 31::; I}.

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O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI

Let

A = {f

E C(X) :

flD

E

A(D)}

and B = C(Y), where CO denotes the algebra of all complex-valued continuous functions on '. Let ¢ be the ring homomorphism from C into C(K) defined in [20, Example 5.3]. For f E A, put

f(2), y=O, p(f)(y) = { ¢(f(2))(1/n), y = lin, f(y - 3), Iy - 31 :::; 1. Then p is a ring homomorphism and satisfies the condition (ii). But (i) is not satisfied and p is not real-linear on A.

4. Ring homomorphisms into algebras of analytic functions Suppose that A is a completely metrizable topological algebra with an identity and r(X, Ox) is the algebra of global sections of a connected reduced complexanalytic space (X, Ox). Becker and Zame [2] proved among other things that if p is a ring homomorphism from A into r(X, Ox) such that the range of p contains a non-constant section, then p is linear or conjugate linear. This is not the case for ring homomorphisms on non-complete algebras. (Suppose that P is the algebra of polynomials on C and H(CC) is the algebra of entire functions. Let T be a nontrivial ring homomorphism on C and define p on P by p(Eanz n ) = ET(an)Zn for every polynomial E anz n . Then p is a ring homomorphism, but it is neither linear nor conjugate linear. By Theorem 5.1 there also exists an injective ring homomorphism from Pinto C since {O} is a prime ideal in P.) Nevertheless we show automatic linearity results for ring homomorphisms on certain non-complete metrizable algebras. THEOREM 4.1. Suppose that A is a complex (commutative or non-commutative) algebra with unit e. Suppose that Y is a non-empty set and B is a complex algebra of complex-valued functions on Y which contains the constant functions. Suppose that for every non-constant function b E B the range of b contains a non-empty open subset of c. Let p be a ring homomorphism from A into B. If there exists an element a in A such that the resolvent set of a contains a non-empty open subset G of C and p( a) is non-constant, then p is linear or conjugate linear. PROOF. It is easy to see that p( e) = 0 or 1 since the range of a non-constant function in B contains a non-empty open set. If p( e) = 0, then p is 0 on A and so p is linear. Suppose that p(e) = 1. In the same way as above, we see that p(ie) = i or -i. We show that p is linear if p(ie) = i. (If p(ie) = -i, then 15 defined by 15(a) = p(a) will be linear by what we will show, so p will be conjugate linear.) We will show that p(,Xe) = ,X for every complex number 'x. First we show that p(Ce) C cc. Suppose not. Then there is a complex number ,x such that p(,Xe) is a non-constant function. Note that Re'x or Im'x is irrational since p(xe) = x for every rational real number x by a simple calculation. Since p('xe)(Y) contains a non-empty open set, there exists r E p('xe)(Y) with Rer and Imr both rational. Then p(,Xe) - r is not an invertible element in B. Therefore (,x - r)e is not invertible in A, that is, ,x = r, which is a contradiction. Thus we have proved that p(Ce) C C, or p induces a ring homomorphism on cc. We denote the induced ring homomorphism also by p.

AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS

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Suppose that p is non-trivial on C. Since p(a) is not a constant, the interior of p( a) (Y) contains a complex number s whose real and imaginary parts are both rational, by our assumption for B. Since p is non-trivial, there exists a sequence {w n } of non-zero complex number such that Ip(wn)1 tends to infinity as n --+ 00 and s+ l/w n E G for every n. Since p(i) = i we see that p(s+ l/wn ) = s+ 1/ p(w n ) and we may assume that

s + l/p(w n ) E p(a)(Y) for every n. Thus p(a) - (s + 1/ p(wn )) is not invertible in B. It follows that a - (s + l/w n )e is not invertible in A, which is a contradiction, proving that p is trivial. Since p(i) = i, we have that p(A) = A for every complex number A. We conclude that p is linear on A. 0 The spectrum of each element in a Banach algebra is compact, so the conditions for A in Theorem 4.1 are satisfied by every Banach algebra with unit. Since the range of non-constant analytic function is a non-empty open subset of C, algebras of global sections on connected, reduced complex-analytic spaces satisfy the condition for B in Theorem 4.1. Thus we have the following, which is a version of a more general result of Becker and Zame [2, Theorem 3.1]. But our proof is considerably simpler. COROLLARY 4.2. Let Ao be a Banach algebra with unit. Suppose that p is a ring homomorphism from Ao into r(X, Ox), the algebra of global sections on a connected, reduced complex-analytic space (X , Ox). If p( Ao) contains a nonconstant section, then p is linear or conjugate linear.

