aa r X i v : . [ m a t h . F A ] O c t A short proof that B ( L ) is not amenable Yemon Choi
Abstract
Non-amenability of B ( E ) has been surprisingly difficult to prove for the classical Ba-nach spaces, but is now known for E = ℓ p and E = L p for all 1 ≤ p < ∞ . However, thearguments are rather indirect: the proof for L goes via non-amenability of ℓ ∞ ( K ( ℓ ))and a transference principle developed by Daws and Runde (Studia Math., 2010).In this note, we provide a short proof that B ( L ) and some of its subalgebras arenon-amenable, which completely bypasses all of this machinery. Our approach is based onclassical properties of the ideal of representable operators on L , and shows that B ( L ) isnot even approximately amenable.Keywords: amenable Banach algebras, Banach spaces, operator ideals, representable op-erators.MSC 2020: 46H10, 47L10 (primary); 46B22, 46G10 (secondary) Throughout this paper: all algebras are associative and taken over the field C , but they neednot have identity elements.The Wedderburn structure theorem implies, with hindsight, that a finite-dimensional alge-bra with homological dimension zero is isomorphic to a sum of full matrix algebras. Amenabil-ity for Banach algebras, introduced in B. E. Johnson’s seminal work [Joh72], may be thoughtof as a weakened version of having homological dimension zero, and the two notions coincidefor finite-dimensional Banach algebras. In particular, finite sums of full matrix algebras areamenable, while the algebra of 2 × E the algebra B ( E ) is amenable. It was soon recognized that for most E the answer should be negative,but that proving this for specific natural E could be very hard. While the Hilbertian casewas known to follow very indirectly from deep results on C ∗ -algebras, no progress was madeon the other classical Banach spaces until C. J. Read’s breakthrough result that B ( ℓ ) is non-amenable [Rea06]. His proof was simplified by G. Pisier [Pis04], and N. Ozawa subsequentlyprovided a unified proof of non-amenability of B ( ℓ ), B ( ℓ ) and B ( c ) [Oza04]. Further his-torical details can be found in V. Runde’s survey article [Run10b], or in the introduction ofhis companion paper [Run10a].The paper [Run10a] contains the strongest general results thus far on non-amenabilityof B ( E ); among other things, it establishes the non-amenability of B ( ℓ p ) and B ( L p ) for all1 ∈ (1 , ∞ ). A key ingredient in the proof is the following “transference principle” developedby M. Daws and V. Runde in [DR08]:– if F is a Banach space, amenability of B ( ℓ p ( F )) implies amenability of ℓ ∞ ( K ( ℓ p ( F )));– if E is an infinite-dimensional L p -space in the sense of Lindenstrauss and Pe lczy´nski,then amenability of ℓ ∞ ( K ( E )) implies amenability of ℓ ∞ ( K ( X )) for every L p -space X .Although the case of L was not resolved in [Run10a], the transference principle remainsvalid for p = 1 (see [DR08, Theorems 1.2 and 4.3]), and so amenability of B ( L ) ∼ = B ( ℓ ( L ))would imply amenability of ℓ ∞ ( K ( ℓ )). Therefore it suffices to prove that the latter algebra isnon-amenable, and this was recently demonstrated in the PhD thesis of E. Aldabbas [Ald17];as in [Run10a], crucial use is made of a technical innovation from [Oza04], which was itselfinspired by the arguments of [Rea06]. At the time of writing, the proof from [Ald17] has notbeen published.Thus, although non-amenability of B ( L ) is now known, the existing proof is both indirect(going via ℓ ∞ ( K ( ℓ ))) and technically complicated (relying on “Ozawa’s lemma” as formulatedin [Run10a]). The purpose of this note is to show that non-amenability of B ( L ) can beproved very quickly by studying a particular closed ideal R ⊳ B ( L ), without any need for thetransference principle or the ideas in [Oza04]. Our method actually proves slightly more: if A ⊆ B ( L ) is a closed subalgebra that contains R , then A is not even approximately amenablein the sense of [GL04, GLZ08].The ideal R occurs very naturally in the study of operators on L , and is related to afactorization result of D. R. Lewis and C. Stegall. Thus our approach has a rather differentflavour