Abstract. In this paper we study conditions on a Banach space X that
Abstract. In this paper we study conditions on a Banach space X that
arXiv:math/9209211v1 [math.FA] 8 Sep 1992AMENABILITY OFBANACH ALGEBRASOFCOMPACT OPERATORSN. Grønbæk,B.
E. Johnson,andG. A. WillisAbstract.
In this paper we study conditions on a Banach space X thatensure that the Banach algebra K(X) of compact operators is amenable. Wegive a symmetrized approximation property of X which is proved to be such acondition.
This property is satisfied by a wide range of Banach spaces includ-ing all the classical spaces. We then investigate which constructions of newBanach spaces from old ones preserve the property of carrying amenable al-gebras of compact operators.
Roughly speaking, dual spaces, predual spacesand certain tensor products do inherit this property and direct sums do not.For direct sums this question is closely related to factorization of linear oper-ators. In the final section we discuss some open questions, in particular, theconverse problem of what properties of X are implied by the amenability ofK(X).0.
IntroductionAmenability is a cohomological property of Banach algebras which wasintroduced in [J]. The definition is given below.
It may be thought of asbeing, in some ways, a weak finiteness condition. For example, amenabilityof C*-algebras is equivalent to nuclearity, see [Haa].
Also, a group algebra,L1(G), is amenable if and only if the locally compact group, G, is amenable,see [J], and many theorems valid for finite or compact groups have weakergeneralizations to amenable groups but to no larger class. This equivalenceis the origin of the term for Banach algebras.
However, in some situationsamenability is not a finiteness condition. For example, a uniform algebra isamenable if and only if it is self-adjoint, see [Sh], and, for finite dimensionalBanach algebras, amenability is equivalent to semisimplicity.The significance of amenability for some classes of Banach algebras sug-gests the question as to what it means for other Banach algebras.
In thisGAW partially supported by SERC grant GR-F-74332.Typeset by AMS-TEX1
2N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISpaper we investigate the amenability of the algebras of compact and of ap-proximable operators on the Banach space X.
This was begun in [J], whereit is shown that K(X) is amenable if X is ℓp, 1 < p < ∞, or C[0, 1]. (K(X)denotes the algebra of compact operators on X and F(X) the algebra ofapproximable operators.) Relevant properties of Banach spaces, such as theapproximation property, are now understood better than they were when [J]was written and so we are able to make more progress.We have not yet found such clear characterizations of amenability for thealgebras of approximable and compact operators as are known for classesof algebras mentioned in the first paragraph.
It does appear though thatamenability of F(X) and K(X) may be equivalent to approximation prop-erties for X. One immediate observation is that, since amenable Banachalgebras have bounded approximate identities, if the algebra of compact op-erators on X is amenable, then, by [D, Theorem 2.6], X has the boundedcompact approximation property and, if the algebra of approximable oper-ators is amenable, then X has the bounded approximation property.
More-over, results in [G&W] and [Sa] show that, if K(X) is amenable, then X∗has the bounded compact approximation property and, if F(X) is amenable,then X∗has the bounded approximation property. It follows that, if F(X)is amenable, then K(X) = F(X).Amenability of F(X) is not equivalent to X or X∗having the boundedapproximation property however, as examples in the paper show.
Some sortof symmetry also seems to be required. In Section 3 we formulate a sym-metrized approximation property, called property (A) , such that, if X hasproperty (A) , then F(X) is amenable.This formulation is an abstractversion of the argument used in [J].
We show that, if X has a shrinking,subsymmetric basis, then it has property (A) and hence F(X) is amenable.Many spaces which do not have such a basis also have property (A) .The necessity of some sort of symmetry becomes apparent when we con-sider the stability of the class of spaces X such that F(X) is amenable.Subject to some restrictions, this class of spaces is closed under tensor prod-ucts and taking duals, as is shown in Sections 2 and 5. However, it is notclosed under direct sums or passing to complemented subspaces, see Section6.
The results in Sections 5 and 6 depend on some new stability propertiesfor amenable Banach algebras which we establish in those sections.Many questions remain to be answered before we understand fully theconnection, if any, between amenability of F(X) or K(X) and approximationproperties of X. These questions are discussed in the last section of the paper.We do not investigate other homological properties of F(X) and K(X).
Oneother such property has been studied in [Ly].We now give the definition of amenability for Banach algebras. It is madein terms of Banach modules and derivations.
Recall that, for a Banach alge-bra A, a Banach space X is a Banach A-bimodule if X is a A-bimodule and
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS3there is a constant K such that ||a.x|| ≤K||a|| ||x|| and ||x.a|| ≤K||a|| ||x||for each a in A and x in X. If X is a Banach A-bimodule, then the dual space,X∗, is a Banach A-bimodule with the actions defined by ⟨a.x∗, x⟩= ⟨x∗, x.a⟩and ⟨x∗.a, x⟩= ⟨x∗, a.x⟩, for a in A, x in X and x∗in X∗.
A derivation into anA-bimodule X is a linear map D : A →X such that D(ab) = a.D(b)+D(a).b,for all a, b in A. If x belongs to X, then the map a 7→a.x−x.a is a derivationinto X.
Such derivations are called inner.Definition 0.1. The Banach algebra A is amenable if, for every BanachA-bimodule X, every continuous derivation D : A →X∗is inner.See [J, Section 5], or [B&D, Definition VI.2].This definition will sometimes be used directly but we will often use an-other characterization of amenability, namely that A is an amenable Banachalgebra if and only if Ab⊗A has an approximate diagonal.
An approximatediagonal is a bounded net, {dλ}λ∈Λ, in Ab⊗A such thatlimλ→∞||a.dλ −dλ.a|| = 0andlimλ→∞||π(dλ)a −a|| = 0,(a ∈A),where π denotes the product map Ab⊗A →A and module actions on Ab⊗Aare defined by a. (b ⊗c) = (ab) ⊗c and (b ⊗c).a = b ⊗(ca), for a, b and c inA.
If we define a product on Ab⊗A by (a ⊗b)(c ⊗d) = ac ⊗db, then the firstof these conditons can also be stated aslimλ→∞∥(a ⊗1 −1 ⊗a)dλ∥= 0(a ∈A),where 1 is a formally adjoined unit. Approximate diagonals are useful, forexample, when we show that, if X has property (A), then F(X) is amenable.1.
NotationWe begin by establishing notation. Throughout, X and Y will denote(infinite dimensional) Banach spaces and X∗the space of bounded linearfunctionals on X with its usual norm.
Small letters x etc. will denote elementsin X, whereas x∗etc.
will denote elements in X∗.We will consider thefollowing classes of operators:F(X, Y ) = {finite rank operators X →Y }N (X, Y ) = {nuclear operators X →Y }F(X, Y ) = uniform closure of F(X, Y )= {approximable operators X →Y }K(X, Y ) = {compact operators X →Y }I(X, Y ) = {integral operators X →Y }B(X, Y ) = {bounded operators X →Y }
4N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISWe shall write F(X) instead of F(X, X) etc.These are all two-sided operator ideals in B(X, Y ),and when X = Y theyare, except F(X), Banach algebras in their natural norms.We refer thereader to any of [D&U], [Pie], [Pis] for details.Finite rank operators will, when convenient, be written as tensors, thatis, if x∗1, .
. ., x∗n ∈X∗and y1, .
. ., yn ∈Y , we shall denote the operatorx →P x∗i (x)yi by P yi ⊗x∗i .If S ∈B(X, Y ) we denote the adjoint map in B(Y ∗, X∗) by Sa, i.e.⟨S(x), y∗⟩= ⟨x, Sa(y∗)⟩(x ∈X, y∗∈Y ∗)If M ⊆B(X, Y ) we define M a ⊆B(Y ∗, X∗) byM a = {T a|T ∈M}This should not be confused with the notation for dual space.
For instance,if X has Grothendieck’s approximation property, then N (X)∗= B(X∗),whereas N (X)a is the set of so-called X-nuclear operators on X∗.We shall use the concepts left approximate identity, bounded left approx-imate identity etc. in accordance with [B&D].2.
Tensor productsIt is of course important to be able to form new Banach spaces from oldones while preserving the property of carrying amenable algebras of compactoperators. The first case to be considered is that of taking tensor productsbecause many important spaces can be viewed as appropriate tensor prod-ucts, for instance Lp-spaces with values in a Banach space.
We shall hereinvestigate whether amenability of K(X) and K(Y ) implies amenability ofK(Z), when Z is the completion of X ⊗Y in some crossnorm topology. Anobvious approach to this problem is to try to show that K(X) ⊗K(Y ) is adense subalgebra of K(Z) and then to deduce amenability of K(Z) from thatof K(X)b⊗K(Y ) by an appeal to [J, Corollary 5.5].
This program is consid-erably easier to carry through if X and Y have the approximation property.However, rather than making this assumption, we prefer to work with ap-proximable operators instead of compact operators. The definition to followdescribes what is needed for above mentioned program to work.Definition 2.1.
Let X and Y be Banach spaces and let α be a crossnormon X ⊗Y . Denote the completion by X ⊗α Y .