Let S be a subset of C. We denote by Rs the algebra of all rational functions on C with poles off S. Although Rs is a wlital algebra, it cannot be a Banach algebra by the Baire category theorem. If S = C, then Rs = P, and so there is a ring homomorphism p on Rc into r( X , Ox) for (X, Ox) = C such that p(Rc) contains a non-constant function, while p is neither linear nor conjugate linear. In the case where C \ S contains an interior point, the situation is different; in this case we prove an automatic linearity result. COROLLARY 4.3. Let S be a subset of C whose complement contains interior points. Suppose that (X, Ox) is a connected, reduced complex-analytic space and r(X, Ox) is the algebra of global sections. Suppose that p is a ring homomorphism from Rs into r (X , Ox). If the range of p contains a non-constant section, then p is linear or conjugate linear. PROOF. In the same way as in the proof of Theorem 4.1 we see that p(C) C C. Suppose that z denotes the identity function: z(w) = w for every complex number w. Then we have that p(z) is non-constant. (Suppose not. Then p(f) is a constant section for every f E Rs.) On the other hand z - A is invertible for every A E C \ S, which contains a non-empty open set. Thus the conditions in Theorem 4.1 are satisfied. It follows by Theorem 4.1 that p is linear or conjugate linear. 0

Note that every ring homomorphism p of R0 into r(X, Ox) is constant-valued for the empty set 0. (We see that p(C) C C as before. Suppose that p(f) is not a constant section for some non-constant rational function f. Then there is a complex number r in p(f)(X) with rational real and imaginary parts. It follows that f - r is not invertible in R0, which is a contradiction.)

212

O. HATORI, T. ISHII, T. MIURA. AND S.-E. TAKAHASI

Let A be one of the algebras P, njj or the disk algebra A(D), where jj denotes the closed unit disk in C, and H(D) the algebra of analytic functions on the open unit disk. Although both P and no are dense in the disk algebra, automatic linearity results for ring homomorphisms on these algebras are different from each other. Suppose that p is a ring homomorphism from A into H(D) such that the range of p contains a non-constant function. If A = njj (resp. A(D), then p is linear or conjugate linear by Corollary 4.3 (resp. Corollary 4.2). But that is not the case for A = P. The ring homomorphisms defined by p(2:a n z n ) = 2:1'(a n )zn for polynomials 2: anz n are neither linear nor conjugate linear for non-trivial ring homomorphisms l' on C.

5. Complex-valued ring homomorphisms In this section we consider ring homomorphisms into the complex number field C. Suppose that A is a complex algebra and p is a non-zero ring homomorphism from A into C. Then the kernel ker p of p is a prime ideal. Recall that a proper ideal I of A is said to be a prime ideal of A if fg E I implies that f E I or gEl. By using well-known results of algebra, we see the converse is also valid; for every prime ideal such that the cardinal number of the quotient algebra of the algebra by the ideal is equal to that of the continuum, there exists a ring homomorphism into C whose kernel coincides with the ideal. Let K be an extension field of a field k. (Here and after a field means a commutative field.) We recall a subset S of K is said to be algebraically independent over k if the set of all finite products of elements in S is linearly independent over k. A subset T of K which is algebraically independent over k and is maximal with respect to the inclusion ordering is said to be a transcendence base of Kover k. By definition, for every transcendence base T of Kover k, K is algebraic over the quotient field k(T) of the polynomial ring of T over k. There exists a transcendence base of Kover k (cf. [11, Theorem X.l.I]). Using the same argument as in [9] we can prove the following (cf. [11, 20]). (This might be a standard fact. But we present here with a proof for the convenient of the readers.) THEOREM 5.l. Let A be a commutative complex algebra with unit e. Suppose that I is a prime ideal of A such that the cardinal number of AI I is that of the continuum c. Then there exists a ring homomorphism p from A into C such that kerp=I. PROOF. The quotient algebra AI I has no non-zero divisor of zero, for I is a prime ideal. We denote by K the field of fractions over AI I. Let Q be the field of complex numbers whose real and imaginary parts are both rational. Let TK be a transcendence base for Kover Q and T a transcendence base for Cover Q. Then the cardinal number of TK (resp. T) is c since that of AI I (resp. q is c. There exists au injection a defined from TK onto T. Since TK is algebraically independent, there is a unique extension from Q(TK ) onto Q(T), which is also denoted by a, and a is a ring homomorphism. Since C is algebraically closed and K is an algebraic extension of Q(TK), there exists an extension of a which defines a ring homomorphism of K into C by Theorem VII.2.8 in [11]. We also denote it bya. Let h be the natural homomorphism of A onto AI I. Put p = a 0 h. Then p is the desired ring homomorphism. D

AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS

213

Note that the corresponding ring homomorphism p is not unique. Let T be any non-zero ring homomorphism on C. Then TOp is a ring homomorphism on A with ker p = ker TOp. As a corollary of Theorem 5.1 we display a pathological feature of ring homomorphisms on algebras of analytic functions into C; even injection can be possible. COROLLARY 5.2. Let A be a unital algebra which consists of holomorphic functions on a domain in Then there exists an injective ring homomorphism of A into Co

cn.

