Completely bounded bimodule maps and spectral synthesis
aa r X i v : . [ m a t h . F A ] J a n COMPLETELY BOUNDED BIMODULE MAPS ANDSPECTRAL SYNTHESIS
M. ALAGHMANDAN, I. G. TODOROV, AND L. TUROWSKA
Abstract.
We initiate the study of the completely bounded multipliersof the Haagerup tensor product A ( G ) ⊗ h A ( G ) of two copies of the Fourieralgebra A ( G ) of a locally compact group G . If E is a closed subset of G we let E ♯ = { ( s, t ) : st ∈ E } and show that if E ♯ is a set of spectralsynthesis for A ( G ) ⊗ h A ( G ) then E is a set of local spectral synthesisfor A ( G ). Conversely, we prove that if E is a set of spectral synthesisfor A ( G ) and G is a Moore group then E ♯ is a set of spectral synthesisfor A ( G ) ⊗ h A ( G ). Using the natural identification of the space of allcompletely bounded weak* continuous VN( G ) ′ -bimodule maps with thedual of A ( G ) ⊗ h A ( G ), we show that, in the case G is weakly amenable,such a map leaves the multiplication algebra of L ∞ ( G ) invariant if andonly if its support is contained in the antidiagonal of G . Contents
1. Introduction 12. Preliminaries 33. Multipliers of bivariate Fourier algebras 64. Spectral synthesis in A h ( G ) 175. The case of virtually abelian groups 296. VN( G ) ′ -bimodule maps and supports 32References 431. Introduction
The connections between Harmonic Analysis and Operator Theory orig-inating from the seminal papers of W. Arveson [2] and N. Varopoulos [35]have been fruitful and far-reaching. A particular instance of this interactionis the relation between Schur and Herz-Schur multipliers [6, 17] that hasbeen prominent in applications, for example to approximation properties ofgroup operator algebras (see e.g. [5]). It is well-known that, given a locallycompact second countable group G , the Schur multipliers on G × G can beidentified with those (completely) bounded weak* continuous maps on thespace B ( L ( G )) of all bounded operators on L ( G ) (here G is equipped withleft Haar measure) that are also bimodular over L ∞ ( G ), where the latter is Date : 31 December 2016. viewed as an algebra of multiplication operators on L ( G ). The right invari-ant part of the space of Schur multipliers (which arises from the functions ϕ on G × G that satisfy the condition ϕ ( sr, tr ) = ϕ ( s, t )) consists preciselyof those maps that, in addition to the aforementioned properties, preservethe von Neumann algebra VN( G ) of G .The original motivation behind the present work was the development ofa counterpart of the latter result in a setting where the places of VN( G )and L ∞ ( G ) are exchanged. The space of all completely bounded weak* con-tinuous VN( G )-bimodule maps on B ( L ( G )) has played a distinctive rolein Operator Algebra Theory and have lately been prominent through thetheory of locally compact quantum groups (see e.g. [20] and [22]). Thosesuch maps that also preserve the multiplication algebra of L ∞ ( G ) have beenstudied since the 1980’s and are known to arise from regular Borel mea-sures on G (see [16, 27, 32]). However, a characterisation, analogous to theright invariance in the context of Schur multipliers – and one that uses onlyharmonic-theoretic properties – was not known. In the present paper, we es-tablish such a characterisation and observe that it can be formulated in thelanguage of spectral synthesis: it is equivalent to the statement that the an-itdiagonal of G is a Helson set with respect to the Haagerup tensor product A ( G ) ⊗ h A ( G ) of two copies of the Fourier algebra A ( G ) of G . Our inves-tigation highlights the connections between completely bounded bimodulemaps and spectral synthesis, which have not received substantial attentionuntil now, despite the importance of both notions in modern Analysis.The aforementioned result required the development of a ground theoryof bivariate Herz-Schur multipliers and served as a motivation to study ques-tions of spectral synthesis in A ( G ) ⊗ h A ( G ). Our results show that, withrespect to spectral synthesis, the latter algebra is better behaved than theseemingly more natural A ( G × G ), and point to substantial distinctions