aa r X i v : . [ m a t h . F A ] A ug Constructing alternating -cocycles on Fourieralgebras Y. Choi
Abstract
Building on recent progress in constructing derivations on Fourier algebras, we providethe first examples of locally compact groups whose Fourier algebras support non-zero,alternating 2-cocycles; this is the first step in a larger project. Although such 2-cocyclescan never be completely bounded, the operator space structure on the Fourier algebraplays a crucial role in our construction, as does the opposite operator space structure.Our construction has two main technical ingredients: we observe that certain estimatesfrom [12] yield derivations that are “co-completely bounded” as maps from various Fourieralgebras to their duals; and we establish a twisted inclusion result for certain operatorspace tensor products, which may be of independent interest.Keywords: alternating cocycle, co-completely bounded, Fourier algebra, Hochschild coho-mology, operator space, opposite operator space, tensor product.MSC 2020: 16E40, 46J10, 46L07 (primary); 43A30, 46M05 (secondary)
Fourier algebras of locally compact groups have been a fertile source of examples in the studyof general Banach function algebras, while also having some important applications to thestudy of operator algebras associated to group representations (see e.g. [8]). One theme witha long history is the study of how properties of a group G are reflected in properties of itsFourier algebra A( G ). For instance, if G is compact and non-abelian and f ∈ A( G ), thematrix-valued Fourier coefficients b f ( π ) must decay at a certain rate as π “tends to infinity”,which intuitively suggests that f should have a degree of differentiability or H¨older continuity.This heuristic underlies a theorem of Johnson that when G is SO(3) or SU(2), there are non-zero derivations from A( G ) to its dual; using general restriction theorems for Fourier algebras,it follows that the same is true for any G that contains a closed copy of SO(3) or SU(2).Johnson’s result went counter to some expectations at the time: for given a Lie group G and some X in its Lie algebra, the Lie derivative along X , viewed as a continuous operator on C ∞ c ( G ), does not extend to a continuous map A( G ) → C b ( G ). (If such an extension existed itwould yield non-zero continuous point derivations on A( G ), contradicting the fact that pointsof G are sets of synthesis for A( G ).) Enlarging the codomain from C b ( G ) to A( G ) ∗ allows ∂ X f to be a distribution rather than a function, but a separate averaging argument is needed For details, see the proof of [9, Theorem 7.4] or the discussion in [2, §
1o explain why f ∂ X f has any chance of being continuous from A( G ) to A( G ) ∗ . Despitethese technical difficulties, Johnson’s result has been greatly extended in recent years, by usingthe operator-valued Fourier transform for certain Type I groups to make explicit calculationsand estimates: see the papers [2], [3] and [12]. These papers were motivated by a conjectureposed by Forrest and Runde in [7], which predicted exactly which groups G allow non-zeroderivations A( G ) → A( G ) ∗ ; the conjecture was confirmed for all Lie groups in [12], and arecent preprint of Losert [14] contains a solution in full generality.For any commutative Banach algebra A , the space of derivations A → A ∗ coincides withthe first Hochschild cohomology group H ( A, A ∗ ). The higher-degree groups H n ( A, A ∗ ) po-tentially capture more information about A , but have proved to be extremely difficult tocalculate except in degenerate cases. A more promising approach to finding computable in-variants, which was first pursued systematically in [10], is to consider the alternating part of H n ( A, A ∗ ); alternating cocycles are more tractable than general ones since they are built fromderivations (and there is also conceptual motivation for singling out this class, see Remark 3.4below). Nevertheless, the existence of non-zero alternating cocycles on a Banach functionalgebra is very sensitive to properties of the given norm, and is not guaranteed by simplyhaving “enough derivations”, as illustrated in Example 3.7 below.Given the recent progress in studying derivations on Fourier algebras, it is natural to turnour attention to alternating cocycles on A( G ). The present paper is the first step in getting thislarger programme off the ground, by producing the first examples of groups whose Fourieralgebras support non-zero alternating 2-cocycles. In fact, we show that not only do suchgroups exist, but they occur in abundance among the classical Lie groups. Theorem . Let n ≥ and let G be one of the groups SU( n ) , SL( n, R ) or Isom( R n ) . Thenthere is a non-zero, continuous, alternating -cocycle on A( G ) . Theorem 1.1 is a special case of a much more general result, which in turn follows bycombining Theorems 5.7 and 6.3 below. The starting point for the proof of Theorem 5.7 isthe canonical identification of A( H × L ) with the operator space projective tensor productof A( H ) and A( L ), valid for any locally compact groups H and L . However, it should beemphasised that our proof requires more than a merely formal use of operator spaces andcompletely bounded maps; one can show that there are no completely bounded, non-zero,alternating cocycles on Fourier algebras.The key extra ingredient in the proof of Theorem 5.7 is the following surprising phe-nomenon, which may have independent interest for those working in operator space theory. Theorem . Let X and Y be operator spaces, and let e Y denote Y equipped with its opposite operator-space structure . Let b ⊗ and q ⊗ denote the projective andinjective tensor products of operator spaces. Then k w k X q ⊗ e Y ≤ k w k X b ⊗ Y for all w ∈ X ⊗ Y .That is: the identity map on X ⊗ Y extends to a contractive linear map θ X,Y : X b ⊗ Y → X q ⊗ e Y . The opposite operator space structure may be thought of, intuitively, as a “mirror image”of the given one. One of the background aims of this paper is to argue that in applyingoperator-space methods to the study of Fourier algebras, it may be useful to work simultane-ously with both the canonical operator space structure on A( G ) and its mirror image.2 .2 An overview of the main technical difficulties We now sketch why the natural attempt to prove Theorem 1.1, by merely following algebraicrecipes in an appropriate functional-analytic category, does not work, and indicate the newideas needed to overcome these difficulties. None of the material here is logically necessaryfor the proof of Theorem 1.1 or Theorem 1.2, and it may be skipped if the reader wishes toget straight to the precise mathematical details.We start with a simple example from commutative algebra.
Example . Consider the C -algebra C [ z, w ] and define F : C [ z, w ] × C [ z, w ] → C [ z, w ] by F ( f , f ) = ∂f ∂w ∂f ∂z − ∂f ∂w ∂f ∂z ( f , f ∈ C [ z, w ]) . Then F is an alternating 2-cocycle (on C [ z, w ] with coefficients in C [ z, w ]). Note that C [ z, w ] ∼ = C [ z ] ⊗ C [ w ], and we may view ∂/∂z as ( d/dz ) ⊗ ι and ∂/∂w as ι ⊗ ( d/dw ).This is a special case of a general algebraic construction: given two commutative C -algebras A and B , a symmetric A -bimodule X and a symmetric B -bimodule Y , and derivations D A : A → X and D B : B → Y , we can “wedge together” the two amplified derivations D A ⊗ ι B : A ⊗ B → X ⊗ B and ι A ⊗ D B : A ⊗ B → A ⊗ Y to obtain an alternating 2-cocycle F : ( A ⊗ B ) × ( A ⊗ B ) → X ⊗ Y . Under mild conditions on D A and D B , F will be non-zero.Therefore, in cases where we have non-zero derivations from A( H ) and A( L ) into ap-propriate modules, one could try to obtain alternating 2-cocycles on A( H × L ) by applyingthe natural analogue of this construction for the category of Banach spaces. Unfortunatelythis would only yield a non-zero 2-cocycle on the Banach space projective tensor productA( H ) b ⊗ γ A( L ), and the natural map A( H ) b ⊗ γ A( L ) → A( H × L ) is never surjective in suchcases. On the other hand, A( H × L ) can be identified with the operator-space projectivetensor product A( H ) b ⊗ A( L ). But in the natural analogue of the algebraic construction forthe category of operator spaces, one must start with completely bounded derivations fromA( H ) and A( L ) into symmetric “c.b.-bimodules”, and it was shown independently by Spronkand Samei that the only such derivations are identically zero: see [17, Theorem 5.2] or [18].To resolve this apparent stalemate we need a new approach, which is to examine whatoccurs if we start from a pair of non-zero derivations D H : A( H ) → A( H ) ∗ and D L : A( L ) → A( L ) ∗ that become completely bounded after changing the operator space structure on thecodomains. Although this breaks certain aspects of the algebraic construction, somethingsurvives if D H and D L are completely bounded when A( H ) ∗ and A( L ) ∗ are equipped withtheir opposite operator space structures (it is an unrecorded observation of the author andGhandehari that this property holds for the derivations constructed in [2]). For then, as weshall show in Proposition 5.2, combining D H and D L and doing some careful book-keepingyields a bounded (but not completely bounded) bilinear map F : A( H × L ) × A( H × L ) −→ A( H ) ∗ b ⊗ [A( L ) ∗ ] ∼ (1.1)which behaves like an alternating 2-cocycle on the dense subalgebra A( H ) ⊗ A( L ).3his is still not enough to obtain Theorem 1.1, since the right-hand side of (1.1) is notan A( H × L )-bimodule in the cases where D H and D L exist with the required properties.The saving grace is Theorem 1.2, which allows us to embed this space continuously (butnot completely boundedly!) into A( H × L ) ∗ . Note that in general X b ⊗ Y and X q ⊗ e Y are incomparable as operator spaces (just take X = C ), and so Theorem 1.2 does not seem tofollow just from the universal/extremal properties of b ⊗ and q ⊗ in the category of operatorspaces. Instead, our proof proceeds by embedding X and Y into B ( E ) and B ( F ) for Hilbertspaces E and F , which allows us to calculate or bound various tensor norms on B ( E ) ⊗ B ( F )by viewing elements of this space as elementary operators on Schatten classes.It is striking that to prove Theorem 1.1, which on the face of it makes no reference tooperator spaces and completely bounded maps, we are driven to make substantial use of suchtechniques. Let us now describe the organization of the rest of this paper. In Section 2 we establish someglobal conventions for our notation, and set up the definition of A( G ) that is most suitable forthis paper. In Section 3 we give the key definitions of derivations and alternating 2-cocyclesfor commutative Banach algebras, illustrating the general definitions with some key examplesthat will motivate our proof of Theorem 5.7. We also record some basic constructions thatwere not mentioned in [10], as they may be useful for subsequent work.Section 4 has two purposes. We introduce the key notion of a co-completely bounded map (co-cb for short) between two given operator spaces; and we collect some results concerningthe canonical operator space structure on A( G ), some of which are only stated implicitly inthe literature. In doing so, we spend time on the crucial notion of the opposite operator spacestructure and its functorial properties; this requires us to set out some basic properties thatdo not seem to be mentioned explicitly in [5] or [15].Section 5 contains the main work needed to establish Theorem 1.1. En route, we give theproof of Theorem 1.2, reducing the problem to a special case which is handled by means of aninterpolation argument. The section ends by stating and proving the main technical theoremof this paper, Theorem 5.7, which says loosely speaking that we can construct non-zero 2-cocycles given enough non-zero co-cb derivations.For some groups where non-zero derivations have been constructed, the co-cb propertycan be read out of the explicit formulas in [2]; but in fact those cases and more besides canbe obtained by repurposing some technical results from [12]. Details are given in Section 6,culminating in Theorem 6.3 which provides co-cb derivations in all cases needed to establishTheorem 1.1. Finally, in Section 7 we make some remarks and pose some questions, with aview to possible avenues for future work.We have attempted to make this paper accessible to workers in the general area of Banachalgebras, or functional analysts interested in structural properties of particular Banach alge-bras. In particular, we have not assumed any prior familiarity with either Fourier algebras oflocally compact groups, or the Hochschild cohomology groups of Banach algebras, and havetried to include a small amount of extra motivation for these objects. On the other hand, wedo assume some previous exposure to the basic language of operator spaces and completelybounded maps. 4 Preliminaries
Throughout this article, all derivations and cocycles from Banach algebras into Banach bimod-ules are tacitly assumed to be norm-continuous. This is the convention adopted, for instance,in [10], and this article will not be concerned with any issues of automatic continuity.The algebraic tensor product of two complex vector spaces E and F is denoted by E ⊗ F .The term “map” is our short-hand for “linear map” or “linear operator”.All Banach spaces are defined over complex scalars. For a Banach space E , B ( E ) denotesthe algebra of bounded linear operators on E . The adjoint of a bounded linear map f : E → F between Banach spaces is denoted by f ∗ : F ∗ → E ∗ , with one important exception: if E and F are Hilbert spaces, then we shall denote the adjoint map F ∗ → E ∗ by f , to avoid confusionwith the adjoint in the sense of operators between Hilbert spaces.One slight departure from usual conventions is that when E is a Hilbert space, we shallformulate various constructions in terms of the dual space E ∗ rather than the conjugatespace E ; of course the two spaces are canonically isomorphic as Banach spaces via the Riesz–Fr´echet theorem. (This decision is motivated by issues concerning operator space structures,but is ultimately only a matter of notational preference.) When E = L (Ω , µ ) for somemeasure space (Ω , µ ) and η ∈ L (Ω , µ ) we shall write ev η for the functionalev η : ξ
7→ h ξ, η i L (Ω ,µ ) = Z Ω ξη dµ (2.1)so that η ev η is a linear isomorphism of Banach spaces L (Ω , µ ) → L (Ω , µ ) ∗ .The projective tensor product of Banach spaces E and F is denoted by E b ⊗ γ F ; theHilbertian tensor product of Hilbert spaces V and W is denoted by V ⊗ W .Our notational conventions for operator spaces and completely bounded maps will be setout in Section 4, since they are not needed until Section 5. Fourier algebras originated from the study of L -group algebras of locally compact abelian (LCA) groups. Given a LCA group Γ with Pontrjagin dual G = b Γ, the Fourier transform F : L (Γ) → C ( G ) is an injective algebra homomorphism, and so one may study the convolutionalgebra L (Γ) by examining the function algebra F ( L (Γ)) equipped with the norm pushedforwards from L (Γ). This function algebra, denoted by A( G ), is now known as the Fourieralgebra of G .Using Bochner’s theorem, one can characterize A( G ) in terms of positive-definite functionson G , without reference to the group Γ. Guided by this philosophy, and following work ofGodement and Stinespring in the unimodular case, Eymard [6] gave a definition of A( G ) thatis valid for any locally compact group G ; details of the foundational results from Eymard’soriginal paper, and much more besides, may be found in the recent book [11]. However, formost of this paper, it is more convenient to work with an alternative description of A( G ). Theone we use is also standard, and can be found in e.g. [19, Defn. VII.3.8], but our presentationhas some cosmetic differences from the usual one since we wish to work with duals of Hilbertspaces rather than their conjugates. 5ix a choice of left Haar measure on G , denoted by ds , and let λ : G → U ( L ( G ))denote the left regular representation of G on L ( G ) defined by [ λ ( x ) f ]( s ) = f ( x − s ). Given ξ, η ∈ L ( G ) and x ∈ G , letΨ( ξ ⊗ ev η )( x ) = h λ ( x ) ξ, η i L ( G ) = Z G ξ ( x − s ) η ( s ) ds . (2.2)This defines a contractive linear map Ψ : L ( G ) b ⊗ γ L ( G ) ∗ → C ( G ) whose range we denoteby A( G ). We equip A( G ) with the quotient norm of L ( G ) b ⊗ γ L ( G ) ∗ / ker(Ψ). Using Fell’sabsorption theorem, it can be shown that A( G ) is closed under pointwise product and thenorm on A( G ) is submultiplicative. (See e.g. [21, § G ) is a Banach algebra of functions on G , called the Fourier algebra of G . Theequivalence of this definition with the original one can be extracted from the results in [6,Chapitre 3] (see also [11, Proposition 2.3.3]). Note that with our definition, if G is a LCAgroup then we recover the isomorphism L ( b G ) ∼ = A( G ) using Parseval’s theorem.Given Banach algebras A and B and a continuous homomorphism A → B , derivations andcocycles on B can be pulled back to give derivations and cocycles on A (a precise statementwill be given in Lemma 3.5). It is therefore useful to identify homomorphic images of A( G )which have a simpler form, so that we can build derivations or cocycles on those algebrasinstead. The following result was proved by Herz in a more general setting, and is usuallyknown as Herz’s restriction theorem . Theorem . If G is a locally compact group and G is a closed sub-group, then restriction of functions C ( G ) → C ( G ) defines a norm-decreasing homomor-phism A( G ) → A( G ) which is a quotient map of Banach spaces. For a detailed proof and some historical comments, see [11, § § G to be non-abelian; Fourier algebras of abelian groupsare amenable (in the Banach-algebraic sense) and hence all cohomology with dual-valuedcoefficients vanishes.In concrete cases, such as those in Theorem 1.1, once we have constructed a well-definedcocycle on a particular Fourier algebra it will be obvious from context that this cocycle is notidentically zero. However, in order to state our main technical theorem (Theorem 5.7) in itsgreatest generality, it will be convenient to use the following lemma as a “soft” work-around. Lemma . Let G be a locally compact group. Then { a : a ∈ A( G ) } and { b : b ∈ A( G ) } bothhave dense linear span in A( G ) .Proof. Let A = A( G ) ∩ C c ( G ). A is a dense subalgebra of A( G ) (this is immediate fromEymard’s original definition, but also follows easily from the one given above). Moreover, foreach f ∈ A there exists g ∈ A such that f g = f ; see [6, Lemme 3.2] or [11, Prop. 2.3.2].Hence, by the usual polarization identity ab = 14 (cid:2) ( a + b ) − ( a − b ) (cid:3) , we have A ⊆ lin { b : b ∈ A } . (The converse inclusion also holds.) Similarly, combining theidentity x y = 124 (cid:2) ( x + y ) + ( x − y ) − ( x + iy ) − ( x − iy ) (cid:3) A ⊆ lin { b : b ∈ A } ⊆ lin { a : a ∈ A } .(Once again the converse inclusion is trivial.) Thus, both lin { a : a ∈ A( G ) } and lin { b : b ∈ A( G ) } contain A and hence are dense in A( G ). -cocycles on commutative Banach algebras We assume familiarity with the basic language of Banach algebras and Banach bimodules overthem. Let A be a Banach algebra and X a Banach A -bimodule. For n ≥ C n ( A, X ) be thespace of bounded n -multilinear maps A ×· · ·× A → X , with the convention that C ( A, X ) = X .There are maps δ n : C n ( A, X ) → C n +1 ( A, X ), called the
Hochschild coboundary operators , thatsatisfy δ n +1 ◦ δ n = 0 for all n ≥
0. We only need the cases n = 1 and n = 2, which have thefollowing explicit form:– for T ∈ C ( A, X ) and a, b ∈ A , δ T ( a, b ) := a · T ( b ) − T ( ab ) + T ( a ) · b ;– for F ∈ C ( A, X ) and a, b, c ∈ A , δ T ( a, b, c ) = a · T ( b, c ) − T ( ab, c ) + T ( a, bc ) − T ( a, b ) · c .Let Z n ( A, X ) := ker δ n ; elements of this space are called n -cocycles. Note that 1-cocyclesare the same as derivations.