We call X ⊗α Y a tight tensorproduct of X and Y , if the following two conditions hold. (i) There is K > 0 so that for all S ∈F(X), T ∈F(Y ) the operator onX ⊗Y given by(S ⊗T)x ⊗y = Sx ⊗Ty(x ∈X, y ∈Y )has α- operator norm not exceeding K∥S∥∥T∥.
(ii) span{S ⊗T | S ∈F(X), T ∈F(Y )} is dense in F(X ⊗α Y )
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS5Remark: The condition (i) is apparently weaker than Grothendieck’s ⊗-norm condition [Gr,Ch.1.3] in that it only concerns finite rank operators ona tensor product into itself.With this definition we have the obvious:Theorem 2.2. Suppose that F(X) and F(Y ) are amenable and that X⊗αYis a tight tensor product.
Then F(X ⊗α Y ) is amenable.Proof. [J,Corollary 5.5]To apply this theorem we need to be able to recognize tight tensor prod-ucts.The following easy proposition is helpful.It shows that, as usualwhen dealing with tensor products, it is important to be able to identify(X ⊗α Y )∗.
We shall view (X ⊗α Y )∗as a subspace of B(Y, X∗) (or equiv-alently of B(X, Y ∗)). We give B(Y, X∗) the canonical structure as a rightBanach module over F(X) and F(Y ), that is, the module actions are therestrictions of the canonical actions of B(X) and B(Y ).Proposition 2.3.
Let X and Y be Banach spaces and let α be a crossnormon X ⊗Y . Then X ⊗α Y is a tight tensor product if and only if the followingtwo conditions hold:(i) (X⊗αY )∗is a right Banach F(X)- and F(Y )- submodule of B(Y, X∗)(ii) X∗⊗Y ∗is norm dense in (X ⊗α Y )∗.Proof.
(i): Let z = P xi ⊗yi ∈X ⊗Y , let Φ ∈(X ⊗α Y )∗, and let S ∈F(X), T ∈F(Y ). Then⟨S ⊗T z, Φ⟩=X⟨Sxi ⊗Tyi, Φ⟩=X⟨Sxi, ΦTyi⟩= ⟨z, SaΦT⟩,so that S ⊗T is α-bounded with ∥S ⊗T∥α ≤K∥S∥∥T∥if and only if theBanach module properties hold with module constants KXKY ≤K.
(ii): We shall use the identification F(Z) = Z∨⊗Z∗.The canonical mapF(X) ⊗F(Y ) →F(X ⊗α Y ) then becomes(x∨⊗x∗) ⊗(y∨⊗y∗) →(x ⊗α y)∨⊗(x∗⊗y∗).Using the injective property of∨⊗, it is now clear that the image of F(X)⊗F(Y ) is dense if and only if X∗⊗Y ∗is dense in (X ⊗α Y )∗.Recall that a crossnorm is called reasonable if the dual norm is also acrossnorm. In this case tightness is particularly easy to describe.
6N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISCorollary 2.4.
Suppose that α is a reasonable crossnorm on X ⊗Y andthat the module property 2.3. (i) holds.
Then X ⊗α Y is tight if and only if(X ⊗α Y )∗= X∗⊗α∗Y ∗,where α∗denotes the dual norm.With Proposition 2.3 at hand we can now give conditions for tightness forsome important tensor products. From 2.3.
(ii) it is not surprising that theRadon-Nikodym property enters the picture.Theorem 2.5. Let X and Y be Banach spaces, let [0, 1] be the unit interval,and let (Ω, Σ, µ) be a σ-finite measure space.
Then(W) The following are equivalent:(i) X∨⊗Y is tight for all X. (ii) C([0, 1], Y ) is a tight tensor product of C[0, 1] and Y(iii) Y ∗has RNP.
(P) X b⊗Y is tight if and only if F(Y, X∗) = B(Y, X∗). (M) Lp(µ, X), 1 ≤p < ∞is a tight tensor product of Lp(µ) and X if andonly if X∗has RNP with respect to µ.Proof.
The identification of (X ⊗α Y )∗with a subspace of B(Y, X∗) givesin the cases (W) and (P) I(Y, X∗) and B(Y, X∗) respectively, so the mod-ule property 2.3. (i) is obvious for these tensor products.Next, let S ∈F(Lp(µ)) and T ∈F(X).
From the proof of Proposition 2.3. (i) it followsthat it is enough to show the submultiplicativity of the module norm forΦ belonging to a norm determining subset of Lp(µ, X)∗.
Let 1p + 1p′ = 1.Since Lp′(µ, X∗) is isometrically embedded in Lp(µ, X)∗and since Lp(µ, X)is isometrically embedded in Lp′(µ, X∗)∗it suffices look at Φ ∈B(Lp(µ), X∗)coming from an element g ∈Lp′(µ, X∗).With the identifications being madewe haveΦ(f) =ZΩfg dµ(f ∈Lp(µ)).Then for S ∈B(Lp(µ)) and T ∈B(X)T aΦS(f) =ZΩS(f)T a ◦g dµ.An appeal to the vector valued version of H¨olders inequality, gives the desirednorm inequality.We now consider the statement 2.3. (ii) in our three cases.
First we lookat (W). Since (X∨⊗Y )∗= I(Y, X∗) we are asking whether the finite rank
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS7operators X∗⊗Y ∗are dense in I(Y, X∗) in the integral norm. The impli-cation (i) ⇒(ii) is obvious and (iii) ⇒(i) is valid because, if Y ∗has RNP,then I(Y, X∗) = N (Y, X∗) isometrically, [D&U,Theorem VI.4.8, CorollaryVIII.2.10].
The implication (ii) ⇒(iii) is true because, under the assumption(ii), F(C[0, 1], Y ∗) is dense in I(C[0, 1], Y ∗). (We are here using the symmet-ric rˆoles of C[0, 1] and Y .) By [D&U,Theorem VI.3.12, Corollary VIII.2.10]every absolutely summing operator C[0, 1] →Y ∗is nuclear since, by Lemma2.8 below, I(C[0, 1], Y ∗) = N (C[0, 1], Y ∗), again using the symmetric rˆolesof C[0, 1] and Y and identifying C[0, 1]∗with M[0, 1], the Banach space ofRadon measures on the unit interval with the total variation norm.. TheRNP of Y ∗is now the content of [D&U, Corollary VI.4.6].In the case (P) we just have to observe that (X b⊗Y )∗= B(Y, X∗) andcl(X∗⊗Y ∗) = F(Y, X∗).Finally, as already noticed, Lp′(µ, X∗) is isometrically isomorphic to asubspace of Lp(µ, X)∗.
As a consequence Lp(µ, X) is tight if and only ifLp(µ, X)∗= Lp′(µ, X∗).But this is equivalent to X∗having RNP withrespect to µ, [D&U,Theorem IV.1.1].¿From a classical theorem by Pitt [Pit] we get an immediate consequenceof Theorem 2.5. (P).Corollary 2.6. ℓp b⊗ℓq is tight if and only if 1p + 1q < 1.Corollary 2.7.
If X∗has RNP and F(X∗) is amenable, then F(F(X)) isamenable.Proof. By Corollary 5.3 below, amenability of F(X∗) forces amenability ofF(X).
The identification F(X) = X∨⊗X∗shows that F(X) is a tight tensorproduct of X and X∗.We have not been able to find the technical observation needed above inthe literature.Lemma 2.8. Let M(K) be the Banach space of Radon measures on a com-pact Hausdorffspace with the total variation norm and let Φ : Y →M(K) bea finite rank operator.
Then the integral and nuclear norms of Φ coincide.Proof. Since M(K) is a L1,1+ε-space (cf.
[ L&P, Definition 3.1 ]) for allε > 0, there is a finite dimensional subspace with rg Φ ⊆V and a projectionP : M(K) →V with ∥P∥≤1 + ε. Since V is finite dimensional we haveN (Y, V ) = I(Y, V ) isometrically.
If ι : V →M(K) is the inclusion map weget∥Φ∥nucl = ∥ιPΦ∥nucl≤∥PΦ∥nucl= ∥PΦ∥int≤(1 + ε)∥Φ∥int,
8N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISso that ∥Φ∥nucl ≤∥Φ∥int.
The reverse inequality is always true.3. Diagonals for Mn(C)For many Banach spaces X, in particular the classical spaces, it is possibleto prove that K(X) is amenable as a consequence of a uniform local structureof X, that is, as a consequence of a property of finite dimensional subspaces.Before we set the scenario in which this approach will work we shall take acloser look at finite dimensional spaces.
It is well known and easy to provethat Mn(C) is amenable. In this section we shall view this in terms of faithfulirreducible representations of finite groups.
However, rather than speakingabout faithful representations we shall consider finite subgroups of Gln(C).Likewise, we shall express irreducibility as a property of the embedding ofthe group into Mn(C).Lemma 3.1. Let D : G →Gln(C) be an n-dimensional representation of agroup G. Then D is irreducible if and only if span D(G) = Mn(C).Proof.