from the combinatorial arguments in [Rea06] and [Oza04], and we hope that thisalternative perspective could be useful for studying the non-amenability problem for B ( E ) onother Banach lattices. Most of our conventions for notation and terminology are either standard in the literatureor clear from context. However, to make our paper more accessible, we have endeavoured toprovide precise references for various “well-known” or “standard” facts about Banach spaces.The term “operator” is synonymous with “bounded linear map” although we shall some-times refer to “bounded operators” just for emphasis. For a Banach space E , B ( E ) denotesthe algebra of bounded operators on E , while K ( E ) denotes the algebra of compact operatorson E .The projective tensor product of Banach spaces E and F is denoted by E b ⊗ F . For ourpurposes, it can be characterized as by the following universal property: whenever E , F and G are Banach spaces and β : E × F → G is a bounded bilinear map, there is a unique boundedlinear map f : E b ⊗ F → G that satisfies f ( x ⊗ y ) = β ( x, y ) for all x ∈ E and y ∈ F . Moreover, k f k = k β k . Note also that for any x ∈ E and y ∈ F we have k x ⊗ y k E b ⊗ F = k x kk y k .Given a measure space (Ω , Σ , µ ) we abbreviate L p (Ω , Σ , µ ) to L p (Ω). In the case of [0 , σ -algebra and Lebesgue measure, we simply write L p ; in the case of N with2he discrete σ -algebra and counting measure, we simply write ℓ p . If A is a Borel subset of[0 ,
1] then | A | denotes its Lebesgue measure.The background we need concerning Banach-space valued integration can be found in anysource that defines the Bochner integral over a finite measure space. By a slight abuse ofterminology, we say that a function from [0 ,
1] to a Banach space X is strongly measurable ifit is strongly measurable with respect to the Borel σ -algebra of [0 , Since this paper is intended for a general rather than a specialist audience, we use this sectionto record some basic definitions and examples from the literature on amenability of Banachalgebras, in order to supply some context for the main result.The following definition of amenability is not the original one given by B. E. Johnson in[Joh72], but is a standard equivalent formulation that is often more useful or more suggestive.
Definition . Let A be a Banach algebra and define π : A b ⊗ A → A by π ( a ⊗ b ) = ab . An approximate diagonal for A is a net ( d α ) in A b ⊗ A such that, for each a ∈ A , we have k a · d α − d α · a k A b ⊗ A → k aπ ( d α ) − a k A → α → ∞ .If the net ( d α ) is bounded then we call it a bounded approximate diagonal . A Banach algebrapossessing a bounded approximate diagonal is said to be amenable .Note that if A is finite-dimensional and amenable, taking a cluster point of the net ( d α )yields an element ∆ ∈ A ⊗ A such that π (∆) = 1 A and a · ∆ = ∆ · a . In (non-Banach)homological algebra such a ∆ is known as a separating idempotent for A and serves as anexplicit witness that A has homological dimension zero.It is well known that finite-dimensional matrix algebras M n ( C ) ≡ B ( C n ) have homologicaldimension-zero, and that an explicit separating idempotent for M n ( C ) is∆ = 1 n n X i,j =1 E ij ⊗ E ji . By an averaging argument, ∆ can be written as an absolutely convex combination of tensorsof the form x ⊗ x − where x is a signed permutation matrix. For details see the proof of[GJW94, Prop. 3.2]. It follows that if A = B ( ℓ np ) for any 1 ≤ p ≤ ∞ , k ∆ k A b ⊗ A = 1. Fromthis, a routine exhaustion argument allows one to construct an explicit bounded approximatediagonal for K ( ℓ p ) when 1 ≤ p < ∞ and K ( c ).Using a more abstract version of this idea, the paper [GJW94] developed a condition ona given Banach space E , called Property ( A ), which is sufficient for amenability of K ( E ).Property ( A ) is studied in detail in that paper; while it is not known to hold for all L p -spaces,it does hold for all L p ( µ )-spaces (1 ≤ p ≤ ∞ ) and their preduals (see [GJW94, Theorem 4.7]and [GJW94, Theorem 4.3]).As mentioned in the introduction, the expectation is that for most E the Banach algebra B ( E ) is in