PROOF. Since the ideal containing only zero is a prime ideal and the cardinality of A is the same as that of the continuum, there exists a ring homomorphism p of A into C whose kernel consists only of zero, by Theorem 5.1. Then p is an injective ring homomorphism. 0 Note that the injective ring homomorphism in Corollary 5.2 can never be surjective if A contains non-constant functions since A is not a field. Note also that every ring homomorphism from a unital commutative C* -algebra into C cannot be injective if the dimension of the algebra is greater than one since {O} is not a prime ideal in this case. Together with the results in the previous sections we give a complete description of ring homomorphisms on the disk algebra A(D). COROLLARY 5.3. Let p be a non-zero ring homomorphism on the disk algebra into itself. Then ker p is a prime ideal. If the range of p contains a non-constant function, then p is linear or conjugate linear; there exists 'P E A(D) with 'P(D) c fJ such that zED, f E A(D) p(f)(z) = f 0 'P(z), or

p(f)(z) = f

0

z E fJ,

'P(z),

f E A(fJ).

On the other hand, suppose that'P E A(D) with 'P(D) a(f)(z) = f

0

'P(z),

c

D. Then

ZED,

f

E A(fJ)

zED,

f

E A(D)

defines a linear ring homomorphism and a(f)(z) = f

0

'P(z),

defines a conjugate linear ring homomorphism.

PROOF. A(D) has no non-zero divisors of zero, so the kernel of any ring homomorphism from complex algebra with unit element into A(D) must be a prime (algebra) ideal. If p(A(D) contains a non-constant function, then by Theorem 4.1 we see that p is linear or conjugate linear. Suppose that p is linear. Then it is well known and easy to prove, since the maximal ideal space of A(D) is the closed unit disk D, that there exists 'P E A(fJ) with 'P(D) c D such that p(f)(z) =

f

0

'P(z)

holds for every f E A(D) and zED. Suppose that p is conjugate linear. Let h : A(fJ) -+ A(fJ) be defined as h(f)(z) = f(2),

f

E A(D),

zED.

214

O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI

Then hop is a linear ring homomorphism on the disk algebra. It follows that there exists r.p E A(D) with r.p(D) c D such that

h 0 p(f)(z)

=

1 0 r.p(z) ,

zED,

I

E

A(D).

Thus we see that

p(f)(z) = 1 0 r.p(z), holds for every I E A(D) and zED. Conversely, suppose that r.p E A(D) with r.p(D) c D. Then it is easy to see that u(f)(z)

=

1 0 r.p(z),

zED,

IE A(D)

zED,

IE A(D)

defines a linear ring homomorphism and

u(f)(z) = 1 0 r.p(z),

defines a conjugate linear ring homomorphism.

o

Let n be a positive integer and An(D) the subalgebra of those I in A(D) whose n-th derivative I(n) on D is continuously extended up to D. An(D) is a unital commutative Banach algebra with the norm IIIlIn = L~=o III(k)lIoo/k! for I E An(D), where II . 1100 is the supremum norm on D. Then Corollary 5.3 is also valid for An(D). Prime ideals in A(D) and An(D) are studied in [16]. (See also [4] for the case of A(D).) Mortini proved that every non-zero prime ideal is contained in a unique maximal ideal. He in fact showed that a non-zero and nonmaximal prime ideal in An(D) (resp. A(D)) is dense in exactly one of the ideals {J E An(D) : 1(>") = 1'(>..) = ... = I(j)(>..) = O} for some 0 ~ j ~ n (resp. {I E A(D) : 1(>") = O}), >.. E aD. We also see by a theorem of Dietrich [4] that the cardinal number of the set of all prime ideals of A(D) which is contained in a maximal ideal {J E A(D) : 1(>") = O}, >.. E aD is 2', the cardinal number of the set of all the subsets of the continuum. Thus we see that there are 2' ring homomorphisms on the disk algebra. Acknowlegement. The authers would like to thank Professor Ken-Ichiroh Kawasaki for his valuable comments. They also would like to thank the referees for their careful reading of the paper and their valuable comments.