We extend the representation to the group algebra CG.Then thelemma is an easy consequence of Jacobson’s density theorem, [B&D, Theorem24.10].Henceforth we shall deal with finite subgroups of Gln(C) spanning thewhole of Mn(C). These we shall call irreducible (n × n)-matrix groups.
Theconnection of such with amenability of Mn(C) is described in the followingproposition. The symbols eij denote as usual the matrix units.Proposition 3.2.
Let G be a finite irreducible (n × n)-matrix group. Then1|G|Pg∈G g⊗g−1 is equal to the canonical diagonal d0 = 1nPni,j=1 eij ⊗eji forMn(C).
The canonical diagonal d0 is the only element of Mn(C) ⊗Mn(C)which is simultaneously a diagonal for Mn(C) and for the opposite algebraMn(C)op.Proof. Let d =1|G|Pg∈G g ⊗g−1.
That d is a diagonal for Mn(C) means(3.1)Xg∈Gag ⊗g−1 =Xg∈Gg ⊗g−1a(a ∈Mn(C))and(3.2)π(d) = I .Likewise, d being a diagonal for Mn(C)op means(3.3)Xg∈Gga ⊗g−1 =Xg∈Gg ⊗ag−1(a ∈Mn(C))
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS9and(3.4)πop(d) = I,where πop is the opposite multiplication πop(a ⊗b) = ba.Since span G = Mn(C) it is enough to consider a ∈G and then exploitlinearity. We prove (3.3):Xg∈Gga ⊗g−1 =Xg∈Gga ⊗a(ga)−1=Xu∈Gau ⊗au−1Since Ga = G, (3.3) follows.
The identity (3.1) is proved similarly, and(3.2) and (3.4) are obvious.Simple computations with matrix units show that d0 satisfies all of (3.1),. .
. ,(3.4).
Now let d = Pi ai ⊗bi be any element satisfying (3.2) and (3.3)and write d0 = Pj a′j ⊗b′j. Thend =Xiai ⊗bi =Xi,jai ⊗b′ja′jbi=Xi,jaia′j ⊗b′jbi=Xi,jaibia′j ⊗b′j=Xja′j ⊗b′j = d0,finishing the proof.
Note that (3.1) and (3.4) follows automatically from (3.2)and (3.3), since d0 satisfies (3.1) and (3.4). (The use of the average1|G|Pg∈G g ⊗g−1 probably dates back to the earlydays of representation theory.It is a refinement of this which gives theequivalence of amenability of group algebras and the existence of invariantmeans, [J, Theorem 2.5])Example 3.3.
We shall several times in the sequel use irreducible matrixgroups of the following kind.Let H be a group of (n × n) permutationmatrices corresponding to a transitive subgroup of the symmetric group Sn.ThenG = {D(t)σ | t ∈{±1}n,σ ∈H}is an irreducible (n × n)-matrix group.If H = Sn, then G is called themonomial group of degree n.
10N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLIS4.
Amenability as a consequence of an approximation propertyIn this section we shall develop a method to lift uniformly the diagonalsof a matrix algebra to form an approximate diagonal for F(X). The ideais illustrated by the example X = Lp(µ).
Locally Lp(µ) looks like ℓnp so wehave ‘local’ diagonals. Furthermore, these diagonals are uniformly bounded(by 1).
Using a direct limit argument we can form an approximate diagonalfor all of F(Lp(µ)).This approach will work for all the classical spaces. The definition below iscustomised to make it work in a rather general situation.
To formulate it letus first look at a finite biorthogonal system {(xi, x∗j) | xi ∈X; x∗j ∈X∗; i, j =1, . .
., n}. Using this system we may define a map E : Mn(C) →F(X) byE((aij)) =Xi,jaij xi ⊗x∗j.By biorthogonality, E is an algebra homomorphism.Definition 4.1.
Let X be a Banach space. We say that X has property (A)if there is a net of finite biorthogonal systems{(xi,λ, x∗j,λ) | xi,λ ∈X; x∗j,λ ∈X∗; i, j = 1, .
. ., nλ}(λ ∈Λ)and corresponding mapsEλ : Mnλ(C) →F(X)(λ ∈Λ)such that with Pλ = Eλ(Inλ) the following holdA(i) Pλ −→1X stronglyA(ii) P aλ −→1X∗stronglyA(iii) For each λ there is an irreducible (nλ × nλ)-matrix group Gλ suchthatsup{∥Eλ(g)∥op | g ∈Gλ, λ ∈Λ} < ∞.We now show how to lift the diagonals of the matrix algebras to F(X).Theorem 4.2.
Suppose X has property (A). Then F(X) is amenable.Proof.
With notation as in the description of property (A), define the net(dλ)λ∈Λ in F(X)b⊗F(X) bydλ =1|Gλ|Xg∈GλEλ(g)b⊗Eλ(g−1)(λ ∈Λ).By assumption this is a bounded net. Observing that π(dλ) = Pλ, we con-clude by A(i) that (π(dλ))λ∈Λ is a bounded left approximate identity forF(X).
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS11Let F ∈F(X). ThenF.dλ −dλ.F = (F −PλFPλ).dλ −dλ.
(F −PλFPλ)+ PλFPλ.dλ −dλ.PλFPλ= (F −PλFPλ).dλ −dλ. (F −PλFPλ),since1|Gλ|Pg∈Gλ g⊗g−1 is a diagonal for Mnλ(C).
By A(ii) (Pλ) is a boundedright approximate identity for F(X), so that F.dλ −dλ.F −→0Remark 4.2.a. The condition of biorthogonality in property (A) is strongerthan necessary.
The following asymptotic trace condition suffices to establishamenability:1nλnλXi⟨xi,λ, x∗i,λ⟩−→1along Λ.With this condition replacing biorthogonality all statements inthis section about property (A) remain valid. We have made no use of thisgreater generality and so do not give the details here.
However, if X hasproperty (A), then A(i) implies that X is a π-space and so probably thereare spaces which satisfy the weaker condition but not property (A). (Notethat apparently there are spaces with the bounded approximation propertywhich are not π-spaces, see the introduction to [C&K].
)Remark 4.2.b Conditions A(i) and A(ii) together imply that X is whatmight be called a “shrinking πλ-space”, compare with the discussion in[G&W]. Thus, if X has a basis and Pn is the projection onto the span ofthe first n basis elements, then (Pn) satisfying A(i) and A(ii) implies thatthe basis is a shrinking basis.
If, furthermore, (Pn) satisfies A(iii) with themonomial group of degree n, then the basis is a symmetric basis, see [L&T,Ch. 3a].
In this case X will have property (A), see also Theorem 4.5 below.Property (A) is preserved for some natural Banach spaces formed from theoriginal space, as set forth in the next two theorems. This will enable us toestablish amenability of F(X) for a large class of Banach spaces, includingall the classical spaces.Theorem 4.3.
Let X be a Banach space. If X∗has property (A), then Xhas property (A).Proof.
Let{(x∗i,λ, x∗∗i,λ)}(λ ∈Λ)be a net of biorthogonal systems satisfying the conditions of (A) with respectto X∗. Let U and V be the sets of all finite dimensional subspaces of X andX∗respectively, and let U ∈U and V ∈V be given.By means of theprinciple of local reflexivity [L&T], choose a linear mapSU,V,λ : span ({x∗∗i,λ | i = 1, .
. ., nλ} ∪U) 7−→X
12N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISsuch that(1) ∥SU,V,λ∥≤2.
(2) SU,V,λ|U = 1U. (3) ⟨SU,V,λx∗∗i,λ, x∗⟩= ⟨x∗, x∗∗i,λ⟩for all x∗∈span({x∗i,λ} ∪V ).We order U and V by containment and U × V × Λ by the product order.By construction{(SU,V,λx∗∗i,λ, x∗j,λ) | i, j = 1, .
. ., nλ}((U, V, λ) ∈U × V × Λ)is a net of finite biorthogonal systems.
We denote the corresponding liftsof matrix algebras by EU,V,λ and the corresponding projections by PU,V,λ.ThenPU,V,λ = SU,V,λP aλιX ,where ιX is the canonical inclusion of X into X∗∗and Pλ’s are the property(A) projections for X∗. Clearly {PU,V,λ} is a bounded set and for x ∈U∥PU,V,λx −x∥= ∥SU,V,λP aλx) −x∥= ∥SU,V,λP aλx −x∥≤2∥P aλx −x∥,where the two last steps follow from (1) and (2) above.
Hence A(i) is satisfied.Similarly for x∗∈VP aU,V,λ(x∗) =Xx∗i,λ ⊗SU,V,λ(x∗∗i,λ(x∗)=X⟨SU,V,λx∗∗i,λ, x∗⟩x∗i,λ=X⟨x∗, x∗∗i,λ⟩x∗i,λ = P aλx∗,using (3), so that A(ii) is satisfied. The supremum in A(iii) is increased byat most a factor 2, using the same irreducible matrix groups: GU,V,λ = Gλ.We have thus found a net of finite biorthogonal systems which satisfiesthe conditions needed for property (A).Property (A) also behaves nicely with respect to tensor products:Theorem 4.4.