some sense too large to be amenable. However, S. A. Argyros and R. Haydonconstructed in [AH11] an infinite-dimensional space X such that every bounded operator on X is a compact perturbation of a multiple of the identity, solving one of the major foundationalproblems of the subject. The nature of their construction also ensures that X ∗ is isomorphic to ℓ ; thus X has Property ( A ), and so K ( X ) is amenable. Since unitizations of amenable Banachalgebras are amenable, it follows that B ( X ) = C I + K ( X ) is amenable ([AH11, Prop. 10.6]).3 .3 Preliminary results needed for our paper The second condition in the definition of a (bounded) approximate diagonal says that the net( π ( d α )) is a (bounded) right approximate identity for A . There is a corresponding notion ofa (bounded) left approximate identity. Crucially, amenability of a Banach algebra not onlyensures bounded left and right approximate identities in the algebra itself, but also in some ofits closed ideals. The following lemma follows from standard results in the theory of amenableBanach algebras and their bimodules. For instance, it is an immediate corollary of [CL89,Theorem 3.7]. Lemma . Let A be an amenable Banach algebra and let J be a closed, -sided ideal in A which has a bounded right approximate identity. Then J has a bounded left approximateidentity. From this we immediately deduce the following result.
Corollary . Let A be a Banach algebra and J a closed -sided ideal. Suppose that • J has a bounded right approximate identity; • there exists x ∈ J such that x / ∈ J x .Then A is not amenable. In the next section, we will introduce a particular closed ideal in B ( L ) and show that itsatisfies both conditions in Corollary 2.3. B ( L ) Given a Banach space E , an operator T : L → E is said to be representable if there exists abounded, strongly measurable function h T : [0 , → E such that T ( f ) = Z f ( s ) h T ( s ) ds for all f ∈ L where the right-hand side is interpreted as an E -valued Bochner integral. (In some sources,such as [Ros75], the terminology “differentiable” is used instead of “representable”.)Note that if such an h T exists, we have k T k ≤ k h T k ∞ by basic properties of the Bochnerintegral; one can show that equality holds, although this is not needed for the present paper.We denote by R the set of all representable operators from L to itself. It follows easilyfrom the definitions that R is a closed left ideal in B ( L ). Therefore, to show that it is alsoa right ideal, it suffices to prove that T S ∈ R for all T ∈ R and all S ∈ B ( L ). This is notobvious from the definition, but is an immediate consequence of the next result which is dueto D. R. Lewis and C. Stegall. Theorem . Let E be a Banach space and let T ∈ B ( L , E ) . Then T isrepresentable if and only if it factors (boundedly) through ℓ . E -valued functions, see [DF93, Appendix C, § Remark . The original paper [LS73] does not make use of the perspective of Bochnerintegrals and vector-valued L p -spaces. Indeed, while Theorem 3.1 was known at the time tofollow from the techniques in [LS73], the result itself is never explicitly stated there, althoughsome version of it appears en passant in the proof of [LS73, Theorem 1]. C.f. the remarks in[Ros75, Appendix A].The first part of the next result is well-known to Banach space theorists, although we arenot aware of a reference. Proposition . There exists T ∈ R \ K ( L ) such that ST ∈ K ( L ) for all S ∈ R . Inparticular, R does not have any left approximate identity (bounded or otherwise).Proof. Let S ∈ R . By the Lewis–Stegall theorem, S factors through ℓ , and hence it mapsweakly convergent sequences to norm convergent sequences (since ℓ has the Schur property).By the Eberlein–ˇSmulian theorem, it follows that S maps relatively weakly compact subsetsof L to totally bounded subsets of L . Moreover, every weakly compact operator on L is representable (see e.g. [DF93, Section C5], [DU77, Chapter III, Lemma 2.9] or [Rya02,Prop. 5.40]). It therefore suffices to choose any T ∈ B ( L ) which is weakly compact but notcompact.There are various indirect ways to show the existence of weakly compact non-compactoperators on L . We describe one easy and explicit construction for the reader’s convenience. Example . Fix a partition of (0 ,