References [1] B. H. Arnold, Rings of opemtors on vector spaces, Ann. of Math. 45(1944), 24-49 [2] J. A. Becker and W. R. Zame, Homomorphisms into analytic rings, Amer. Jour. Math. 101(1979), 1103-1122 [3] L. Bers, On rings of analytic junctions, Bull. Amer. Math. Soc. 54(1948), 311-315 [4] W. E. Dietrich, Jr., Prime ideals in uniform algebms, Proc. Amer. Math. Soc. 42(1974), 171-174 [5] M. Eidelheit, On isomorphisms of rings of linear opemtors, Studia Math. 9(1940),97-105 [6] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981 [7] H. Iss'sa, On the meromorphic junction field of a Stein variety, Ann. Math. 83(1966), 34-46 [8] 1. Kaplansky, Ring isomorphisms of Banach algebms, Canadian J. Math. 6(1954),374-381 [9] H. Kestelman, Automorphisms of the field of complex numbers, Proc. London Math. Soc. 53(1951), 1-12 [10] 1. Kra, On the ring of holomorphic functions on an open Riemann surface, Trans. Amer. Math. Soc. 132(1968),231-244 [11] S. Lang, Algebm (second edition), Addison-Wesley, California, 1984. [12] M. H. Lebesgue, Sur les tmnsformations ponctuelles, tmnsformaant les plans en plans, qu'on peut definir par des procedes analytiques, Atti della R. Acc. delle Scienze di Torino 42(1907), 532-539

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[13J T. Miura, Star ring homomorphisms between commutative Banach algebrns, Proc. Amer. Math. Soc. 129(2001), 2005-2010 [14J L. Molnar, The rnnge of a ring homomorphism from a commutative C' -algebrn, Proc. Amer. Math. Soc. 124(1996), 1789-1794 [15J L. Molnar, Automatic surjectivity of ring homomorphisms on H* -algebrns and algebrnic differences among some group algebrns of compact groups, Proc. Amer. Math. Soc. 128(2000), 125-134 [16J R. Mortini, Prime ideals in the algebrn An(D), Complex Variables Theory AppJ. 6(1986), 337-345 [17J M. Nakai, On rings of analytic and meromorphic functions, Proc. Japan Acad. 39(1963), 79-84 [18J W. Rudin, An algebrnic charncterization of conformal equivalence, Bull. Amer. Math. Soc. 61(1955), 543 [19J S. de Corrado Segre, Un nuovo campo di ricerche geometriche, Atti della R. Acc. delle Scienze di Torino 25(1889), 276-301 [20J P. Semrl, Non-linear perturbations of homomorphisms on C(X), Quart. J. Math. Oxford (2) 50(1999),87-109 [21J S.-E. Takahasi and O. Hatori, A structure of ring homomorphisms on commutative Banach algebrns, Proc. Amer. Math. Soc. 127(1999),2283-2288 DEPARTMENT OF MATHEMATICAL SCIENCE, GRADUATE SCHOOL OF SCIENCE AND TECHNOL-

950-2181 JAPAN E-mail address:hatorilDmath.se.niigata-u.ae.jp

OGY, NIIGATA UNIVERSITY, NIIGATA

NIIGATA CHUO HIGH SCHOOL, NIIGATA

951-8126 JAPAN

DEPARTMENT OF BASIC TECHNOLOGY, ApPLIED MATHEMATICS AND PHISICS, YAMAGATA UNI-

992-8510 JAPAN E-mail address:miura«lyz.yamagata-u.ae.jp

VERSITY, YONEZAWA

DEPARTMENT OF BASIC TECHNOLOGY, ApPLIED MATHEMATICS AND PHISICS, YAMAGATA UNI-

992-8510 JAPAN E-mail address:sin-eiOemperor.yz.yamagata-u.ae.jp

VERSITY, YONEZAWA

Contemporary :r...1athematics Volume 328. 2003

Carleson Embeddings for Weighted Bergman Spaces Hans Jarchow and Urs Kollbrunner ABSTRACT. We are going to discuss Carleson measures for the standard weighted Bergman spaces A~ (-1 < a < 00, 0 < p < (0). These are finite, positive Borel measures J.L on the unit disk in IC such that, given 0 < q < 00, A~ embeds, as a set, continuously into Lq(J.L). Such measures have been closely investigated by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking [11], [12]. We complement their results, in particular by characterizing compactness, order boundedness and related (absolutely) summing properties of the canonical embedding A~ <-t Lq(J.L).