Let X and Y be Banach spaces and let Z be a tight tensorproduct of X and Y . If X and Y have property (A), then so does Z.Proof.
We write Z = X ⊗α Y . Let (Oλ)λ∈Λ and (Rµ)µ∈M be property (A)nets of biorthogonal systems for X and Y respectively.
We define the tensorproduct (Oλ ⊗Rµ)(λ,µ)∈Λ×M to be the product ordered net of biorthogonalsystems for X ⊗α Y given asOλ ⊗Rµ = {(x ⊗y, x∗⊗y∗) | (x, x∗) ∈Oλ, (y, y∗) ∈Rµ}.
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS13Using the identification Mn(C)⊗Mp(C) = Mnp(C), one checks easily thatthe property (A) liftsE(λ,µ) : Mnλnµ(C) →F(Z)are nothing but E(λ,µ) = Cλ ⊗Dµ, where Cλ and Dµ are the lifts belongingto X and Y respectively.Hence A(i) holds for Z and, since X∗⊗Y ∗isdense in Z∗, we also have A(ii). To obtain A(iii) it suffices to notice that, ifG and H are irreducible (m × m)- and (n × n)- matrix groups, then G ⊗His an irreducible (mn × mn)- matrix group.We shall now give some concrete examples of spaces with property (A).The first is very much in the spirit of [J, Proposition 6.1].
Recall that a basis(xn)n∈N in a Banach space X is called subsymmetric if (xn)n∈N is uncon-ditional and equivalent to the basis sequence (xni)i∈N for every increasingsequence (ni)i∈N, see [L&T, Ch. 3.a] and [Si, Ch.
21].Theorem 4.5. Suppose that X has a subsymmetric and shrinking basis.Then X has property (A).Proof.
Let (xn)n∈N be a subsymmetric and shrinking basis and let (x∗n)n∈Nbe the associated sequence of coordinate functionals. Then{(xi, x∗j) | i, j = 1, .
. ., n}(n ∈N)is a sequence of finite biorthogonal systems satisfying the conditions of prop-erty (A).
A(i) is immediate, A(ii) follows from the basis being shrinking. Toprove A(iii) we shall use the following observations.Since (xn)n∈N is unconditional, the family of operators of the form(4.1)U ∞Xn∈Nanxn=Xn∈ηs(n)anxn,where η ⊆N and s ∈{±1}N, is uniformly bounded, say by K > 0.
Thesubsymmetry means that the family of operators of the form(4.2)A(mi)(ni)(x) =∞Xi=1x∗mi(x)xniis uniformly bounded, say by M > 0.Here (mi)i∈N and (ni)i∈N are twoarbitrary increasing sequences of integers.Let Gn be the subgroup of the monomial group of degree n defined by thepermutation matrix σ corresponding to the cyclic permutation (12 · · ·n), i.e.Gn = {D(t)σk | t ∈{±1}n, k = 0, . .
., n −1},
14N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISBy Lemma 3.1 Gn is an irreducible (n × n)-matrix group.
We write elementsin X as sequences. Then for g = D(t)σk we have:E(g)(ξ1, ξ2, .
. ., ξn, .
. .) =(t(1)ξn+1−k, .
. ., t(k)ξn, t(k + 1)ξ1, .
. ., t(n)ξn−k, 0, .
. .
)We see that E(g) has the form E(g) = A1U1 + A2U2 for appropriate choicesof operators Ui of type (4.1) and Ai of type (4.2). Hence the supremum inA(iii) does not exceed 2KM.Corollary 4.6.
Let X be a reflexive Orlicz sequence space or a reflexiveLorentz sequence space. Then X has property (A) and so F(X) is amenable.Proof.
See [L&T, Ch.3.a] for a discussion showing that these spaces satisfythe hypotheses of Theorem 4.5.We shall now give substance to the remark that the setup of property (A)is customized to deal with the classical spaces.Theorem 4.7. Let K be a compact Hausdorffspace and let (Ω, Σ, µ) be ameasure space.
Then C(K) and Lp(µ) , 1 ≤p ≤∞, have property (A).Proof. Since C(K)∗= L1(µK) for a suitable measure space (ΩK, ΣK, µK)and L∞(µ) = C(Kµ) for a suitable compact space Kµ, it follows from The-orem 4.3 that it is enough to consider Lp(µ) for 1 ≤p < ∞.
We shall givethe proof in detail in the case of a probability space, cf. the remark below.Let S be a finite collection of disjoint measurable subsets of Ωwhose unionis all of Ω.
As it is customary in integration theory we order such dissectionsby S1 ≺S2 if every set in S1 is a union of sets from S2. We define thebiorthogonal systems in Lp(µ) × Lp′(µ) 1p + 1p′ = 1 byOS = {(1µ(L) 1pχL,1µ(M) 1p′χM) | L, M ∈S},where χ•’s denote indicator functions.
It is now a routine matter to verifyproperty (A).Let PS be the property (A) projections.For an indicatorfunction χM we havePS(χM) = χMP aS(χM) = χMwhenever {M} ≺S, so A(i) and A(ii) are immediate. To prove A(iii), con-sider S = {M1, .
. ., Mn} and define GS to be the monomial group of degreen.
Let g = D(t)σ where t ∈{±1}n and σ is a permutation matrix. Firstnotice that∥nXi=1ai1µ(Mi) 1pχMi∥p =nXi=1|ai|p.
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS15Using this we get for an arbitrary f ∈Lp(µ)∥ES(g)f∥p = ∥nXi=1t(i)1µ(Mi) 1p′ ZMif dµ1µ(Mσ(i)) 1pχMσ(i)∥p=nXi=11µ(Mi) pp′ |ZMif dµ|p≤nXi=11µ(Mi) pp′µ(Mi)pp′ZMi|f|p dµ(H¨older Inequality)=nXi=1ZMi|f|p dµ≤∥f∥p.Remark 4.7.a. A proof of the general case can be given along the samelines but with added minor technicalities.
Alternatively, we may reduce itto the special case. We are interested only in finite-dimensional subspaces.Functions in such a subspace are supported on a σ-finite measure space.
Thecorresponding complemented subspaces of Lp(µ) have projection constantsuniformly bounded by 1 and are isometrically isomorphic to Lp-spaces ofprobability measures.Combining this with Theorem 2.5 and Theorem 4.4 we get a large collec-tion of Banach spaces carrying amenable algebras.Corollary 4.8. Suppose that X has property (A).
If X∗has RNP, thenC(K, X) has property (A). If X∗has RNP with respect to µ, then Lp(µ, X)has property (A) for 1 ≤p ≤∞.Corollary 4.9.
F(ℓp b⊗ℓq) is amenable if and only if 1p + 1q < 1.Proof. By Corollary 2.6 F(ℓp b⊗ℓq) is amenable for 1p + 1q < 1.
In [A&F] itis shown that, if r ≤s, then B(ℓr, ℓs) = (ℓr b⊗ℓs′)∗contains a complementedcopy of B(ℓ2) and thus fails the approximation property, [Sz]. Hence, when1p + 1q ≥1, then F(ℓp b⊗ℓq) does not have a bounded right approximate identityand is consequently not amenable.Probably it is too much to hope that amenability of F(X) is equivalentto X having property (A).
Since the approximate diagonal stemming fromproperty (A) is obtained by means of lifts of the canonical diagonals of matrixalgebras, it will have the approximate versions of the extra properties (3.3)and (3.4). It seems unlikely that such approximate diagonals should alwaysexist, once amenability of F(X) is established.In Section 6 we will see
16N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISexamples of spaces for which F(X) is amenable but for which we do notknow whether X has property (A) or even the weaker property mentionedin Remark 4.2.a.5.
Dual spacesWe have seen that property (A) passes from a dual Banach space to itspredual. The following stability property for amenability implies an extensionof this fact, namely, that if the algebra of approximable operators on a dualspace is amenable, then the algebra of approximable operators on any predualof the space is amenable.
It also implies a similar, but weaker, result for thealgebra of compact operators.Theorem 5.1. Let A be an amenable Banach algebra and I be a closed, leftideal in A which has a bounded two-sided approximate identity.
Then I isamenable.Proof. It is convenient to define a new product on Ab⊗A by (a ⊗b) • (c ⊗d) = ac ⊗bd, that is, • is the usual product on Ab⊗A.