1] into countably many subsets with strictly positivemeasure (e.g. intervals (2 − n , − n ] for n ∈ N ) and let P : L → ℓ be the associated conditionalexpectation. Let ι , : ℓ → ℓ be the canonical embedding; and fix an injection j : ℓ → L with closed range (for instance, using Rademacher functions). Then T := jι , P is non-compact, since P is an open mapping, ι , is non-compact, and j is bounded below. Onthe other hand, T is weakly compact since it factors through ℓ . Note also that by Pitt’stheorem we get a direct proof that ST ∈ K ( L ) for all S : L → ℓ , without requiring theLewis–Stegall theorem or the fact that weakly compact operators on L are representable.W. B. Johnson has informed the author that in forthcoming work with N. C. Phillips andG. Schechtman, they show that for 1 ≤ p < ∞ the only closed ideal in B ( L p ) with a boundedleft approximate identity is K ( L p ). In the same work, they also establish the following result,which is the key ingredient needed for the present paper. Proposition . R has a bounded right approximate iden-tity. Moreover, we can choose this net to consist of norm-one idempotents. Since the work of Johnson–Phillips–Schechtman is still unpublished at time of writing,we include a self-contained proof of Proposition 3.5. The argument originally shown to theauthor by W. B. Johnson used ideas from [LS73] and some auxiliary results on K ( L ). Ourapproach uses the perspective of vector-valued L ∞ -spaces, and is based on a suggestion ofM. Daws (personal communication). 5 emma . Let E be a Banach space and let h : [0 , → E be strongly measurable. For any ε > , there is a (strongly) measurable h ε : [0 , → E whose range is countable and whichsatisfies k h − h ε k ∞ ≤ ε . We omit the proof of this lemma, which is a variation on the usual argument for scalar-valued functions. It is usually found in the literature as part of the proof of the Pettismeasurability criterion (see e.g. [DF93, Theorem B11] or the proof of [Rya02, Prop. 2.15]).For an explicit reference with a full proof, see [HvNVW16, Lemma 2.1.4].
Proof of Proposition 3.5.
Let R be the set of operators L → L that are represented bybounded and countably-valued measurable functions [0 , → L . Then R is a left ideal in B ( L ) and by Lemma 3.6 it is dense in R . Hence, by a 3-epsilon argument, it suffices to provethat R has a bounded right approximate identity consisting of norm-one idempotents.Given a partition of [0 ,
1] as a countable disjoint union of measurable subsets, [0 ,
1] = F ∞ n =1 A n , define a corresponding conditional expectation P : L → L by the formula P ( f )( t ) = 1 | A n | Z A n f if t ∈ A n , ( ∗ )with the convention that if | A n | = 0 we interpret | A n | − R A n f as zero. Then P is a norm-oneidempotent in B ( L ), which belongs to R since P is represented by h P := P ∞ n =1 | A n | − A n .If h : [0 , → L is constant on each A n , with h ( A n ) = { c n } say, then the operator T ∈ R represented by h satisfies T P ( f ) = ∞ X n =1 Z A n h · ( P f ) = ∞ X n =1 c n Z A n P f = ∞ X n =1 c n Z A n f = T ( f ) ( f ∈ L );that is, T P = T . Now, given T , . . . , T m ∈ R , represented by bounded functions h , . . . , h m :[0 , → L respectively, note that there is a countable partition [0 ,
1] = F ∞ n =1 A n such thateach h j is constant on each A n . Defining P by the formula ( ∗ ), the previous calculation nowgives T j P = T j for all j = 1 , . . . , m .Therefore, if we order the set of countable partitions of [0 ,