Let B n ( A, X ) = im δ n − ; elements of this space are called n -coboundaries. The quotient space H n ( A, X ) := Z n ( A, X ) / B n ( A, X ) is the n th Hochschildcohomology group. For the rest of this section, we only consider commutative Banach algebras A , and thoseBanach A -bimodules X which are symmetric in the sense that a · x = x · a for all a ∈ A and x ∈ X . Note that if A is commutative, then both A itself and its dual A ∗ are symmetricBanach A -bimodules.Let S n denote the symmetric group on { , . . . , n } , and let σ ( − σ denote the signaturehomomorphism S n → {± } . A given T ∈ C n ( A, X ) is called symmetric if T ( a , . . . , a n ) = T ( a σ (1) , . . . , a σ ( n ) ) for all a , . . . , a n ∈ A and σ ∈ S n and alternating if T ( a , . . . , a n ) = ( − σ T ( a σ (1) , . . . , a σ ( n ) ) for all a , . . . , a n ∈ A and σ ∈ S n .We write Z n alt ( A, X ) for the space of all alternating n -cocycles. Remark . In the case n = 2, every T ∈ C ( A, X ) is the sum of a symmetric part and analternating part. Also, every 2-coboundary is symmetric (since A is commutative and X issymmetric). Thus the natural map Z ( A, X ) → H ( A, X ) is actually an injection .Following [10, Definition 2.2], we say that T ∈ C n ( A, X ) is an n -derivation if it is aderivation in each variable separately. If T is either symmetric or alternating, then to verifythe n -derivation property it suffices to check that T is a derivation in the first variable, i.e. tocheck the identity T ( bc, a , . . . , a n ) = b · T ( c, a , . . . , a n ) + T ( b, a , . . . , a n ) · c for all b, c, a , . . . a, n ∈ A .7iven our earlier definitions, a straightforward calculation shows that every 2-derivation is a2-cocycle. (It is crucial here that A is commutative and X is symmetric.) In particular, everyalternating 2-derivation defines an element of Z ( A, X ).The alternating 2-cocycles that we shall construct when proving Theorem 1.1 are createdas alternating 2-derivations. For sake of completeness, we mention that every alternating2-cocycle turns out to be a derivation in the first variable, and hence (by the remarks above)is an alternating 2-derivation. We omit the details, since this result is the n = 2 case of[10, Theorem 2.5]. Note also that in the introduction, and in the statement of Theorem 1.1,we restricted ourselves to cocycles on A taking values in A ∗ . This is no loss of generality:by the n = 2 case of [10, Corollary 2.11], if there exists some X with Z ( A, X ) = 0 then Z ( A, A ∗ ) = 0.The following example illustrates what the abstract definitions above mean in practice,and provides motivation for later constructions. Example C ( T ) and C ( T )) . (i) Define a linear map D : C ( T ) → C ( T ) ∗ by D ( f )( f ) = Z T ∂f ∂θ ( p ) f ( p ) dp where p = e πiθ and dp denotes the usual uniform measure on T . It follows immediatelyfrom the product rule that D is a derivation.(ii) Define a bilinear map F : C ( T ) × C ( T ) → C ( T ) ∗ by F ( f , f )( f ) = Z T (cid:18) ∂f ∂θ ( p ) ∂f ∂θ ( p ) − ∂f ∂θ ( p ) ∂f ∂θ ( p ) (cid:19) f ( p ) dp where p = ( e πiθ , e πiθ ) and dp denotes the usual uniform measure on T . Clearly F is alternating as a bilinear map, and a similar calculation to part (i) shows it is aderivation in the first variable; thus it is an alternating 2-cocycle. Remark . We noted earlier that an alternating 2-cocycle is a coboundary if and only if itis identically zero. This is true for all n ≥ injection of the vector space Z n alt ( A, X ) into the n th Hochschild cohomology group H n ( A, X ).The range of this injection is one summand in a canonical decomposition of H n ( A, X ) into n pieces; for further discussion of this decomposition, see e.g. [1, §
3] and [20, § Remark . We briefly leave the world of Banach algebras to mention some importantcontext from algebraic geometry. If R is the coordinate ring of a smooth complex variety,every cocycle on R with symmetric coefficients is equivalent to an alternating one; this is oneversion of the Hochschild–Kostant–Rosenberg theorem.
For instance, this applies to the complexco-ordinate ring of the algebraic group SL , which occurs naturally as a dense subalgebra ofA(SU(2)). While the HKR theorem itself does not seem to extend to the Banach-algebraicsetting, it suggests that the alternating cocycles on commutative Banach algebras have somedeeper meaning, rather than being ad hoc definitions, and hence deserve further study.8 .2 Tools for constructing alternating cocycles Following a strategy analogous to those used in [2, 3, 12], we shall prove Theorem 1.1 by estab-lishing the non-vanishing of Z (A( G ) , A( G ) ∗ ) for some judiciously chosen closed subgroup G ⊂ G . We record the following lemma for later reference. Lemma . Let A and B be commutative Banach algebras and let θ : A → B be a continuoushomomorphism. Then for any F ∈ Z ( B , B ∗ ) , the induced map θ ∗ F defined by θ ∗ F ( a , a )( a ) := F ( θ ( a ) , θ ( a ))( θ ( a )) belongs to Z ( A , A ∗ ) . If F = 0 and θ has dense range, then θ ∗ F = 0 . The proof follows easily from the definitions and we omit the details.As mentioned in Section 1.2, in commutative algebra there is a standard procedure forconstructing alternating 2-cocycles on a tensor product of two algebras, given a pair of deriva-tions on the respective algebras. With minor modifications, one can do the same in the settingof commutative Banach algebras and symmetric Banach bimodules. This observation is surelyknown to specialists in the cohomology of Banach algebras, but we have not found an explicitstatement in the literature; this is somewhat surprising since it provides a natural converseto a special case of [10, Theorem 3.6]),
Lemma . Given commutative Banach algebras A and B , symmetricBanach bimodules X and Y over A and B respectively, and derivations D A : A → X , D B : B → Y , the formula F ( a ⊗ b , a ⊗ b ) := [ D A ( a ) · a ] ⊗ [ b · D B ( b )] − [ a · D A ( a ) ⊗ D B ( b ) · b ] (3.1) defines an alternating -cocycle F : A b ⊗ γ B × A b ⊗ γ B → X b ⊗ γ Y . Since some of the relevant calculations will recur when we come to prove Theorem 5.7, weprovide a detailed proof.
Proof.