1. Introduction

The Carleson measures under investigation are finite, positive Borel measures Jl. on the open unit disk l[J in the complex plane such that, given -1

<

0

<

00

and 0 < p, q < 00, the (classical) weighted Bergman space A~ is a subset of Lq(J.l) and the embedding is bounded. Such measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15J, W.W. Hastings [6J and D.H. Luecking [11], [12J. Our first main topic is to complement their results by characterizing when the canonical embedding I : A~ ~ Lq(J.l) is compact. We will see that this is always the case if p > q. Our second main topic is to characterize when I is order bounded, that is, the unit ball of A~ is a subset of some order interval in Lq(J.l). As a consequence, we obtain necessary and sufficient conditions for I to have specific (absolutely) summing properties. Our results extend corresponding ones for composition operators which have been obtained e.g. in [3]' [4], [20]' [21J. In fact, they can also be viewed as results on composition operators which may have rather unusual range spaces. They apply, for example, to pointwise multipliers. Many of the results to be presented remain valid for measures J.l on D for which f f-> f induces just a bounded linear map A~ ~ Lq(J.l) (not necessarily injective). This is rather straightforward; precise formulations, however, require somewhat 1991 Mathematics Subject Classification. Primary 46 E 15,47 B 38, 47 B 10; Secondary 46 B 25, 30 H 05, 32 H 10. Key words and phrases. Weighted Bergman spaces, Carleson measures, composition operators, compactness, order boundedness, absolutely summing operators. The results of this paper are part of the dissertation of the second named author written at the University of Ziirich under the supervision of the first.

© 217

2003 Alnerican Mathematical Society

HANS JARCHOW AND URS KOLLBRUNNER

218

clumsy notation. Also, there is an immediate extension to complex Borel measures on 1lJ whose variation is (a, p, q) - Carleson. We are indebted to the referee for providing Example 8 and for bringing to our attention the paper [15] by V.L. Oleinikov and B.S. Pavlov. 2. Weighted Bergman spaces Throughout the paper, we will use standard results and notation from (quasi-) Banach space theory. We will work on the open unit disk 1lJ = {z E C: Izl < I} in the complex plane. The space 1i(1lJ) of all analytic functions 1lJ -> C is a F'rechet space with respect to the topology of uniform convergence on compact subsets of 1lJ. Let da be normalized area measure on 1lJ. For each a > -1,

dao(z) :=

(a + 1) (1 -lzI 2 )O da(z)

is a probability measure on 1lJ. For each 0 < p < Bergman space is defined to be A~ := A~

00,

the corresponding weighted

1i(1lJ) n P(ao ).

is closed in LP (a 0); it is a Banach space if p 2:: 1 and a p - Banach space if Its (p- ) norm will be denoted by II . Ilo,p. A~ is a Hilbert space and has a reproducing kernel:

o < p < 1.

Ko(z, w) = K(z, w)o+2; here K(z,w) = (1 - ZW)-l is the reproducing kernel for the Hardy space H2. For reasons like this, the scale of Hardy spaces is often considered as the scale of weighted Bergman spaces which corresponds to a = -1. Some of the results below actually remain true for this case, and some can even be extended to analytic Besov spaces B~ (f E B~ {::} f' E A~+p). Nevertheless, in this paper we will only deal with the case -1 < a < 00. 3. Carleson measures All measures on 1lJ will be finite, positive Borel measures. Let -1 < a < 00 and 00 be given. We say that a measure /L on 1lJ is an (a, p, q) - Carleson measure if A~ c Lq(/L) and the embedding A~ '---+ Lq(/L) is continuous: there is a constant C> 0 such that IIfIILq(~) ::; C ·lIfIIA~ "If E A~. Given an (a, p, q) - Carleson measure, the canonical embedding I : A~ -> Lq(/L) will be referred to as a Carleson embedding. As mentioned in the introduction, a number of the results to follow remain true if we just require that f 1-+ f induces a bounded linear map A~ -> Lq(/L). Also, complex measures whose variation is (a, p, q) - Carleson can be incorporated. Moreover, there are extensions to analytic functions of several variables. However, we are not going to discuss such generalizations in this paper. We say that /L is a compact (a,p, q) - Carleson measure if the embedding A~ '---+ Lq(/L) exists and is compact. For example, an a.e. positive function h E Lq(a/3) defines the bounded multiplier Mh : A~ -> Lq(a/3) : f 1-+ fh iff the measure h q da/3 is (a,p, q) - Carleson. Moreover, discrete (a, p, q) - Carleson measures on 1lJ can be defined using appropriate versions of 'sampling sequences', etc.

o < p, q <

CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES

219

An important example is obtained by looking at the composition operator C", : J 0 'P induced by a non-constant analytic function 'P : 10 ---> 10. Clearly, C",: A~ ---> A~ exists iff a o o'P- 1 is (o:,p,q)-Carleson. More generally, an arbitrary measure J.l on 10 is (0:, p, q) - Carleson if and only if, for every analytic map 'P : 10 ---> 10, CI{J maps A~ boundedly into Aq(J.l) := 1i(1O) n Lq(J.l). In fact, the condition applied to the identity of 10 shows that J.l is (o:,p,q)Carleson. On the other hand, if J.l is (0:, p, q) - Carleson and 'P : 10 ---> 10 is analytic, then CI{J : A~ ---> Aq(J.l) is well-defined and bounded. For non-constant functions 'P, the condition is further equivalent to J.l 0 'P -1 being (0:, p, q) - Carleson. This allows an interpretation of Carleson embeddings, and in particular of multipliers as above, as composition operators. However, in such a general setting the range space of a composition operator might be unpleasent, and desirable properties may not be available. For example, Aq(J.l) embeds continuously into 1i(1O) if and only if Aq(J.l) is a closed subspace of Lq(J.l) and all point evaluations Aq(J.l) ---> C : J f-+ J(z), z E 10, are continuous. (0:, p, q) - Carleson measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking

J

f-+

[11],[12]. The hyperbolic metric on 10 is given by

. lJd(J "I 1 _ J(J

e(z, w) := l~f

where the infimum extends over all smooth curves 'Y in 10 joining z and w. For w E 10 and r > 0, let Br (w ) = {z E 10 : e( z, w) < r} be the corresponding hyperbolic disk. Actually, the particular choice of r > 0 doesn't really matter in our context. Let us agree to write A~ '---+ Lq(J.l) if A~ is a subset of U(J.l) and the embedding is continuous. Similar for other function spaces. The Carleson measures under consideration can be characterized in terms of the function

THEOREM

3.1. Let -1 < 0: <

00

and 0 < p, q <

00

be given.

(a) IJp::; q, then A~

'---+

Lq(J.l) iJ and only iJ Ho,p,q(w) is bounded on 10.

(b) IJp> q, then A~

'---+

Lq(J.l) if and only if Ho,p,q

E

L-/!!q(A).

Here dA(z) = da(z)(I-JzJ2)-2 is the Mobius invariant measure on 10. In [12] Luecking presents an interesting proof of (b) which is based on the inequalities of Khinchin and Kahane for Rademacher functions (see e.g. [2]). It is well-known (compare K. Zhu [22]) that there is a constant C = C(o:, r) > 0 such that 1 C . ao(Br(w)) ::; (I-JwJ2)o+2 ::; C· ao(Br(w)) 'Vw E 10 . Therefore we may also say that Theorem 3.1 refers to properties of the function w f-+ J.l(Br (W))I/ q(1 _JWJ2)-(o+2)/p .

220

HANS JARCHOW AND URS KOLLBRUNNER

It also follows that Ha,p,q E L~(A) if and only if J-t(B r (·))/C7 a (B r (·)) is in LP/(p-q) (C7 a ). This will be used in the proof of Theorem 4.3 below. If p ~ q; then the relevant parameter in Theorem 3.1 is q (0: + 2)/p, whereas for p > q and fixed 0:, dependence is on p / q. As a first immediate consequence we may state: COROLLARY

if and only if A~

3.2. For any -1 < 0: < "---> Ltq(J-t).

00,0

< p,q < 00 and t > 0,

A~ "--->

Lq(J-t)

In turn, this leads to: 3.3. Let -1 < 0:,0:' < 00 and 0 < p, pi, q, q' < 00 be given. (a) Ifp ~ q, pi ~ q' and q. (0: + 2)/p = q'. (0: ' + 2)/p', then A~ "---> Lq(J-t) iff A~, "---> Lq' (J-t).

COROLLARY

(b) Ifp> q and p/q = pi /q', then A~

"--->

Lq(J-t) iff A~

"--->

Lq' (J-t).

For a large range of parameters, this allows a reduction to Hilbert spaces as follows: 3.4. Suppose that -1 < 0:,0:' < 00 and 0 < p, q < 00. (a) Ifp ~ q and 0:' + 2 = q. (0: + 2)/p, then A~ "---> Lq(J-t) iff A~,

COROLLARY

(b) If p > q and pi /2 = 2/q' = p/q, then A~ A~ "---> L 2 (J-t). A special known case occurs when we take J-t Horowitz [7]).

"--->

Lq(J-t) iff A~

3.5. Suppose that -1 < 0:, (3 < 00 andO < p, q < (a) If p ~ q, then A~ "---> A~ iff (0: + 2)/p ~ ((3 + 2)/q. "--->

A~ iff (0: + l)/p

"--->

L 2 (J-t).

Lq' (J-t) iff

= C7fJ for some (3 > -1 (see C.

COROLLARY

(b) If p > q, then A~

"--->

00.

< ((3 + l)/q.

There are several ways to modify the domain space of a composition operator. In a systematic fashion, we may proceed as follows; cf. [4]. Each of the kernel functions K a ( Z, .) is bounded (z E lV), and ._ ( 1 - IzI2 ) (a+2)/p

(1 _

ka,p,z(w),has (p-) norm one in representation

A~

(0 < p <

00).