Let (dα)α∈A be anapproximate diagonal for A, let (eβ)β∈B be a bounded two-sided approximateidentity for I, and putpαβγ = dα • (eβ ⊗eγ)(α ∈A; β, γ ∈B).Then, since I is a left ideal and (dα)α∈A and (eβ)β∈B are bounded nets, pαβγbelongs to a bounded subset of I b⊗I.For each c in I we havelim supγ∥c.pαβγ −pαβγ.c∥=lim supγ∥(c ⊗1) • dα • (eβ ⊗eγ) −dα • (eβ ⊗eγ) • (1 ⊗c)∥=lim supγ∥((c ⊗1) • dα −dα • (1 ⊗c)) • (eβ ⊗eγ)∥=lim supγ∥(c.dα −dα.c) • (eβ ⊗eγ)∥,using limγ(eγc −ceγ) = 0Since (dα)α∈A is an approximate diagonal and (eβ)β∈B is bounded we getfrom the inequality||(cdα −dαc) • eβ ⊗eγ|| ≤||(cdα −dαc)|| ||eβ|| ||eγ||thatlimα lim supβlim supγ||(cdα −dαc) • eβ ⊗eγ|| = 0 ,
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS17and solimα lim supβlim supγ∥c.pαβγ −pαβγ.c∥= 0.Furthermore,limα limβ limγ π(pαβγ)c = limα limβ limγ π(dα • (eβ ⊗eγ))c= limα limβ π(dα • (eβ ⊗1))c= limα π(dα)c= c,where the second and third equality follow from (eβ)β∈B being a left ap-proximate identity for I and I being a left ideal, and the last from (dα)α∈Abeing an approximate diagonal. It follows that we may choose a net from{pαβγ | α ∈A; β, γ ∈B} which is an approximate diagonal for I. ThereforeI is amenable.This theorem is an improvement on the last assertion in Proposition 5.1in [J].
It may also be shown, by a similar argument but with dα(eβ ⊗eγ) inplace of dα • (eβ ⊗eγ), that, if A is an amenable Banach algebra and I isa two-sided ideal in A with a bounded left approximate identity, then I isamenable.Corollary 5.2. Let X be a Banach space such that K(X∗) is amenableand K(X) has a bounded two-sided approximate identity.Then K(X) isamenable.Proof.
K(X)a, which is anti-isomorphic to K(X), is a closed left ideal inK(X∗) and has a bounded two-sided approximate identity.Example 4.3 in [G&W] provides a Banach space, X, such that K(X∗) hasa bounded two-sided approximate identity but K(X) does not. This examplesuggests that the hypothesis that K(X) has a bounded two-sided approximateidentity is necessary.
However, if X has the approximation property it is not.Corollary 5.3. Let X be a Banach space such that F(X∗) is amenable.Then F(X) is amenable.Proof.
Since F(X∗) has a bounded left approximate identity, X∗has thebounded approximation property, by [D, Theorem 2.6]. Hence, by [G&W,Theorem 3.3], F(X) has a bounded two-sided approximate identity.It is an open question, which is discussed further in Section 7, whetheramenability of K(X) implies that X has the approximation property.The converse to Corollary 5.2 holds if K(X∗) has a bounded two-sidedapproximate identity.
This fact will follow from another stability propertyof amenability.
18N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISTheorem 5.4.
Let A be a Banach algebra which has a bounded two-sidedapproximate identity and let I be a closed, left ideal in A which is amenableand has a bounded left approximate identity for A. Then A is amenable.Proof.
By Proposition 1.8 in [J], it will suffice to check that all derivationsfrom A into duals of essential A-bimodules are inner. (An A-bimodule Y isessential if Y = span{a.y.b : a, b ∈A; y ∈Y }, because, with the hypothesisof a bounded approximate identity, this last space is closed.
)Let D : A →Y ∗be a derivation, where Y is an essential A-bimodule.Since I is amenable, there is y∗in Y ∗such that Da = a.y∗−y∗.a for everya in I. Then the map, δ : A →Y ∗, defined by δa = a.y∗−y∗.a is an innerderivation from A and so D −δ is a derivation from A whose restriction toI is zero.Now let a and b belong to A and let (eλ)λ∈Λ be a bounded net in I whichis a left approximate identity for A.
Then, since I is a left ideal,0 = limλD −δ(aeλ).b= limλD −δ(a).eλb=D −δ(a).b,where the two first identities are true because D −δ is a derivation whichannihilates I, and the third because (eλ)λ∈Λ is a left approximate identity.It follows that ⟨b.y,D −δ(a)⟩= 0 for every y in Y and a and b in A.Since Y is essential, D = δ and is thus inner.The next result may now be proved in a similar way to Corollary 5.2.Corollary 5.5. Let X be a Banach space such that K(X) is amenableand K(X∗) has a bounded two-sided approximate identity.
Then K(X∗) isamenable.The argument of Proposition 6.1 in [J] shows, without change, that K(c0)is amenable. It follows from this corollary and the fact that K(ℓ1) has abounded two-sided approximate identity that K(ℓ1) is amenable.
Proposition6.1 in [J] does not yield this fact about ℓ1 directly, although we have shownit in Section 2 by modifying the argument in [J] suitably.Example 5.6.The requirement in Corollary 5.5 that K(X∗) have abounded two-sided approximate identity is necessary. Since ℓ2 is reflexiveit has the RNP.
Hence, by Theorem 2.5, ℓ2∨⊗ℓ2 is a tight tensor product andso, by Theorem 2.2, F(ℓ2∨⊗ℓ2) is amenable. Now (ℓ2∨⊗ℓ2)∗is isomorphicto ℓ2 b⊗ℓ2 and (ℓ2∨⊗ℓ2)∗∗to B(ℓ2).
Since F(ℓ2∨⊗ℓ2) has a bounded two-sidedapproximate identity, F(ℓ2b⊗ℓ2) has a bounded left approximate identity, see
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS19[G&W, Theorem 3.3]. However, B(H) does not have the approximation prop-erty (see [Sz]) and so F(ℓ2b⊗ℓ2) does not have a bounded right approximateidentity.
Therefore, F((ℓ2∨⊗ℓ2)∗) is not amenable.6. Direct sumsIn the following it is necessary to use the algebra of double multiplierson a Banach algebra A.
A double multiplier on A is a pair of boundedoperators, (L, R), on A which commute and satisfy, for all a and b in A :L(ab) = L(a)b; R(ab) = aR(b); and aL(b) = R(a)b. Denote the set of alldouble multipliers on A by M(A). Then M(A) is a Banach space with theobvious norm and sum and becomes a Banach algebra when equipped withthe product (L1, R1)(L2, R2) = (L1L2, R2R1).
If T = (L, R) is a doublemultiplier on A, then L(a) will be denoted by Ta and R(a) by aT.Each element, a, of A determines a double multiplier, (La, Ra), where Laand Ra are respectively the operators on A of left and right multiplication bya. Similarly, if A is embedded as an ideal in a Banach algebra B, then eachelement of B determines a double multiplier on A.
Thus each operator onthe Banach space X determines a double multiplier on K(X) and on F(X).Note also that there is always an identity, I, in M(A).Now let P1 be an idempotent in M(A) and put P2 = I −P1 and Aij =PiAPj, i, j = 1, 2. Next put A◦11 = π(A12b⊗A21) and A◦22 = π(A21b⊗A12),where π denotes the product in A.
Then A◦ii is isomorphic, as a linear space,to the quotient of Aij b⊗Aji, j ̸= i, by ker(π) ∩(Aij b⊗Aji). Let ∥.∥◦denotethe quotient norm on A◦ii.In this section we prove a couple of abstract results about the stabilityof amenability when A is cut down to A11 by an idempotent in M(A) andthen apply them to the case where A = K(X) for some Banach space X andP1 is determined by a projection on X.
We will thus establish some stabilityproperties of amenability of K(X) under direct sums of Banach spaces.Proposition 6.1. Let A and Aij, i, j = 1, 2, be as above.Then A hasa bounded two-sided approximate identity if and only if A11 and A22 havebounded two-sided approximate identities and Aij is an essential left Aii- andright Ajj-module, i, j = 1, 2.Proof.
Let {eλ}λ∈Λ be a bounded net in A. Then {P1eλP1 + P2eλP2}λ∈Λ isa two-sided approximate identity if and only if {PieλPi}λ∈Λ is a two-sidedapproximate identity in Aii, a left approximate identity for Aij and a rightapproximate identity for Aji, i = 1, 2; j ̸= i.The first of the abstract results is the followingTheorem 6.2.
Let A and Aij, i, j = 1, 2, be as above and suppose that Ahas a bounded two-sided approximate identity and that A22 = A◦22. Then Ais amenable if and only if A11 is amenable.
20N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISProof.
The inclusion map A◦22 →A22 is continuous and is also a surjection.Hence, by the open mapping theorem, ∥.∥◦is equivalent to the given normon A22. Furthermore, since A has a bounded two-sided approximate identity,Proposition 6.1 shows that A22 also has a bounded two-sided approximateidentity.
Therefore there is a bounded net {cβ}β∈B in A21 b⊗A12 such that{π(cβ)}β∈B is a bounded approximate identity for A22. The elements of thisnet have the form cβ = Pi rβi ⊗sβi .Now suppose that A11 is amenable and let {dα11}α∈A be an approximatediagonal for A11.
We will show that A is amenable by showing that it hasan approximate diagonal consisting of elements of the formdα,β = dα11 + cβdα11.Here we have equipped Ab⊗A with the product (a ⊗b)(c ⊗d) = ac ⊗dbas described in the introduction. Note first of all that the set of all suchelements is bounded because ∥dα,β∥≤∥dα11∥(1 + ∥cβ∥).