Since D A and D B are bounded, F extends to a continuous bilinear map( A b ⊗ γ B ) × ( A b ⊗ γ B ) → X b ⊗ γ Y , by the universal property of b ⊗ γ . Given a , a , a ∈ A and b , b , b ∈ B , direct calculation yields F ( a ⊗ b , a ⊗ b ) + F ( a ⊗ b , a ⊗ b ) = 0(using the fact that X and Y are symmetric bimodules) and F ( a a ⊗ b b , a ⊗ b ) = ( [ D A ( a a ) · a ] ⊗ [ b b · D B ( b )] − [ a a · D A ( a ) ⊗ D B ( b b ) · b ]= [ D A ( a ) · a a ] ⊗ [ b b · D B ( b )]+[ D A ( a ) · a a ] ⊗ [ b b · D B ( b )] − [ a a · D A ( a ) ⊗ D B ( b ) · b b ] − [ a a · D A ( a ) ⊗ D B ( b ) · b b ]= F ( a ⊗ b , a a ⊗ b b ) + F ( a ⊗ b , a a ⊗ b b )9using the fact that D A and D B are derivations into symmetric bimodules). Hence, by linearityand continuity, F is both alternating and a derivation in the first variable, so by the earlierremarks in this section it is an alternating 2-cocycle as required.The formula (3.1) should be compared with Example 3.2. In that example, F was obtainedby “wedging together” two derivations defined on the dense subalgebra C ( T ) ⊗ C ( T ), onebeing a copy of D acting in the θ direction and the other being a copy of D acting in the θ direction. Lemma 3.6 may be regarded as an abstract analogue of this construction. However,since C ( T ) b ⊗ γ C ( T ) = C ( T ), the lemma does not suffice on its own to construct alternating2-cocycles on C ( T ). Indeed, the next example shows that for some function algebras on T ,the approach suggested by Lemma 3.6 cannot possibly work. Example . For α ∈ (0 ,
1) and n ∈ N , consider the little Lipschitz algebra lip α ( T n ) ≡ lip( T n , d ( · ) α ). Classical Fourier analysis tells us that the trigonometric polynomials are densein lip α ( T n ). Moreover, given f and f in lip α ( T ), a straightforward calculation shows that f ⊗ f ∈ lip α ( T ). Thus we may identify the algebraic tensor product lip α ( T ) ⊗ lip α ( T ) witha dense subalgebra R ⊂ lip α ( T ). 0 < α ≤ / / < α ≤ / / ≤ α < Z (lip α ( T ) , lip α ( T ) ∗ ) zero non-zero non-zero Z (lip α ( T ) , lip α ( T ) ∗ ) zero zero zero Z (lip α ( T ) , lip α ( T ) ∗ ) zero non-zero non-zero Z (lip α ( T ) , lip α ( T ) ∗ ) zero zero non-zeroFigure 1: 1-cocycles and 2-cocycles for lip α ( T ) and lip α ( T )We now appeal to some consequences of [10, Corollary 4.4]; the relevant information isdisplayed in Figure 3.7. When 1 / < α <
1, lip α ( T ) has a non-zero derivation into its dual,and so using Lemma 3.6 we can construct an F ∈ Z (lip α ( T ) b ⊗ γ lip α ( T ) , lip α ( T ) ∗ b ⊗ γ lip α ( T ) ∗ ),which by inspection is non-zero when restricted to R .Since lip α ( T ) ∗ b ⊗ γ lip α ( T ) ∗ embeds continuously in lip α ( T ) ∗ , we therefore have a natural,non-zero, densely-defined alternating 2-cocycle on lip α ( T ) for all α ∈ (1 / , α ∈ (1 / , / Z (lip α ( T ) , lip α ( T ) ∗ ), sincethe latter space vanishes.Lemma 3.6 can still be applied to produce alternating 2-cocycles on A( H ) b ⊗ γ A( L ). How-ever, this is not enough to produce cocycles on A( H × L ), because of the following two facts.1) The natural map A( H ) b ⊗ γ A( L ) → A( H × L ) is surjective if and only if either H or L has an abelian subgroup of finite index [13]. (See also [11, § b ⊗ instead of b ⊗ γ .) This embedding extends to a continous map φ : lip α ( T ) b ⊗ γ lip α ( T ) → lip α ( T ). In fact, φ is injective, butthe proof requires some technical facts from Banach space theory which would distract us here.
10) If H has an abelian subgroup of finite index, then the only (continuous) derivationA( H ) → A( H ) ∗ is the zero map. (In fact it suffices that the connected component of H be abelian; see [7, Theorem 3.3] or [11, § This section is devoted to the infrastructure needed for the proof of Theorem 5.7. We payparticular attention to issues of functoriality; the reason for introducing operator space tensorproducts and the opposite operator space structure is not just to equip Banach spaces withextra structure, but to be able to combine linear maps that respect this extra structure.
All concepts not defined explicitly here can be found in standard sources, such as the earlychapters of [5] or [15].Henceforth, we abbreviate the phrase “operator space structure” to o.s.s. Given operatorspaces X and Y , CB ( X, Y ) denotes the space of completely bounded maps X → Y ; note thatthis space has a canonical o.s.s., defined via the identification M n CB ( X, Y ) ∼ = CB ( X, M n Y ).Whenever H is a Hilbert space and we refer to B ( H ) as an operator space, we assume(unless explicitly stated otherwise) that it is equipped with its usual, canonical o.s.s.; notethat if we do this, then there is a natural and completely isometric identification of B ( H )with CB ( COL H ), where COL H denotes H equipped with the column o.s.s. Tensor products and tensor norms.
The projective and injective tensor products of operatorspaces are denoted by b ⊗ and q ⊗ respectively (this is the notation of [5], rather than that of [15]).Note that if E and F are Hilbert spaces then the underlying Banach space of COL E b ⊗ ( COL F ) ∗ is E b ⊗ γ F ∗ , but the underlying Banach space of COL E b ⊗ COL F is E ⊗ F .Given operator spaces V , W and X , we say that a bilinear map V × W → X is jointlycompletely bounded (j.c.b.) if it extends to a completely bounded map V b ⊗ W → X .This is equivalent to saying that the “curried map” V → L ( W, X ) extends to a completelybounded map V → CB ( W, X ), or the same with V and W interchanged. Indeed V b ⊗ W maybe characterized, up to completely isometric isomorphism, as the completion of V ⊗ W thatsatisfies CB ( V b ⊗ W, X ) ∼ = CB ( V, CB ( W, X )) ∼ = CB ( W, CB ( V, X )) completely isometrically. ( ♦ )If f ∈ CB ( E, X ) and g ∈ CB ( F, Y ) then by tensoring we obtain completely boundedmaps E b ⊗ F → X b ⊗ Y and E q ⊗ F → X q ⊗ Y ; for extra emphasis, these maps will be denotedby f b ⊗ g and f q ⊗ g respectively.We shall make passing use of the Haagerup tensor norm, but only “at level 1”, and weonly require the following facts:1) for any w ∈ E ⊗ F we have k w k E q ⊗ F ≤ k w k E ⊗ h F ≤ k w k E b ⊗ F ; This seems to now be the accepted terminology, and agrees with [15]. One should beware that in [5] suchbilinear maps are called “completely bounded”, which in most later sources is instead used for those mapslinearized by the Haagerup tensor product.
11) if A and B are C ∗ -algebras and w ∈ A ⊗ B , then k w k A ⊗ h B = inf n ∈ N inf (cid:26)(cid:13)(cid:13)(cid:13)X nj =1 a ∗ j a j (cid:13)(cid:13)(cid:13) / (cid:13)(cid:13)(cid:13)X nj =1 b j b ∗ j (cid:13)(cid:13)(cid:13) / (cid:27) where the inner infimum is over all representations of w as P nj =1 a j ⊗ b j .For a proof of 1) see e.g. [5, Theorem 9.2.1]; for 2), see e.g. [15, Ch. 5]. The opposite operator space structure.
Given an operator space W , one may define a newsequence of matrix norms on W by (cid:13)(cid:13)X i a i ⊗ w i (cid:13)(cid:13) ( n ) , opp := k X i a ⊤ i ⊗ w i k ( n ) ( a i ∈ M n , w i ∈ W ) . (C.f. [15, § W with a new o.s.s., whichwe call the opposite o.s.s. ; the resulting operator space will be denoted by f W . In longerexpressions, when considering the opposite operator space, we use the notation ( . . . ) ∼ ; forinstance B ( H ) ∼ denotes B ( H ) equipped with the opposite of its usual o.s.s.Note that for a Hilbert space H , ( COL H ) ∼ = ROW H and ( ROW H ) ∼ = COL H . Moregenerally, ( f W ) ∼ = W . Remark . f W is often denoted in the literature by W op . We have chosen different notationbecause there is a potential conflict with the usage of A op to denote the “opposite algebra”,i.e. the algebra with the same underlying vector space but with reversed product. The twoconventions match happily if A = B ( H ) but are at odds if A = A( G ).It is easily checked that if f : X → Y is completely bounded, then so is f : e X → e Y , withthe same cb-norm. To emphasise the functorial behaviour we write this as e f : e X → e Y . Thesame calculation gives, with some book-keeping, a more precise result: we omit the details. Lemma . Given operator spaces X and Y , the assignment f e f defines a completely iso-metric isomorphism CB ( X, Y ) ∼ ∼ = CB ( e X, e Y ) . In particular, we can identify ( X ∗ ) ∼ with ( e X ) ∗ . Note that for any operator spaces V and W , the identity map on V ⊗ W extends toa completely isometric isomorphism e V b ⊗ f W ∼ = ( V b ⊗ W ) ∼ . One can show this using theexplicit definition of the matrix norms associated to b ⊗ , but it can also be deduced from thecharacterization in ( ♦ ), combined with repeated application of Lemma 4.2. Co-cb maps between operator spaces.