ZW)2

The functions

f

E A~ which admit a

00

f(w) =

L

an

Vw E lV ,

ka,p,zn(w)

n=l

where the scalars linear space, say

an

satisfy

En lanl <

00

and the

Zn's

are taken from lV, form a

This is a Banach space with norm

IlfIIA~)

00

:= inf

{L lanl:

(*) holds} .

n=l

In fact, the map e1(lV)

--+

A~) : (az)zEllJ

1--4

EZEllJ

azka,p,z is a metric surjection.

CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES

221

It is immediate that

• if 0
00

are such that (a+2)/p

= (/3+2)/q,

Atomic decomposition is available for weighted Bergman spaces (e.g. [10], [1], [22]), hence

• A~

S:!

A~l) (with equivalent norms).

In particular, if a> p-2, then A~), alias Ab with /3+2 = (a+2)/p, is isomorphic to fl. Moreover: • If p 2: 1 then A~) '---+ A~ (boundedly and densely).

A~) = Ab (boundedly and densely). In fact, it was shown by J.H. Shapiro [19] that in the latter case Ab is the Banach • If 0

< p < 1, then A~

'---+

envelope of A~, that is, the Banach space which is obtained by completing A~ with respect to the biggest norm which is smaller than the given p-norm. PROPOSITION 3.6. Let -1 < a < 00 and 0 < p ::; q < 00, q 2: 1. The following are equivalent: (i) J.L is an (a,p, q) - Carleson measure. (ii) A~)

'---+

Lq(J.L) (boundedly).

(iii) SUPzEV IIko,p,zlbu.t) < 00. PROOF. (i) =? (ii) is obvious if p 2: 1 and follows from the above result of J.H. Shapiro [19] if p < 1 ::; q: A~) is the Banach envelope of A~, and so the convex hull of BAP is dense in B A(p), (ii) =? (iii) is trivial, and (iii) =? (i) is immediate from the followi~g estimate in ;hich C is a constant depending only on 0 < r < 1: J.L(B r (W))l/ q 1 )l/q Br{w) dJ.L(z) (1 - IwI 2){o+2)/p = (1 -lwI 2){o+2)/p . - (1 -

<

-

12){o+2)/p

I

-

W

C· (

.

(1 (1

Br{w)

1 d ( )) l/q (1 -lwI2)2q{o+2)/p J.L Z

r1(1(1 -- wz)2{o+2)/p I

IwI 2){o+2)/p qd

lv

Z) l/q

J.L()

= C ·lIko,p,wIILq{~).

o

Now apply Theorem 3.1(a).

Actually, in the last step, no restriction on p and q is needed: Ho,p,q(w) is bounded whenever A~) '---+ Lq(J.L). There is another interesting consequence of Theorem 3.1, Corollary 3.2, and Proposition 3.6: COROLLARY 3.7. If 1 ::; q < p < 00 and (p/q) - 2 < a < L~(A) implies Ho,p,q(w) E LOO(A).

00,

then Ho,p,q(w) E

PROOF. Define a' > -1 by a' + 2 = q(a + 2)/p. Then, with £T = £T(A),

Ho,p,q E Lpq/{p-q) <=> A~

'---+

Lq(J.L)

=? A~/q) = A~, '---+

<=> H O ',l,l E L oo <=> Ho,p,q E L oo

.

L 1 (J.L)

o

HANS JARCHOW AND URS KOLLBRUNNER

222

EXAMPLE 3.8. (a) Let a, p, q be as in Corollary 3.7, let (an) be a sequence in foo \ fP/(p-q) , and let (1]n) be a sequence in llJ such that (!(1]n,1]k) :::: r· 8nk for all n, k, and such that llJ = Un Br(1]n). Consider the measure JL = En bn 811n on llJ where bn = lanl' (1-I1]nI 2 )q("'+2)/p and note that (b n ) E fl. A calculation reveals that H""p,q is in LOO(A) but not in uq/(p-q) (A). We conclude that the converse in Corollary 3.7 doesn't hold. (b) In Proposition 3.6, (ii) {::} (iii) is true for arbitrary 0 < p, q < 00, and (i) ~ (ii) holds trivially whenever q :::: 1. But (iii) ~ (i) fails for 1 ~ q < p and a + 2 > p. In fact, if JL is as in (a), then A~ 'f+ Lq(JL) since H""p,q tfLpq/(p-q)(A), but H-y,l,q = H""p,q E LOO(A) if we put "I = (a + 2)/p - 2. From A~) = A~l) = A~ we conclude that A~ '----> Lq(JL). 4. Compactness We shall frequently make use of the following classical result: THEOREM 4.1 (Pitt's Theorem). If 0 < p < q < 00 then every operator f q --+ fP is compact. This was obtained in 1936 by H.R. Pitt [16] for p :::: 1. For an extension see H.P. Rosenthal [17]. The result as stated was proved recently by E. Oja [14]. By atomic decomposition, A~ and fP are isomorphic (see [10], [1]). Combining this with Pitt's theorem we see that in particular the embedding in Corollary 3.5.(b) is compact. It will follow from the next theorem that the embedding in 3.5.(a) is compact iff (a + 2)/p < (f3 + 2)/q. Our characterization of compactness of Carleson measures splits into two parts. We consider the case p ~ q first. Here the characterization is as expected: THEOREM 4.2. Let -1 < a < 00 and 0 < p ~ q < statements are equivalent: (i) JL is a compact (a,p, q) - Carles on measure. (ii) A~)