In order to provethat an approximate diagonal can be constructed, we shall use the following(6.1)limα [(π(c) ⊗1)(a21 ⊗1) −(1 ⊗a21)c]dα11 =limα [(a12 ⊗1)c −(1 ⊗π(c))(1 ⊗a12)]dα11 = 0,for each c ∈A21 b⊗A12 and aij ∈Aij. It is enough to prove (6.1) for c anelementary tensor b21⊗b12.
Then the first expression equals (b21⊗1)(b12a21⊗1 −1 ⊗b12a21)dα11, which tends to zero, because (dα11) is an approximatediagonal for A11. The other limit is obtained analogously.We will show that(6.2)limβ limα π(dα,β)a = a(a ∈A),and(6.3)limβlim supα∥(a ⊗1 −1 ⊗a)dα,β∥= 0(a ∈A).This will imply that an approximate diagonal can be constructed from thedα,β’s.First we prove (6.2).
If a is in A, then a = P1a + P2a and (6.2) followsbecause we havelimα π(dα,β)P1a = limα π(dα11)P1a = P1a,since π(cβdα11) ∈A21π(dα11)A12 and π(dα11) is a bounded approximate identityfor A11. Likewiselimβ limα π(dα,β)P2a = limβ limα π(cβdα11)P2a= limβ π(cβ)P2a = P2a,
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS21again since π(cβdα11) ∈A21π(dα11)A12.Now we prove (6.3). Since a = a11 + a12 + a21 + a22, where aij is in Aij,we may treat these terms separately.
We have(a11 ⊗1 −1 ⊗a11)dα,β = (a11 ⊗1 −1 ⊗a11)dα11(a12 ⊗1 −1 ⊗a12)dα,β = (a12 ⊗1)cβdα11 −(1 ⊗a12)dα11(a21 ⊗1 −1 ⊗a21)dα,β = (a21 ⊗1)dα11 −(1 ⊗a21)cβdα11(a22 ⊗1 −1 ⊗a22)dα,β = (a22 ⊗1 −1 ⊗a22)cβdα,βClearly the first term tends to 0 as α −→∞. The second term may berewritten as(a12 ⊗1)cβ −(1 ⊗π(cβ))(1 ⊗a12)dα11 + 1 ⊗(a12π(cβ) −a12)dα11so that, using (6.1) and thatπ(cβ)β∈B is a bounded right approximateidentity for A12, the statement (6.3) is true in this case.The third term may be rewritten as(π(cβ) ⊗1)(a21 ⊗1) −(1 ⊗a21)cβdα11 +(a21 −π(cβ)a21) ⊗1dα11and treated analogously.For the fourth term it is enough to look at elements of the form a22 =b21b12 since by assumption these elements span a dense subset of A22 andwe are working with bounded nets.
We then get(a22 ⊗1 −1 ⊗a22)dα,β =(b21 ⊗1)(b12 ⊗1 −1 ⊗b12)dα,β + (1 ⊗b12)(b21 ⊗1 −1 ⊗b21)dα,β,so that this case follows from the two previous cases.To prove the converse, suppose now that A is amenable and let {dα}α∈Abe an approximate diagonal for A. Using the multiplier multiplication (Pi ⊗Pj)(a ⊗b) = Pia ⊗bPj and (a ⊗b)(Pi ⊗Pj) = aPi ⊗Pjb, we definedα,β11 = (P1 ⊗P1)dα(P1 ⊗P1 + cβ)(α ∈A , β ∈B).First note that for an elementary tensor we havelimβ π((a ⊗b)cβ) = limβ π((a ⊗b)(P2 ⊗P2)cβ)= limβ aP2π(cβ)P2a= aP2b= π((a ⊗b)(P2 ⊗P2))
22N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISso thatlimβ π(dα,β11 ) = P1 limβ π(dα(P1 ⊗P1) + cβ)P1= P1π(dα(P1 ⊗P1 + P2 ⊗P2))P1= P1π(dα)P1,which is a bounded left approximate identity for A11, directed over α ∈A.For a11 ∈A11 we have(a11 ⊗1 −1 ⊗a11)dα,β11 = (a11 ⊗1 −1 ⊗a11)(P1 ⊗P1)dα(P1 ⊗P1 + cβ)= (P1 ⊗P1)(a11 ⊗1 −1 ⊗a11)dα(P1 ⊗P1 + cβ),which tends to 0 as α →∞, because (dα)α∈A is an approximate diagonal forA.
This concludes the proof of the theorem.We now give some applications of this theorem in the case when A =K(X).Theorem 6.3. Let X be a Banach space.
Then K(X) is amenable if andonly if K(X ⊕C) is amenable.Proof. Let A = K(X ⊕C) and P1 be the projection of X ⊕C onto X withkernel C. Then P2 = I −P1 is the rank one projection onto C with kernel X.Hence A22 = P2K(X ⊕C)P2 is the one-dimensional algebra spanned by P2.It is easily seen that A22 = A◦22.
Furthermore, since A22 has an identity, Ahas a bounded two-sided approximate identity if either A or A11 is amenable.Theorem 6.2 now applies.Many of the classical Banach spaces are isomorphic to their direct sumwith the one-dimensional space and are also isomorphic to their hyperplanes.For some time it was an unsolved problem, the so-called ‘hyperplane prob-lem’, whether every Banach space has this property.However, it is nowknown ([G&M]) that there is a Banach space which is not isomorphic to anyproper subspace and so the above theorem has some content.An important class of Banach spaces is the class of Lp-spaces, where 1 ≤p ≤∞, which were introduced in [L&P]. The Banach space X is said to be anLp,λ-space if there is a constant λ > 0 such that for every finite dimensionalsubspace, B, of X there is a finite dimensional subspace, C, of X such thatB ⊆C and d(C, ℓnp) ≤λ, where n = dim C. (If Y and Z are isomorphicBanach spaces, then d(Y, Z) is inf(∥T∥, ∥T −1∥), where the infimum is overall invertible operators, T, from Y onto Z.) Some examples of Lp-spaces areℓp and Lp(0, 1).
We have already seen in Theorem 4.7 that the algebras ofcompact operators on these examples are amenable.Theorem III(c) in [L&R] shows that Lp-spaces satisfy stronger conditionsthan they are defined to have. Thus, if X is an Lp-space, then there is a
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS23constant λ′ > 0 such that for every finite dimensional subspace, B, of X thereare a finite dimensional subspace, C, of X and a projection, P, of X onto Csuch that: B ⊆C, d(C, ℓnp) ≤λ′, where n = dim C; and ∥P∥< λ′. It followsthat every Lp-space has the approximation property, and so K(X) = F(X)whenever X is an Lp-space.
It follows also that, if X and Y are infinitedimensional Lp-spaces, then every T in F(X) is a product T = UV, whereU : X →Y and V : Y →X are compact operators. Furthermore, if X isan Lp-space, then X∗is an Lq-space, where q−1 + p−1 = 1 ([L&R, TheoremIII(a)]).
Hence X∗has the bounded approximation property and so F(X)has a bounded two-sided approximate identity ([G&W, Theorem 3.3]). Weare now ready to proveTheorem 6.4.
Let 1 ≤p ≤∞and let X be an Lp-space. Then F(X) isamenable.Proof.
Let A = F(ℓp ⊕X) and let P1 be the the idempotent in M(A) de-termined by the projection onto ℓp with kernel X. Then A has a boundedtwo-sided approximate identity because A11 = F(ℓp) and A22 = F(X) do.Also, since each compact operator on X factors through ℓp, A◦22 = A22.Therefore, since F(ℓp) is amenable, F(ℓp ⊕X) is amenable by Theorem 6.2.That F(X) is amenable now follows from another application of Theorem6.2 because F(ℓp ⊕X) has a bounded two-sided approximate identity andevery compact operator on ℓp factors through X.The finite rank projections on Lp-spaces which were described above al-most show that these spaces have property (A).
The projections may be usedto produce a net of biorthogonal systems satisfying A(i) and A(iii). However,it is not clear that the net will satisfy A(ii).
If it could be shown that Lp-spaces in fact have property (A), then there would be a direct proof of theamenability of F(X) for these spaces. It seems that indirect arguments areneeded to establish many of the properties of Lp-spaces, see the remark afterthe statement of Theorem III in [L&R], and so it may be that they do nothave property (A) .
Some specific examples for which this may be tested arethe spaces ℓ2 ⊕ℓp. For 1 < p < ∞, ℓ2 ⊕ℓp is a Lp-space, see [L&P], example8.2, but it is not clear that it has property (A).Theorem 6.2 may be used to show that F(X) is amenable for some otherspaces which may fail to have property (A).
Let {nk}∞k=1 be a sequence ofpositive integers and choose p and q with 1 ≤p, q < ∞and p ̸= q. PutX = (⊕∞k=1ℓnkp )ℓq. Then X has the bounded approximation property and soK(X) = F(X).
If {nk}∞k=1 is bounded, then X is isomorphic to ℓq and sowe will suppose that {nk}∞k=1 is not bounded. Clearly X is isomorphic to acomplemented subspace of (⊕∞k=1ℓp)ℓq and so every T in F(X) is a productT = UV, where U is in F(X, (⊕∞k=1ℓp)ℓq) and V in F((⊕∞k=1ℓp)ℓq, X).We also have that every T in F((⊕∞k=1ℓp)ℓq) is a product T = UV, whereU is in F((⊕∞k=1ℓp)ℓq, X) and V in F(X, (⊕∞k=1ℓp)ℓq).
To see this, for each
24N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISr let Pr be the natural rank r2 projection of (⊕∞k=1ℓp)ℓq onto (⊕rk=1ℓrp)ℓq.Then {Pr}∞r=1 is a bounded left approximate identity for F((⊕∞k=1ℓp)ℓq) andso T = P∞r=1 PrTr, where P∞r=1 ||Tr|| < ∞.
Since {nk}∞k=1 is not bounded,X has a complemented subspace isomorphic to⊕∞r=1(⊕rk=1ℓrp)ℓqℓq. Hencewe have for each r that Pr = UrVr, where Ur is in F((⊕∞k=1ℓp)ℓq, X) and Vrin F(X, (⊕∞k=1ℓp)ℓq), ||Ur|| = 1 = ||Vr|| and UrVs = 0 if r ̸= s. It follows thatT factors as required.Now F((⊕∞k=1ℓp)ℓq) is amenable, see Corollary 4.8.
The above remarksabout factoring approximable operators therefore allow us to apply Theorem6.2 to proveTheorem 6.5. Let {nk}∞k=1 be a sequence of positive integers.
Then thealgebra F((⊕∞k=1ℓnkp )ℓq) is amenable.It was remarked above that F(ℓp ⊕ℓ2) is amenable. This suggests thatF(ℓp ⊕ℓq) may be amenable for all p and q.
That this is not so will followfrom a further general result about amenable Banach algebras.Definition 6.6. A Banach algebra B has trivial virtual centre if, for each b∗∗in B∗∗with bb∗∗= b∗∗b for all b in B, there is λ in C with bb∗∗= λb = b∗∗bfor all b in B.The algebras in which we are interested have this property.Proposition 6.7.
Let X be a Banach space. Then F(X) has trivial virtualcentre.Proof.
Let P be a rank one projection on X. Then PF(X)P = CP.
Supposethat B∗∗0in F(X)∗∗satisfies BB∗∗0 = B∗∗0 B for all B in F(X). Then PB∗∗0 =P 2B∗∗0 = PB∗∗0 P. Since the map B∗∗7→PB∗∗P is the second adjoint of themap B 7→PBP, it follows that there is λ0 in C such that PB∗∗0= λ0P.Consequently {T ∈F(X) | TB∗∗0= λ0T} is a non-zero closed two-sidedideal in the simple Banach algebra F(X).
Therefore TB∗∗0= λ0T for all Tin F(X).For the next theorem let A and Aii be as above.Theorem 6.8. Suppose that A is amenable, that A11 and A22 have trivialvirtual centre and that A21 and A12 are not both zero.
Then Ajj = A◦jj forat least one value of j.Proof. Denote A◦= {a ∈A | PiaPi ∈A◦ii, i = 1, 2}.
On A◦define thenorm ∥a∥◦= max{∥P1aP1∥◦, ∥P1aP2∥, ∥P2aP1∥, ∥P2aP2∥◦}. Then, for a ∈A, a◦∈A◦we have ∥aa◦∥◦≤2∥a∥∥a◦∥◦and ∥a◦a∥◦≤2∥a∥∥a◦∥◦.
Hence(A◦, ∥.∥◦) is a Banach A-bimodule.The map a 7→P1aP2 −P2aP1 = P1a −aP1 is a derivation from A into A◦and so there is C in (A◦)∗∗such that P1aP2 −P2aP1 = aC −Ca for all a in
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS25A. Since A◦= ⊕i,j=1,2PiA◦Pj, we have C = Pi,j=1,2 Cij, where Cij belongsto (PiA◦Pj)∗∗.If aii belongs to Aii, then aiiC−Caii = 0.
In particular, aiiCii−Ciiaii = 0for each aii in Aii, where Cii belongs to (A◦ii)∗∗. Now the second adjoint ofthe inclusion map A◦ii →Aii embeds (A◦ii)∗∗in (Aii)∗∗and so, since Aiihas trivial virtual centre for each i, there are λ1 and λ2 in C such thataiiCii = λiaii = Ciiaii for aii in Aii, i = 1, 2.Suppose, without loss of generality, that A12 is not zero and choose a12 ̸= 0in A12.
Since A is amenable, it has a bounded two-sided approximate identityand so, by Proposition 6.1, A12 is an essential left A11- and essential rightA22-module. Hence there are a11 in A11, a22 in A22 and ˜a12 in A12 with a12 =a11˜a12a22.
We have a12 = a12C −Ca12 = a12C22 −C11a12. Substituting fora12 we get a12 = a11˜a12a22C22−C11a11˜a12a22 = λ2a11˜a12a22−λ1a11˜a12a22 =(λ2 −λ1)a12.
Therefore λ2 −λ1 = 1 and so at least one of λ1 and λ2 is notzero.Suppose that λ1 ̸= 0 and put b∗∗= λ−11 C11. Then b∗∗belongs to (A◦11)∗∗and a11b∗∗= a11 for every a11 in A11.
Hence, if {bα}α∈A is a boundednet in A◦11 which converges to b∗∗in the weak∗- topology, then {a11bα}α∈Aconverges weakly to a11. It follows, as in [B&D,Proposition 11.4], that thereis a net {eβ}β∈B, each eβ being a convex combination of bα’s, which is aright approximate identity for A11.
The approximate identity {eβ}β∈B isbounded, by ∥b∗∗∥, in (A◦11, ∥.∥◦) and A◦11 is an ideal in A11. Hence for eacha11 in A11 and each ǫ > 0 there is c = a11eβ with ∥a11 −c∥< ǫ and∥c∥◦< 2∥b∗∗∥∥a11∥.
Consequently, for each a11 in A11, there is a series Pi ciin A◦11 with Pi ∥ci∥◦< ∞and Pi ci = a11. Therefore A◦11 = A11.These last two results may be reformulated to say that the spaces X withF(X) amenable have a property which is a little like being primary.
Recallthat a Banach space, X, is primary if, for every bounded projection Q on X,either QX or (I−Q)X is isomorphic to X, see [L& T, Definition 3.b.7]. Let ussay that X is approximately primary if, for every bounded projection Q on X,at least one of the product maps π : F(X, QX)b⊗F(QX, X) →F(X) or π :F(X, (I −Q)X)b⊗F((I −Q)X, X) →F(X) is surjective.
Then every primaryspace is approximately primary as is every space with a subsymmetric basis,see [L&T, Proposition 3.b.8].Now put A = F(X) and suppose that A is amenable.Let P1 be theidempotent in M(A) determined by a bounded projection Q on X. Then A11is isomorphic to F(QX) and A22 to F((I −Q)X).
Hence, by Proposition 6.7,Aii has trivial virtual centre for i = 1, 2. Clearly, A12 is not zero and so, byTheorem 6.8, A◦ii = Aii for at least one value of i.
It follows that, if F(X) isamenable, then X is approximately primary.Theorem 6.9. If 1 < p, q < ∞, p ̸= q and neither p nor q is equal to 2,then F(ℓp ⊕ℓq) is not amenable.
26N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISProof.
In view of 6.7, 6.8 and the remarks following it suffices to show that,if 1 < p, q < ∞, p ̸= q and neither is equal to 2, then the product mapπ : F(ℓp, ℓq)b⊗F(ℓq, ℓp) →F(ℓp) is not surjective.Suppose that π is surjective. Then, by the open mapping theorem, there isa K > 0 such that for each T in F(ℓp) we have T = π(P∞n=1 Un ⊗Vn), whereP∞n=1 ||Un||||Vn|| < K||T||.
It follows, since ℓq is isomorphic to (⊕∞n=1ℓq)ℓq,that T = UV, where U is in F(ℓp, ℓq), V in F(ℓq, ℓp) and ||U||||V || < K||T||.Let Pj be the projection onto the span of the first j vectors of the standardbasis for ℓp. Then, since we are supposing that π is surjective, Pj = UjVjwhere ||Uj||||Vj|| < K. Put Qj = VjUj.
Then Qj is a projection on ℓq and||Qj|| < K. Defining U ′j = PjUjQj and V ′j = QjVjPj, we have that U ′j is anisomorphism from the range of Qj to the range of Pj, V ′j is the inverse of U ′jand ||U ′j||||V ′j || < K3. Hence, if π is surjective, then ℓp is finitely representablein ℓq, see [Wo, Definition II.E.15].
It is known that this is not so if p ̸= q andneither is equal to 2. There are several cases.First, suppose that p < 2 < q.
If ℓp were finitely representable in ℓq, then,since ℓq is of type 2, ℓp would be of type 2. (See [Wo, Definition III.A.17 andTheorem III.A.23] ) That is not so.