The next definition is non-standard (though it hassome precedent in [16]) but will be extremely useful for statements and calculations later on. Definition . Let V and W be operator spaces and let f : V → W be a linear map. Note that f is c.b. from V to f W if and only if it is c.b. from e V to W ;in either case we say that f : V → W is co-completely bounded ( co-cb for short). Similarly, f is a complete isometry from V to f W if and only if it is a complete isometry from e V to W ; wethen say that f : V → W is a co-complete isometry. In [16] this concept is called “completely co-bounded”, but then abbreviated to “co-cb” just as we havedone. Our terminology is chosen to avoid potential confusion with cohomology theory (“cobounded”) oroperator theory (“coisometry”).
Lemma . Let F be a Hilbert space. The C -linear map B ( F ) → B ( F ∗ ) that sends anoperator b ∈ B ( F ) to its Banach-space adjoint b : F ∗ → F ∗ is a co-complete isometry. Warning: as we will see in the proof, our chosen notational conventions are important: B ( F ∗ ) is given the o.s.s. of CB ( COL F ∗ ) rather than CB (( COL F ) ∗ ) = CB ( ROW F ∗ ). Proof.
For any operator spaces X and Y , the map CB ( X, Y ) → CB ( Y ∗ , X ∗ ) defined bytaking adjoints is a complete isometry. Thus b b defines a complete isometry B ( F ) ≡ CB ( COL F ) → CB (( COL F ) ∗ ) = CB ( ROW F ∗ ) = CB (( COL F ∗ ) ∼ )Taking opposites and applying Lemma 4.2, the result follows. Remark . Consider L (Ω) for some measure space(Ω , µ ) (we suppress mention of the measure µ for notational convenience), and let α : η ev η denote the canonical, linear , isometric isomorphism L (Ω) → L (Ω) ∗ that is definedvia Equation (2.1). This defines a normal ∗ -isomorphism Ad α : B ( L (Ω) ∗ ) → B ( L (Ω)).Calculation shows that composing Ad α with the map B ( L (Ω)) → B ( L (Ω) ∗ ) fromLemma 4.4 yields a linear , co-complete isometry ⊤ : B ( L (Ω) → B ( L (Ω)) , b b ⊤ , where b ⊤ ( ξ ) := b ∗ ξ for each ξ ∈ L (Ω).Note that if Ω is countable and infinite, equipped with counting measure, then ⊤ is justthe usual “transpose” operator for an infinite matrix. The results in this section are all known to specialists, but are included here for the reader’sconvenience, and to ensure that we have consistent notation and conventions.Given a Hilbert space H and a “concrete” von Neumann algebra M ⊂ B ( H ): the predual M ∗ is a natural quotient of B ( H ) ∗ = COL H b ⊗ ( COL H ) ∗ , and may thus be equipped withthe quotient o.s.s.; moreover, if we take the dual of this o.s.s. on M ∗ , we recover the originalsubspace o.s.s. on M . (See [5, Prop. 4.2.2].) Thus ( M ∗ ) ∗ ∼ = M completely isometrically.In particular, consider the group von Neumann algebra VN( G ) ⊂ B ( L ( G )). Then one canidentify A( G ) with the unique isometric predual of VN( G ). Our chosen definition for A( G )allows one to deduce this quickly by considering the adjoint of the map in Equation (2.2),Ψ ∗ : A( G ) ∗ → ( L ( G ) b ⊗ γ L ( G ) ∗ ) ∗ ∼ = B ( L ( G )) , observing that for each s ∈ G , Ψ ∗ maps the character ev s to the translation operator λ ( s ).See [11, Lemma 2.8.2] for details. We now take, as our canonical o.s.s. on A( G ), the oneinduced by Ψ from VN( G ) ∗ . This is standard knowledge but we could not locate an explicit statement of this result in the literature. Itfollows easily from the completely isometric identifications CB ( Y ∗ , X ∗ ) ∼ = CB ( Y ∗ b ⊗ X, C ) ∼ = CB ( X, Y ∗∗ ), sincethe map CB ( X, Y ) → CB ( Y ∗ , X ∗ ) then corresponds to the canonical inclusion CB ( X, Y ) → CB ( X, Y ∗∗ ). emark . Care is needed when combining parts of the literature. Somesources, following the general framework of locally compact quantum groups, define A( G )to be the subspace of C ( G ) obtained by identifying a vector functional ω ξ,η ∈ VN( G ) ∗ with the function s
7→ h λ ( s − ) ξ, η i L ( G ) . While this gives the same Banach function algebraA( G ) ⊂ C ( G ) as in this paper, it yields the opposite o.s.s. to the one we have just defined.This is related to Proposition 4.8(d) below.Given f ∈ C G , let f ∨ ( x ) = f ( x − ). For any ξ, η ∈ L ( G ) and x ∈ G we haveΨ( ξ ⊗ ev η )( x − ) = Z G ξ ( xs ) η ( s ) ds = Z G η ( x − s ) ξ ( s ) ds = Ψ( η ⊗ ev ξ )( x ) . It follows that the map f f ∨ defines a contractive involution on A( G ) (which must thereforebe isometric). This is known as the flip map or check map on A( G ). Remark . Direct calculations show that when we identify A( G ) ∗ with VN( G ), the adjointof the check map on A( G ) coincides with the restriction to VN( G ) of the transpose operator ⊤ : B ( L ( G )) → B ( L ( G )). This observation is folklore (and a similar calculation may befound in the proof of [7, Prop. 1.5(ii)]). Proposition . Let G , G and G be locally compact groups. (a) The natural inclusion A( G ) ⊗ A( G ) → A( G × G ) extends to a completely isometricisomorphism A( G ) b ⊗ A( G ) ∼ = A( G × G ) . (b) The natural inclusion
VN( G ) ⊗ VN( G ) → VN( G × G ) extends to a complete isometry VN( G ) q ⊗ VN( G ) → VN( G × G ) , whose range is the minimal C ∗ -tensor product VN( G ) ⊗ min VN( G ) . (c) The check map on A( G ) is completely bounded if and only if G has an (open) abeliansubgroup of finite index. (d) The check map on A( G ) is a co-complete isometry.Proof. Parts (a) and (b) are general results about von Neumann algebra preduals and C ∗ -algebras; see e.g. [5, Theorem 7.2.4] and [5, Proposition 8.1.6]) respectively. Part (c) is statedas [7, Prop. 1.5(ii)] where the authors give a complete proof, while making it clear that theresult was already known to previous specialists. Part (d) holds because the adjoint of thecheck map coincides with the transpose operator (see Remark 4.7), which is a co-completeisometry by Remark 4.5. -cocycles from co-cb derivations We start in some generality, since the preliminary results may be useful in subsequent work.The following terminology is not entirely standard, but is analogous to the more fa-miliar notions of completely contractive Banach algebra and completely contractive Banach(bi)module that have appeared in the literature. By a cb-Banach algebra , we mean an opera-tor space A equipped with a bilinear, j.c.b. and associative map A × A → A . Given such an A ,we define a cb-Banach A -bimodule to be an operator space X , equipped with an A -bimodulestructure such that the left action A × X → X and the right action X × A → X are both j.c.b.14learly A itself is a cb-Banach A -bimodule; it is also routine to check that if X is a cb-Banach A -bimodule, so is X ∗ when equipped with the dual o.s.s. These notions also interactwell with the “opposite o.s.s. functor”. If A is a cb-Banach algebra then so is e A ; and if X isa cb-Banach A -bimodule, e X is a cb-Banach e A -bimodule. Remark . Given a cb-Banach algebra A , the class of cb-Banach A -bimodules is usuallynot closed under the operation of taking opposites. For instance, suppose A is unital. If e A were a cb-Banach A -bimodule, then for each x ∈ A the orbit map a ax would be completelybounded from A to e A . Taking x = 1 A we conclude that the identity map on A is co-cb. Inparticular, if A = A( G ) for G compact, this would force G to be virtually abelian (combineparts (c) and (d) of Proposition 4.8). Proposition . Let A and B be cb-Banach algebras; let X be a cb-Banach A -bimodule and Y a cb-Banach B -bimodule. Let T A ∈ CB ( A, e X ) and T B ∈ CB ( B, e Y ) . Then, if we define F , F : ( A ⊗ B ) × ( A ⊗ B ) → X ⊗ Y by F ( a ⊗ b , a ⊗ b ) = [ T A ( a ) · a ] ⊗ [ b · T B ( b )] ,F ( a ⊗ b , a ⊗ b ) = [ a · T A ( a )] ⊗ [ T B ( b ) · b ] , both F and F extend to bounded bilinear maps ( A b ⊗ B ) × ( A b ⊗ B ) → X b ⊗ e Y .Proof. We will only give the proof for F ; the proof for F is very similar.Since T A : A → e X is completely bounded, so is f T A : e A → X . Therefore, if we put S ( a ⊗ b ⊗ a ⊗ b ) := T A ( a ) ⊗ b ⊗ a ⊗ T B ( b )we obtain a complete contraction S = f T A b ⊗ ι e B b ⊗ ι A b ⊗ T B : e A b ⊗ e B b ⊗ A b ⊗ B −→ X b ⊗ e B b ⊗ A b ⊗ e Y .