'---->

with 1 ~ q. The following

Lq(JL) compactly.

(iii) z-+ limlllk""p,zIILq(ll) (iv)

00

= O.

lim Ho.,p,q(w) = O.

Iwl-+l

PROOF. (i) ~ (ii): If p :::: 1 then nothing is to prove since A~) '----> A~. If p < 1 ~ q then, by [19], A~) is the Banach envelope of A~, and the convex hull of B A~ is dense in B A!:)' Hence relative compactness of B A~ in the Banach space Lq(JL) entails relative compactness of B A<.!) in Lq(JL). - Here we have used Bx to denote the unit ball of a (quasi-) Banach space X. (ii) ~ (iii): Suppose that (iii) doesn't hold. Then there exist an c > 0 and a sequence (zn) in llJ such that limn--+ oo IZnl = 1 and Ilko.,p,zn IIL-(Il) > c for all n. By (ii), we may assume that (k""p,zn)n converges to some f E Lq(JL). But lim n -+ oo IZn I = 1 implies f = 0 since clearly (k""p,zn) tends to zero pointwise: contradiction. (iii) ~ (iv): The estimate proved in (iii) ~ (i) of Proposition 3.6 provides us with a constant C = C(r) > 0 such that JL(Br(W))l/q ~ C, (1 - IwI 2 )(",+2)/p . IIk""p,w IIL-(Il) for all W E llJ.

CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES

223

(iv) =} (i): We apply Lemma 4.3.6 of K. Zhu [22]: there exists an integer N such that for sufficiently small r there is a sequence (TJn) in llJ having the following properties:

(1) llJ = U~=l Br(TJn), (2) B r / 4 (TJm) n B r / 4 (TJn) = 0 whenever m -# n, (3) Every Z E llJ is contained in at most N of the sets B 2r (TJn)' Note that limn-+oo ITJnl = 1 follows from (2). Let now (In) be a sequence in BA~' By Montel's theorem, some subsequence (Ink) converges uniformly on compact sets to some I E 1i(llJ). By Fatou's lemma, even I E BA~' Put gk := I - Ink' Vk E N. By our hypothesis there exists, for any given c > 0, an integer ne such that J.L(Br(TJn)) :::; c' (1 - ITJnI 2)q(a+2)/p if n ~ ne' Therefore, with constants depending only on the indicated parameters,

n~, Lr(T/n) Igk(ZW dJ.L(z)

=

n~, L,.(T/,.) (lgk(ZW)q/PdJ.L(z)

< C(r)· n~E Lr(T/n) COa(B:r(TJn))

L2,0(T/n)

r < C(r)· C(r, 0)' "~ (1 _IJ.L(B 2(17n)) )q(a+2)/p 1

n~nE

< C(r)·C(r,o)·c·

TJn

L ({ n~nE J

IgdwWdaa(w)) q/p dJ.L(z)

(1

p

Igk(W)1 daa(w)

)q/P

B 2r (T/n)

Igk (wWda a (w)r/ p

B2r(T/n)

< C(r)· C(r, 0) . c· (

L1

Igk(wWdaa(w) riP

(L

Igk (wWda a (w) ) q/p :::;

n~n.

< C(1')' C(r, 0) . Nq/p . c·

B2r(T/n)

c· c ;

here C = C(r,o,N,p,q). If we choose now ke EN such that Ln
Iv

A slight modification of the argument used to prove (i) =} (ii) shows that compactness of A~ <--+ Lq(J.L) implies compactness of A~, <--+ Lq(J.L) if -1 < 0,0' < 00, 0< p < p' :::; q and (0 + 2)/p = (0' + 2)/p'. It can be shown that (i) {::} (iii) {::} (iv) is true even for arbitrary 0 < q < 00. In the case p > q, we can prove: THEOREM 4.3. Suppose that -1 < 0 < 00 and 0 < q < p < 00. Regardless 01 the (0, p, q) - Carleson measure J.L, the embedding A~ <--+ Lq (J.L) is always compact. For composition operators C