1] by refinement, we obtain a netof norm-one idempotents in R , each having the form ( ∗ ), which serves as a right approximateidentity for R .Combining Proposition 3.3 and Proposition 3.5, we see that R satisfies the conditions ofCorollary 2.3, and therefore B ( L ) is not amenable. In fact, the corollary rules out amenabilityfor every closed subalgebra A ⊆ B ( L ) that contains R .It is notable that for Proposition 3.3, the key feature of R was that every T ∈ R fac-tors through ℓ , while for Proposition 3.5 it seems vital to have the description in terms ofrepresentability by strongly measurable functions on [0 , Remark . In this section we chose to work with R and its properties because it is anideal with intrinsic interest, regardless of the application to non-amenability. One can bypassexplicit mention of R and extract a slightly more direct proof that B ( L ) is non-amenable,by combining specific properties of the operator T in Example 3.4 with calculations in Ap-pendix A. However, this direct approach still seems to require the result that every operator L → ℓ is representable (the “easy direction” of the Lewis–Stegall theorem), and so we donot include the details here. 6 Related examples and variations
Corollary 2.3 can be applied to prove non-amenability of B ( E ) for some other Banach spaces E ,provided we make a left-right switch. Since a Banach algebra A is amenable if and only ifthe opposite algebra A op is, Lemma 2.2 remains true when the words “left” or “right” areinterchanged. Therefore, if a Banach algebra A possesses a closed ideal J that has a boundedleft approximate identity but no bounded right appproximate idenity, A cannot be amenable. Example . Let E be a Banach space and let A ( E ) denote the algebra of approximableoperators on E ; this is a closed ideal in B ( E ). By results of N. Grønbæk and G. A. Willis, A ( E ) has a bounded left approximate identity if and only if E has the bounded approximationproperty, but has a bounded right approximate identity if and only if E ∗ has the boundedapproximation property. (See [GW93, Theorem 2.4 and Theorem 3.3].)Hence, by our previous remarks, if E has the bounded approximation property and E ∗ does not then B ( E ) is non-amenable. This applies for instance when E = ℓ b ⊗ ℓ .It is natural to wonder if the techniques in this paper can be adapted to give an alternativeproof of the non-amenability of B ( ℓ ). In this context, note that by [DR08, Theorem 1.2],amenability of B ( ℓ ) would imply amenability of ℓ ∞ ( B ( ℓ )) and hence amenability of anyultrapower B ( ℓ ) U ; such an ultrapower can be represented as an algebra of operators on someabstract L -space E , and if we can find an operator on E analogous to the operator T inExample 3.4 then it may be possible to run similar arguments to the ones in this paper. Weleave this as a problem for possible future investigation.We briefly comment on approximate amenability, although this was not the main focusof the present work. Given a Banach algebra A let A denote its forced unitization. Wesay that A is approximately amenable if A has an approximate diagonal. This is not theoriginal definition from [GL04]; strictly speaking, what we have just defined is “approximatecontractibility” of A , but the two concepts were shown to coincide in [GLZ08, Theorem 2.1].By [GL04, Corollary 2.4], one has an analogue of Lemma 2.2:if A is approximately amenable and J is a closed ideal in A possessing a bounded right approximate identity, then J has a left approximate identity (not necessarilybounded).For an outline of a direct proof, see the appendix. From this result, we see that Corollary 2.3remains valid if we weaken the hypothesis from amenability to approximate amenability.Hence, by the results of Section 3, every closed subalgebra of B ( L ) which contains R fails tobe aproximately amenable. Acknowledgements
The question of whether the ideal R has either a left or right bounded approximate identityemerged from discussions with Matt Daws and Jon Bannon about a completely differentproblem in von Neumann algebras, while the latter was visiting Lancaster University in 2019as a Fulbright Visiting Scholar.The author thanks Bence Horv´ath and Niels Laustsen for spurring him to learn moreabout ideals in B ( L p ), Bill Johnson for answering his original questions about R and clarifyingsome of the relevant history, Matt Daws for helpful discussions concerning Lebesgue–Bochnerspaces, and Jon Bannon for reminding him that there must be better songs to sing than this.7 A direct proof of Lemma 2.2