Also, since X is a cb-Banach A -bimodule and e Y is a cb-Banach e B -bimodule, putting R ( x ⊗ b ⊗ a ⊗ y ) = ( x · a ) ⊗ ( b · y )defines a complete contraction R : X b ⊗ e B b ⊗ A b ⊗ e Y → X b ⊗ e Y .Since e A b ⊗ e B = ( A b ⊗ B ) ∼ , the composite map RS defines a j.c.b. bilinear map from( A b ⊗ B ) ∼ × ( A b ⊗ B ) to X b ⊗ e Y , which agrees with F on ( A ⊗ B ) × ( A ⊗ B ). In particular, F extends to a bounded bilinear map (no longer completely bounded!) from ( A b ⊗ B ) × ( A b ⊗ B )to X b ⊗ e Y . Remark . The proof of Proposition 5.2 would have been much easier if T A ⊗ ι B ⊗ ι A ⊗ T B extended to a continuous linear map from ( A b ⊗ B ) b ⊗ γ ( A b ⊗ B ) to ( X b ⊗ A ) b ⊗ γ ( B b ⊗ Y ). However,we see no reason why this should always hold, as it requires an interchange/distributivity resultfor b ⊗ and b ⊗ γ .In view of the earlier formula (3.1), one would like to apply Proposition 5.2 with T A and T B being co-cb derivations into symmetric cb-bimodules. However, this stops short of producinggenuine 2-cocycles: the resulting bilinear map merely takes values in X b ⊗ e Y , and in view ofRemark 5.1 there is no reason to suppose that this is even a Banach A b ⊗ B -bimodule. (It is aBanach A b ⊗ γ B -bimodule, but that does not help us.) To go further, we need to move from X b ⊗ e Y to X q ⊗ Y , and this is where we require Theorem 1.2, whose proof we now turn to. Forconvenience we recall the statement of the theorem.15 heorem 1.2 ( reprise ). Let X and Y be operator spaces. Then k w k X q ⊗ e Y ≤ k w k X b ⊗ Y for all w ∈ X ⊗ Y , and so the identity map on X ⊗ Y extends to a contraction θ X,Y : X b ⊗ Y → X q ⊗ e Y .Proof of Theorem 1.2. We start by reducing to a special case.
Step 1.
Given a pair of operator spaces X and Y , let j X : X → B ( E ) and j Y : Y → B ( F )be completely isometric embeddings, for some Hilbert spaces E and F . Note that f j Y : e Y → B ( Y ) ∼ is also a complete isometry.Suppose we know Theorem 1.2 holds for the particular operator spaces B ( E ) and B ( F ).Then we have a diagram as shown in Figure 2, in which the left-hand vertical arrow is a(complete) contraction, while the right-hand vertical arrow is a (complete) isometry (since q ⊗ respects complete isometries). B ( E ) b ⊗ B ( F ) ∼ θ B ( E ) ,B ( F ) ✲ B ( E ) q ⊗ B ( F ) X b ⊗ e Yj X b ⊗ f j Y ✻ ................................................... ✲ X q ⊗ Yj X q ⊗ j Y ✻ Figure 2: An embedding trickMoreover, the diagram in Figure 2 “commutes on elementary tensors”. Hence, for any z ∈ X ⊗ Y , we have k z k X q ⊗ e Y = k ( j X ⊗ j Y )( z ) k B ( E ) q ⊗ B ( F ) ∼ ≤ k ( j X ⊗ j Y )( z ) k B ( E ) b ⊗ B ( F ) ≤ k z k X b ⊗ Y . Thus: if Theorem 1.2 holds for B ( E ) and B ( F ) , then it holds for all operator spaces. Step 2.
Observe that if E and F are Hilbert spaces and w ∈ B ( E ) ⊗ B ( F ), the norm of w in B ( E ) q ⊗ B ( F ) ∼ coincides with the norm of the associated elementary operator on S ( F, E ),the space of Hilbert-Schmidt operators F → E . This is a variation on a well-known fact inC ∗ -algebra theory that can be found in various sources; to avoid any notational ambiguity,we give a precise statement in the following lemma. Lemma . Define Φ : B ( E ) ⊗ B ( F ) → B ( S ( F, E )) by Φ ( a ⊗ b )( c ) = acb . Then for all w ∈ B ( E ) ⊗ B ( F ) we have k Φ ( w ) k B ( S ( F,E )) = k w k B ( E ) q ⊗ B ( F ) ∼ . (5.1) Proof.
Recall that if b ∈ B ( F ) then b : B ( F ∗ ) → B ( F ∗ ) denotes its Banach-space adjoint.Now we make two observations. Firstly: by Lemma 4.4 and the fact q ⊗ respects completeisometries, ι ⊗ B ( E ) q ⊗ B ( F ) ∼ onto B ( E ) q ⊗ B ( F ∗ ). Secondly:there is an injective ∗ -homomorphism θ : B ( E ) ⊗ min B ( F ∗ ) → B ( S ( F, E )), which on elemen-tary tensors satisfies θ ( a ⊗ b )( c ) = acb . Since Φ = θ ◦ ( ι ⊗ α : E ⊗ F ∗ → S ( F, E ) be theHilbert-space isomorphism which sends x ⊗ φ to y φ ( y ) x ; then α intertwines the natural ∗ -representation of the incomplete algebra B ( E ) ⊗ B ( F ∗ ) on E ⊗ F ∗ with the map θ . See also[15, Prop. 2.9.1] or the calculations preceding [5, Eqn. (3.5.1)].)16 tep 3. Combining Steps 1 and 2, we see that Theorem 1.2 will follow if we can prove thefollowing claim: given E , F and Φ as in Step 2, the function Φ extends to a contractivelinear map B ( E ) b ⊗ B ( F ) → B ( S ( F, E )) . (We remind the reader that if E and F are infinite-dimensional, one cannot expect this map to be completely bounded.)Our proof of the claim is based on an interpolation argument. For p ∈ [1 , ∞ ] let S p ( F, E )denote the space of Schatten- p operators from F → E , equipped with its standard norm. Weadopt the convention that S ∞ ( F, E ) = K ( F, E ), the space of all compact operators F → E ,equipped with the operator norm. Define Φ p : B ( E ) ⊗ B ( F ) → B ( S p ( F, E )) by the formulaΦ p ( a ⊗ b )( c ) = acb ( a ∈ B ( E ), c ∈ S p ( F, E ), b ∈ B ( F )) . When p = 2 this is consistent with our earlier notation. Lemma . Let E and F be Hilbert spaces, and let w ∈ B ( E ) ⊗ B ( F ) . Let σ : B ( E ) ⊗ B ( F ) → B ( F ) ⊗ B ( E ) denote the flip map x ⊗ y y ⊗ x . (i) k Φ ∞ ( w ) : S ∞ ( F, E ) → S ∞ ( F, E ) k ≤ k w k B ( E ) ⊗ h B ( F ) . (ii) k Φ ( w ) : S ( F, E ) → S ( F, E ) k ≤ k σ ( w ) k B ( F ) ⊗ h B ( E ) . (iii) k Φ ( w ) : S ( F, E ) → S ( F, E ) k ≤ (cid:16) k w k B ( E ) ⊗ h B ( F ) k σ ( w ) k B ( F ) ⊗ h B ( E ) (cid:17) / .Proof. Part (i) follows immediately by quoting Haagerup’s theorem that Φ ∞ extends to a(complete) isometry from B ( E ) ⊗ h B ( F ) to CB ( S ∞ ( F, E )). However, there is also a directeasy proof: given w = P nj =1 a j ⊗ b j , it suffices to show that (cid:13)(cid:13)X nj =1 a j cb j (cid:13)(cid:13) ≤ k c k (cid:13)(cid:13)X nj =1 a j a ∗ j (cid:13)(cid:13) / (cid:13)(cid:13)X nj =1 b ∗ j b j (cid:13)(cid:13) / . This follows from standard calculations with “row” and “column” block matrices: for details,see e.g.[15, Remark 1.13], in particular the formula (1.12) in [15].Part (ii) follows from part (i) and duality. In more detail: given w = P nj =1 a j ⊗ b j ∈ B ( E ) ⊗ B ( F ), consider the elementary operator Φ ∞ ( σ ( w )) defined on S ∞ ( E, F ) by d P nj =1 b j da j .By part (i), applied with the roles of E and F reversed, k Φ ∞ ( σ ( w )) k ≤ k σ ( w ) k h . On the otherhand, consider the standard trace pairing between S ( F, E ) and S ∞ ( E, F ), where s ∈ S ( F, E )acts as the functional t Tr( st ). Straightforward calculations show that with respect to thispairing, the Banach-space adjoint of Φ ∞ ( σ ( w )) : S ∞ ( E, F ) → S ∞ ( E, F ) is the elementaryoperator Φ ( w ) : S ( F, E ) → S ( F, E ). Since a linear map and its adjoint have the samenorm, (ii) is proved.Finally, note that the elementary operator defined by w acts simultaneously on all S p ( F, E )for p ∈ [1 , ∞ ]. Since S ∞ and S form an interpolation couple with ( S , S ∞ ) / = S , part (iii)now follows from parts (i) and (ii) by applying the Riesz–Thorin interpolation theorem.To finish off, note that the os-projective tensor norm dominates both the Haagerup tensornorm and the “reversed” Haagerup tensor norm. More precisely: for arbitrary operator spaces V and V and w ∈ V ⊗ V , we have k w k V ⊗ h V ≤ k w k V b ⊗ V and k σ ( w ) k V ⊗ h V ≤ k σ ( w ) k V b ⊗ V = k w k V b ⊗ V . Combining these inequalities with Lemma 5.5(iii), we have verified the claim at the beginningof Step 3, and this completes the proof of Theorem 1.2.17 emark . We can strengthen the conclusion of Theorem 1.2, although at present wedo not have applications of the stronger version. Going back to Step 1, we can run thesame argument with the os-projective tensor product replaced by either the Haagerup tensorproduct or its reversed version. Combining this with Step 2 and Lemma 5.5(iii), we concludethat for arbitrary operator spaces X and Y and w ∈ X ⊗ Y , k w k X q ⊗ e Y ≤ (cid:16) k w k X ⊗ h Y k σ ( w ) k Y ⊗ h X (cid:17) / ≤ (cid:16) k w k X ⊗ h Y + k σ ( w ) k Y ⊗ h X (cid:17) . (5.2)We now have the necessary ingredients for our main technical theorem. Theorem . Let H and L be locally compact groups. Suppose that there exist non-zero,co-cb derivations D H : A( H ) → A( H ) ∗ and D L : A( L ) → A( L ) ∗ . Then the bilinear map F : (A( H ) ⊗ A( L )) × (A( H ) ⊗ A( L )) −→ A( H ) ∗ ⊗ A( L ) ∗ defined by F ( a ⊗ b , a ⊗ b ) := [ D A ( a ) · a ] ⊗ [ b · D B ( b )] − [ a · D A ( a ) ⊗ D B ( b ) · b ] extends to a non-zero alternating -cocycle F : A( H × L ) × A( H × L ) → A( H × L ) ∗ .Consequently, for any locally compact group G which contains a closed isomorphic copy of H × L , we have Z (A( G ) , A( G ) ∗ ) = 0 .Proof. For this proof only, just to ease notation slightly, we denote the Fourier algebras of G , H , L and H × L by A G , A H , A L and A H × L respectively; it is convenient to sometimes denotetheir duals by V G , V H , V L and V H × L respectively.Let D H : A H → V H and D L : A L → V ∗ L be non-zero, co-cb derivations, and let F : ( A H ⊗ A L ) × ( A H ⊗ A L ) → V H ⊗ V L be as defined in the statement of the theorem. By Proposition 5.2, F extends to a boundedbilinear map from ( A H b ⊗ A L ) × ( A H b ⊗ A L ) to V H b ⊗ V L ∼ . Applying Theorem 1.2 with X = V H and Y = V L ∼ , we obtain a bounded bilinear map F : ( A H b ⊗ A L ) × ( A H b ⊗ A L ) → V H q ⊗ V L that extends F . Recall (Proposition 4.8) that the natural map A H b ⊗ A L → A H × L is a(completely isometric) isomorphism and that V H q ⊗ V L embeds (completely isometrically) in V H × L = ( A H × L ) ∗ . Thus F can be viewed as a bilinear map A H × L × A H × L → ( A H × L ) ∗ .As in the proof of Lemma 3.6, the defining formula for F shows that F is an alternating2-derivation on the dense subalgebra A H ⊗ A L ⊂ A H × L . By the usual continuity argumentwe deduce that F ∈ Z ( A H × L , A H × L ∗ ).We now show that F is not identically zero. Since F takes values in V H ⊗ V L and thenatural map V H ⊗ V L → V H × L is injective, it suffices to show that F is not identically zero.Observe that if a ∈ A H and b ∈ A L we have F ( a ⊗ b, a ⊗ b ) = [ D H ( a ) · a ] ⊗ [ b · D L ( b )] − [ a · D H ( a )] ⊗ [ D L ( b ) · b ]= 2 a · D H ( a ) ⊗ b · D L ( b )= 14 D H ( a ) ⊗ D L ( b ) .
18y Lemma 2.2, elements of the form a span a dense subspace of A H , and elements of theform b span a dense subspace of A L . Therefore, since D H is continuous and non-zero, thereexists a ∈ A H such that D H ( a ) = 0; similarly, there exists b ∈ A L such that D L ( b ) = 0. Weconclude that F ( a ⊗ b, a ⊗ b ) = 0, as required.This proves the first part of the theorem. The second part follows by pulling back thenon-zero 2-cocycle F ∈ Z ( A H × L , A H × L ∗ ) along the restriction homomorphism A G → A H × L (see Lemma 3.5).To use Theorem 5.7 effectively, we need to know examples of H for which such a D H exists. It turns out that the very first non-zero derivation constructed from a Fourier algebrato its dual, which was produced by Johnson in [9], can be shown with hindsight to be co-cb!In fact, during the writing of [2], the present author and Ghandehari had already observedthat if H is one of the groups(a) SU(2) or SO(3);(b) the real ax + b group (the connected component of R ⋊ R ∗ );(c) the reduced Heisenberg group (the quotient of the 3-dimensional real Heisenberg groupby a central copy of Z );then in each case, the explicit non-zero derivation D H : A( H ) → A( H ) ∗ that is described in[2] turns out to be co-cb. Showing this requires some work, but is mostly just a matter ofcomposing D H with (the adjoint of) the check map and using the Plancherel theorem for eachgroup, c.f. the formulas and remarks in [2, § n ≥
4, we may embed GL ( C ) × GL ( C ) as a closed subgroup of GL n ( C ) bysending ( g , g ) to the block-diagonal matrix diag( g , g , I n − ); the same construction workswith C replaced by R . If H n denotes any of SU( n ), SL( n, R ) or Isom( R n ), then our embeddingmaps H × H onto a closed subgroup of H n . Note also that the real ax + b group is isomorphicto the standard parabolic subgroup of SL(2 , R ). Therefore, we may combine Theorem 5.7 withthe examples (a) and (b) mentioned above.To obtain the remaining case of Theorem 1.1, it would suffice to exhibit a non-zero co-cbderivation from A(Isom( R )) to its dual. For this, the results of [2, 3] are insufficient and werequire results from the subsequent paper [12]. In fact, one can use results from that paperto obtain alternative proofs for the cases (a) (b) and (c), and so for clarity of exposition wedevote the next section to summarizing and making use of the relevant parts of [12]. Remark . An alternative proof that the Fourier algebra of the real ax + b group supportsa non-zero co-cb derivation, independent of both [2] and [12], will appear as part of theforthcoming work [4]. In this section, we show in Theorem 6.3 that there is a plentiful supply of non-zero co-cbderivations from Fourier algebras to their duals. For the strongest results in this direction, we19ake use of the hard work done by the authors of [12] in proving the Lie case of the Forrest–Runde conjecture. While Theorem 6.3 is not hard to invent if one reads [12] in its entirety,it is never actually stated in that paper. We shall therefore extract some of the componentswhich are used to prove [12, Theorem 3.2], and reassemble them into a “black box” that willbe more suitable for our purposes.For G a Lie group, let C ∞ c ( G ) denote the space of compactly supported smooth functionson G . This is contained in A( G ) by [6, (3.26)] and is easily seen to be dense in A( G ) (since C ∞ c ( G ) is dense in L ( G ) and is closed under convolution). Proposition . Let H be any one of the following (con-nected, real) Lie groups: (a) SU(2) or SO(3) ; (b) the real ax + b group (the connected component of R ⋊ R ∗ ); (c) the reduced Heisenberg group (the quotient of the -dimensional real Heisenberg groupby a central copy of Z ); (d) the Euclidean motion group of R ; (e) the “Gr´elaud groups” G θ (certain semidirect products R ⋊ θ R where θ parametrizes theeigenvalues of the corresponding action of R on the Lie algebra of R )Then there exist a weight function v ∈ L ( H ) , not identically zero, and an element X of the Liealgebra of H , such that when we take the corresponding Lie derivative ∂ X : C ∞ c ( H ) → C ∞ c ( H ) , (cid:12)(cid:12)(cid:12)(cid:12)Z H ( ∂ X ⊗ ι ) u ( s, s − ) v ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ k u k A( H × H ) for all u ∈ C ∞ c ( H × H ) . (6.1) Proof.