Therefore ℓp is not finitely representablein ℓq. The case q < 2 < p is dual to this case.Next, suppose that 2 < q < p. If ℓp were finitely representable in ℓq, then,since ℓq is of cotype q, ℓp would be of cotype q.
Since ℓp is not of cotype q,ℓp is not finitely representable in ℓq. The case p < q < 2 is dual to this case.Finally, suppose that 2 < p < q.
If ℓp were finitely representable in ℓq,then ℓp would be isomorphic to a subspace of Lq(µ) for some measure µ, seeProposition 7.1 in [L&P]. It would then follow, by Corollary 2 in [K&P], thatℓp had a complemented subspace isomorphic to ℓ2 or ℓq.
However, that is notpossible because, by Proposition 2.c.3 in [L&T], every operator from ℓp to ℓ2and every operator from ℓq to ℓp is compact. The case q < p < 2 is dual tothis case.
This argument is also sketched on [Wo] pages 104 and 107.The above proof also shows that F(c0 ⊕ℓp) is not amenable when p < 2but does not treat the case p > 2. Similarly, F(ℓ1 ⊕ℓp) is not amenable whenp > 2.We conclude this section with a result which shows that amenability ofF(X) is partially preserved on complemented subspaces of X.6.10 Theorem.
Let X and Y be Banach spaces and suppose that F(X ⊕Y )is amenable. Then at least one of F(X) and F(Y ) is amenable.Proof.
Put A = F(X⊕Y ) and let P1 be the idempotent in M(A) determinedby the projection onto X with kernel Y. Then, by 6.7 and 6.8, Ajj = A◦jj forat least one value of j.
By Theorem 6.2, it follows that at least one of A11 andA22 is amenable. Since A11 is isomorphic to F(X) and A22 is isomorphic toF(Y ), the result follows.
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS27The conclusion of this last theorem is the best possible, that is, there arespaces X and Y such that F(X ⊕Y ) is amenable but F(X) is not. Forexample, let X = c0 ⊕ℓ1 and Y = ℓ1(c0) =⊕∞n=1c0ℓ1.
Then X ⊕Y isisomorphic to Y. Hence F(X ⊕Y ) and F(Y ) are amenable by Corollary 4.8.On the other hand, F(X) is not amenable by Theorem 6.9.7.
Open questions and conclusionThe name ‘amenable’ is used for a Banach algebra A satisfying the co-homological condition H1(A, X∗) = 0 for all Banach- A-modules X, see [J],because of the theorem that a group algebra L1(G) satisfies this condition ifand only if the locally compact group G is amenable, [J]. Amenability is animportant property of groups which has many characterizations.
As well asthe cohomological characterization of the group algebra, it may be describedin terms of group representations, fixed points of group actions, translationinvariant functionals and in other ways. The Følner conditions on compactsubsets of the group characterize amenability in terms of properties intrinsicto the group.
Alternative characterizations of the amenability of K(X) andF(X) would help us to have a better understanding of its significance. Weare thus led to askQuestion 7.1.
What are the intrinsic properties of the Banach space Xwhich are equivalent to amenability of K(X) and F(X)?The results we have obtained so far suggest that amenability of K(X) andF(X) may be equivalent to some sort of approximation property for X. Suchan approximation property, if it were to exist, would be the analogue of theFølner conditions.Approximation properties are certainly necessary. Since an amenable al-gebra has a bounded two-sided approximate identity, if K(X) is amenable,then X∗has what is called in [G&W] the B −K(X)a- AP, and in [Sa] the ∗-b.c.a.p., that is, the identity operator on X∗is approximable in the topologyof convergence on compacta by operators which are adjoints of compact op-erators on X.
It also follows, by [D, Theorem 2.6], that X has the boundedcompact approximation property. Similarly, if F(X) is amenable, then Xand X∗have the bounded approximation property.
However, the relation-ship between amenability of K(X) and the approximation property is notclear.Question 7.2. Does amenability of K(X) imply that X has the approxima-tion property?If there should be a Banach space X which does not have the approximationproperty but is such that K(X) is amenable , then K(X)/F(X) would be aradical, amenable Banach algebra.
At present no example of such a Banachalgebra is known.
28N. GRØNBÆK,B.
E. JOHNSON,ANDG. A. WILLISTheorem 6.9 shows that for F(X) to be amenable it does not suffice thatX∗have the B−F(X)a- AP.
It seems necessary for there also to be some sortof symmetrization of the approximation property. We have seen, in Section4, a symmetrized approximation property, property (A), which forces theamenability of F(X).
This property was used to show that F(X) is amenablefor many of the classical Banach spaces and for spaces with a shrinking,subsymmetric basis.Question 7.3. Is property (A) or some similar symmetrized approximationproperty equivalent to amenability of F(X) or K(X)?In order to determine how close this property is to being equivalent tothe amenability of F(X), it would be useful to investigate whether F(X) isamenable if X is a space which is clearly unlikely to have this symmetrizedapproximation property.
Examples that come to mind are the James space,which does not have an unconditional basis ([L&T, 1.d.2]), and the Tsirelsonspace, which contains no subsymmetric basic sequence ([L&T, p. 132]).Question 7.4. Is F(X) amenable if X is the James space or the Tsirelsonspace?We have seen that the class of spaces, X, such that F(X) is amenable isnot closed under direct sums or under passing to complemented subspaces.However, any space, X, such that F(X) is amenable has the property that X∗satisfies the B −F(X)a- AP and this property is inherited by complementedsubspaces of X and is preserved under direct sums.
Perhaps this is the mostthat can be said about such spaces.Question 7.5. Is the smallest space ideal containing all spaces, X, such thatF(X) is amenable equal to the class of all Banach spaces, X, such that X∗has the B −F(X)a- AP?Recall from [Pie, Definition 2.1.1], that a space idealis a class of Banachspaces which contains the finite dimensional spaces and is closed under directsums and taking complemented subspaces.
It is clear that the class of spacessuch that X∗has the B −F(X)a- AP is a space ideal. Should the answerto 7.2 be ‘no’, an obvious further question would be whether the class of allBanach spaces whose duals have the B −K(X)a- AP is equal to the smallestspace ideal containing all spaces, X, such that K(X) is amenable.It was shown by J. Lindenstrauss, see [L&T, Theorem 3.b.1], that everyBanach space with an unconditional basis is isomorphic to a complementedsubspace of a space with a symmetric basis.
In view of the possible equiva-lence of the amenability of F(X) with some symmetric approximation prop-erty, this suggests the following refinement of 7.5.Question 7.6. Is every Banach space, X, such that X∗has the B −F(X)a-AP isomorphic to a complemented subspace of a space, Y, such that F(Y ) isamenable?
AMENABILITY OF BANACH ALGEBRAS OF COMPACT OPERATORS29The spaces Cp, 1 ≤p < ∞, introduced by W. B. Johnson [Jo1] willprovide the answer to this last question.The space Cp is the ℓp directsum of a sequence of finite dimensional spaces which is dense in the set ofall finite dimensional spaces. It has the property that every approximableoperator factors through it and, for 1 < p < ∞, F(Cp) has a bounded two-sided approximate identity.
Now let X be any space such that X∗has theB−F(X)a- AP. Then, since Cp has the above properties, Theorem 5.2 impliesthat F(X ⊕Cp) is amenable if and only if F(Cp) is amenable.
Therefore theanswer to 7.6 is ‘yes’ if F(Cp) is amenable.On the other hand, if Cp isisomorphic to a complemented subspace of some space, Y, such that F(Y ) isamenable, that is, if the answer to 7.6 is ‘yes’ when X = Cp, then F(Cp) isamenable.Question 7.7. Is F(Cp) amenable for any, and hence all, 1 < p < ∞?Note that C∗1 does not have the approximation property, see [Jo2, Theorem3], and so F(C1) is not amenable.Another theorem, similar to that of Lindenstrauss, is proved in [J,R&Z]and [P], see [L&T, Theorem 1.e.13].It says that any separable Banachspace with the B.A.P.
is isomorphic to a complemented subspace of a Banachspace with a basis. There is an even stronger theorem, see [L&T, Theorems2.d.8 and 2.d.10], that there is a Banach space, U, with basis such that anyseparable Banach space with the B.A.P.
is isomorphic to a complementedsubspace of U and that U is determined uniquely up to isomorphism by thisproperty. The space U is said to be complementably universal for the spaceswith the B.A.P.
Now if X has a shrinking basis, then X∗has the B −F(X)a-AP. This suggestsQuestion 7.8.
(a) Is there a Banach space, V, with a shrinking basis whichis complementably universal for the spaces, X, such that X∗has the B −F(X)a- AP? (b) If so, is F(V ) amenable?Acknowledgements.
We are indebted to D. J. H. Garling for his help witha proof of Theorem 6.9 and to P. G. Casazza, J. Partington and A. Pe lczy´nskifor valuable discussions. We would also like to thank L. Tzafriri for havingbrought the reference [A&F] to our attention.This work was initiated while the first and the third author were visitingUniversity of Leeds.
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