For sake of brevity, we justified the claim in Lemma 2.2 by appealing to more general resultsin [CL89]. Specifically, we were invoking the following standard result:if A is an amenable Banach algebra and J ⊳ A is a closed ideal that is weaklycomplemented in A as a Banach space, then J has a bounded approximate identity.This result implies Lemma 2.2 because in any Banach algebra (regardless of amenability), aclosed ideal with a bounded left-or-right approximate identity is weakly complemented.The proof of the general result is somewhat abstract: one starts with a bounded linearprojection from A ∗ onto J ⊥ , and then uses amenability to average this projection to an A -bimodule map, from which one extracts a left identity for J ∗∗ equipped with the first Arensproduct. It is therefore instructive to have a more direct proof of Lemma 2.2, since thissheds more light on possible refinements of Corollary 2.3. We provide details below, since wehave not seen such a proof written down explicitly. No novelty is claimed for the followingarguments.Let FIN( J ) denote the set of finite subsets of J . Our bounded left approximate identitywill be indexed by FIN( J ) × (0 , ∞ ), given the following partial order: ( F, ε ) (cid:22) ( F ′ , ε ′ ) if F ⊆ F ′ and ε ≥ ε ′ . Thus, fix some F ∈ FIN( J ) and ε >
0; it suffices to find v ∈ J such thatmax x ∈ F k x − vx k < ε , and such that k v k is bounded above by a constant independent of F and ε .The hypotheses of Lemma 2.2 ensure that for some constant C > A has an approximatediagonal bounded in norm by C and J has a right approximate identity bounded in normby C . Let δ > C and ε . Perturbing thebounded approximate diagonal slightly, we obtain ∆ ∈ A ⊗ A with k ∆ k A b ⊗ A ≤ C + 1 and k x · ∆ − ∆ · x k A b ⊗ A ≤ δ and k x − xπ (∆) k ≤ δ for all x ∈ F . (A.1)By definition of the projective tensor norm, we can assume that ∆ = P mi =1 a i ⊗ b i where P mi =1 k a i kk b i k ≤ C + 1. Since { xa i : x ∈ F, ≤ i ≤ m } is a finite subset of J , there existssome f ∈ J with k f k ≤ C and k xa i − xa i f k ≤ δ k a i k for all x ∈ F and all 1 ≤ i ≤ m . (A.2)We put v := P mi =1 a i f b i ∈ J , which satisfies k v k ≤ k ∆ k A b ⊗ A k f k ≤ C . For each x ∈ F , k x − vx k ≤ k x − xπ (∆) k + k xπ (∆) − xv k + k xv − vx k . (A.3)The first term on the right-hand side is bounded above by δ . The second term is boundedabove by (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x m X i =1 a i b i − x m X i =1 a i f b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ m X i =1 k xa i b i − xa i f b i k ≤ X i =1 δ k a i kk b i k ≤ δ ( C + 1) . (A.4)To control the third term in (A.3), note that the map θ : A b ⊗ A → B ( A ) defined by θ ( a ⊗ b )( z ) = azb is contractive. Therefore, since xv = θ ( x · ∆)( f ) and vx = θ (∆ · x )( f ), k xv − vx k = k θ ( x · ∆ − ∆ · x )( f ) k ≤ k x · ∆ − ∆ · x k A b ⊗ A k f k ≤ δC. (A.5)8ence k x − vx k ≤ C + 1) δ ; provided that we chose δ to ensure 2( C + 1) δ ≤ ε , we haveobtained the desired v = v F,ε . This completes the proof of Lemma 2.2. (cid:3)
We briefly indicate how one can adapt this argument to prove the “approximately amenableversion” of Lemma 2.2 that was stated in Section 4. First, note that if J is a closed ideal in A then it remains a closed ideal in the unitization A ♯ ; therefore, by replacing A with A ♯ ifnecessary, we may assume that A has an approximate diagonal. Assume as before that J hasa right approximate identity bounded in norm by some constant C > ∈ A ⊗ A satisfying (A.1) and (A.5), although we have no control on the norm of ∆ itself.Nevertheless, writing ∆ = P mi =1 a i ⊗ b i , we may choose an f ∈ J with k f k ≤ C such that v := P mi =1 a i f b i satisfies k xπ (∆) − xv k ≤ δ for all x ∈ F . Hence, using (A.3), we have k x − vx k ≤ ( C + 2) δ for all x ∈ F , which is enough to obtain a left approximate identity( v F,ε ) for J . References [AH11] S. A. Argyros and R. G. Haydon. A hereditarily indecomposable L ∞ -space thatsolves the scalar-plus-compact problem. Acta Math. , 206(1):1–54, 2011.[Ald17] E. Aldabbas.
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Yemon Choi, Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF,United Kingdom. [email protected]@lancaster.ac.uk