In each case, there is a calculation in [12] that provides suitable X and v . (Strictlyspeaking, X and v are chosen together with S = S X,v ∈ VN( H × H ) such that the integral in(6.1) agrees with h S, u i for all u ∈ C ∞ c ( H × H ). By density and continuity arguments, if suchan S exists it is uniquely determined, and by rescaling the weight function v we can alwaysarrange that k S k ≤ Remark . In [12] the results assembled in Proposition 6.1 were used as follows. Recallthat a commutative Banach algebra A is said to be weakly amenable if Z ( A, A ∗ ) = 0. Let H be one of the groups in Proposition 6.1: then by [12, Lemma 2.1], the functional on A( H × H )that is uniquely defined by (6.1) serves as a witness that H has the following property:the anti-diagonal in H × H is not a set of local smooth synthesis. (AD)By [12, Theorem 1.6], the universal cover of a connected Lie group L has property (AD) ifand only if L does; and by [12, Theorem 1.3], if L has property (AD) then A( L ) is not weaklyamenable. Now by the structure theory of real Lie algebras, every non-abelian connected Lie20roup G contains a closed Lie subgroup H with the same Lie algebra as one of the groupslisted in Proposition 6.1; it follows that A( H ), and hence A( G ), fails to be weakly amenable,as predicted by the Forrest–Runde conjecture.As indicated by the previous remark, the authors of [12] did not pursue Proposition 6.1with the goal of constructing explicit derivations on Fourier algebras, as they aimed to establishstronger structural properties for a wider class of groups. For the present paper, what mattersis the following consequence of Proposition 6.1, which appears to be a new observation. Theorem . Let H be one of the groups listed in (a)–(e) of Proposition 6.1, and let X and v be as provided by that proposition. Then there is a unique bounded linear map D : A( H ) → A( H ) ∗ that satisfies D ( f )( f ) = Z H ( ∂ X f )( s ) f ( s ) v ( s ) ds for all f , f ∈ C ∞ c ( H ) . (6.2) Furthermore, D is a non-zero co-cb derivation.Proof. First, observe that there is at most one (norm-)continuous function D : A( H ) → A( H ) ∗ satisfying (6.2), because C ∞ c ( H ) is dense in A( H ).Since X and v are chosen so that the inequality (6.1) is satisfied, there exists a unique ψ ∈ A( H × H ) ∗ satisfying ψ ( f ⊗ g ) = Z H ( ∂ X ⊗ ι )( f ⊗ g )( s, s − ) v ( s ) ds = Z H ( ∂ X f )( s ) g ( s − ) v ( s ) ds for all f, g ∈ C ∞ c ( H ). ( ∗ )Let T : A( H ) → A( H ) ∗ be the completely bounded map corresponding to ψ , and define D : A( H ) → A( H ) ∗ by D ( f )( f ) := T ( f )( f ∨ ) for f , f ∈ A( H ). Since the check map is aco-complete isometry (Proposition 4.8(d)), D : A( H ) → A( H ) ∗ is co-cb. Moreover, since ψ satisfies ( ∗ ), D satisfies (6.2).It remains to show that D is a derivation. Since C ∞ c ( H ) is dense in A( H ) and D isnorm-continuous, it suffices to check that D ( g g )( g ) = D ( g )( g g ) + D ( g )( g g ) for all g , g , g ∈ C ∞ c ( H ). This follows from (6.2) and the fact that ∂ X : C ∞ c ( H ) → C ∞ c ( H ) is aderivation.Theorem 1.1 now follows by combining Theorem 5.7 with the cases (a), (b) and (d) ofTheorem 6.3, following the argument described at the end of Section 5. Remark . Inspecting the results used to proveProposition 6.1, one sees that for each of the solvable cases, and an appropriately chosen X ,the set of v which “work” contains an infinite-dimensional vector space. It follows that inTheorem 6.3 one obtains not just a single D of the desired form, but an infinite-dimensional vector space of such derivations. Thus for these groups, the suggestion at the end of [12, § This also fits with what one expects from considering derivations on more general Banach function alge-bras A ; often there exists a dense subalgebra A ⊂ A such that the natural predual of H ( A, A ∗ ) contains afree A -module in the algebraic sense, forcing H ( A, A ∗ ) to be infinite-dimensional as a vector space. Avenues for further work
In future work, we intend to set out a more systematic study of the higher-degree alternatingcocycles on Fourier algebras, with the intention of exploring an associated numerical invariantthat can be viewed as a kind of “dimension” associated to such algebras. Since one wouldlike to calcuate or estimate this numerical invariant for as many small examples as possible,progress on the following natural question could be a useful guide for future work.
Question . Does Theorem 1.1 remain true for n = 2 or n = 3?Currently our guess is that the answer is negative for SU(2) and SL(2 , R ), and positivefor SU(3), SL(3 , R ), Isom( R ) and Isom( R ), but there is insufficient evidence to support anyfirm conjectures at this stage.Turning to Theorem 1.2: one would like to understand better the comparison map θ X,Y : X b ⊗ Y → X q ⊗ e Y , perhaps by making greater use of the sharper result outlined in Re-mark 5.6. Indeed, a natural next step is to repeat the (complex) interpolation argumentused in Lemma 5.5 at the level of operator spaces and cb-norms of elementary operators, tosee what θ X,Y looks like at higher matrix levels.The co-cb derivations that are crucial to proving Theorem 1.1 provide natural examples ofc.b. maps from A( G ) ∼ to VN( G ) that behave like noncommutative Fourier multipliers (thisis not immediately apparent from what is stated in Section 6, but can be seen by inspectingthe details in [2] and [12].) Question . Given that A( G ) ∼ and VN( G ) are the endpoints of the scale of noncommuta-tive L p -spaces associated to VN( G ), are there other Fourier multipliers from L p (VN( G )) → L r (VN( G )) which satisfy some form of the Leibniz identity? For fixed p and r , what can wesay about the space of such multipliers?We finish with some natural questions concerning co-cb derivations on Fourier algebras,which are all aimed at strengthening or sharperning the conclusion of Theorem 6.3. Question . The derivations constructed in [2] for SU(2), the real ax + b group and thereduced Heisenberg group are all cyclic and co-cb (c.f. the construction in [4]). Is every cyclicderivation on a Fourier algebra automatically co-cb? Question . Let G be the 3-dimensional real Heisenberg group. The results of [3] constructa non-zero derivation D from A( G ) to a certain symmetric Banach A( G )-bimodule W . Can W be made into a cb-Banach A( G )-bimodule in such a way that D : A( G ) → W is co-cb? Question . It was shown in [12] that the property (AD) for a Lie group G , mentioned inRemark 6.2, ensures that there is a non-zero derivation D : A( G ) → A( G ) ∗ . Does it alsoguarantee that one can choose D to be co-cb?In view of the good hereditary properties of (AD), a positive answer to Q5 would allow usto transfer co-completely bounded derivations between Fourier algebras of Lie groups whichhave the same universal cover, and hence by using the strategy outlined in Remark 6.2 onecould strengthen Theorem 6.3 to the following result: every non-abelian connected Lie group H has non-zero co-cb derivations from A( H ) to A( H ) ∗ . Question . Can the explicit derivations constructed by Losert [14] on connected groups thatare not necessarily Lie, be made into co-cb derivations from Fourier algebras into cb-Banachbimodules? 22he constructions in [14] are closer in spirit to [3] than to [12], and so Question 4 wouldserve as a warm-up for Question 6.
Acknowledgments
A preliminary announcement of some of these results, with a different emphasis and lesscomprehensive results, was circulated in 2016 as an unpublished preprint; the author thanksM. Daws and A. Skalski for several comments and corrections on that document. He alsothanks the authors of [12] for useful discussions about technical aspects of their paper, andto V. Losert for sharing a copy of the preprint [14].Less direct, but no less important, thanks are due to M. Ghandehari and E. Samei fortheir interest and encouragement over several years concerning alternating cocycles on Fourieralgebras, and to M. Whittaker for a conversation at the British Mathematical Colloquium 2019in Lancaster which prompted the author to finally write up this work as a proper paper.Some of these results were presented in conference talks at the Abstract Harmonic Anal-ysis meeting in Kaohsiung, 2018, and the International Workshop on Harmonic Analyis andOperator Theory, Istanbul, 2019. The author thanks the organizers of these meetings for theirrespective invitations to present this work, and for bringing together speakers from a broadrange of specialist interests for enjoyable discussions.Most of the writing of this article was done during the COVID-19 pandemic, under a periodof lockdown conditions in England. The author would therefore like to thank various colleaguesat Lancaster University for an online mixture of camaraderie, commiseration, complaints, andcomputer support during these months, which has gone some way towards replicating a normalworking environment. He hopes to one day visit Barnard Castle.
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C. Zwarich , Von Neumann algebras for harmonic analysis , Master’s thesis, Universityof Waterloo, 2008.Department of Mathematics and StatisticsFylde College, Lancaster UniversityLancaster, United Kingdom LA1 4YFEmail: [email protected]@lancaster.ac.uk