be-tween these two algebras. Indeed, for a vast class of groups we establishtransference of spectral synthesis between A ( G ) and A ( G ) ⊗ h A ( G ), whilesuch result does not hold for A ( G × G ) unless G is virtually abelian.In more detail, the paper is organised as follows. After collecting pre-liminaries and setting notation in Section 2, we study, in Section 3, thebivariate Fourier algebra A h ( G ) def = A ( G ) ⊗ h A ( G ) and establish some if itsbasic properties, highlighting the rather well-known fact that it is a regularcommutative semi-simple Banach algebra with Gelfand spectrum G × G .Viewing A h ( G ) as a function algebra, we examine the space of its com-pletely bounded multipliers, which can be thought of as a bivariate versionof Herz-Schur multipliers, and show, among other things, that this algebrais weakly amenable if and only if the group G is weakly amenable. We ob-tain a characterisation of the completely bounded multipliers of A h ( G ) interms of (bounded) multipliers on products with finite groups, providing aversion of a result from [6] (see Proposition 3.8). We show that the elements OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 3 of the extended Haagerup tensor product A ( G ) ⊗ eh A ( G ) can be viewed asseparately continuous functions, an identification needed thereafter.In Section 4, we study the question of spectral synthesis for A h ( G ). Notethat the dual of A h ( G ) coincides with the extended Haagerup tensor productVN eh ( G ) def = VN( G ) ⊗ eh VN( G ) which, in turn, can be canonically identi-fied, via a classical result of U. Haagerup’s [17], with the space of com-pletely bounded weak* continuous VN( G ) ′ -bimodule maps (here VN( G ) ′ denotes the commutant of VN( G )). Thus, the classical theory of commuta-tive Banach algebras allows us to associate to each such map its support,a closed subset of G × G . Viewing VN eh ( G ) as a (completely contractive)module over A h ( G ), we obtain bivariate versions of some classical resultsof P. Eymard [13]. The main results in Section 4 are related to trans-ference of spectral synthesis: associating to a subset E ⊆ G the subset E ♯ = { ( s, t ) ∈ G × G : st ∈ E } of G × G , we show that if E ♯ is is a setof spectral synthesis for A h ( G ) then E is a set of local spectral synthesisfor A ( G ). Conversely, if E is a set of spectral synthesis for A ( G ) and G is a Moore group then E ♯ is a set of spectral synthesis for A h ( G ). Thus,for Moore groups, the sets E and E ♯ satisfy spectral synthesis simultane-nously. These results should be compared with other transference results inthe literature, see e.g. [24], [30] and [31], and are a part of a programme ofrelating harmonic analytic, one-variable, properties, to operator theoretic,two-variable, ones [33].In Section 5, we assume that G is a virtually abelian group, and showthat, in this case, transference carries over to the set E ∗ = { ( s, t ) ∈ G × G : ts − ∈ E } . This is obtained as a consequence of the fact that, for suchgroups, the flip of variables is a well-defined bounded map on A h ( G ).Section 6 is focused around the question of how the support of a maparising from an element of VN eh ( G ) influences the structure of the map.Our results demonstrate that the support contains information about theinvariant subspaces of the map (see Theorem 6.6 and Corollaries 6.7 and 6.8).As a consequence, we show that a completely bounded weak* continuousVN( G ) ′ -bimodule map leaves the multiplication algebra of L ∞ ( G ) invariantif and only if its support is contained in the antidiagonal of G . This givesan intrinsic, harmonic analytic, characterisation of this class of maps.Operator space tensor products and, more generally, operator space theo-retic concepts and results, play a prominent role in our approach. Our mainreferences in this direction are [3] and [10]. In addition, we use in a crucialway results and techniques about masa-bimodules in B ( L ( G )), whose basictheory was developed in [2] and [12].2. Preliminaries
In this section, we introduce some basic concepts that will be needed inthe sequel and set notation. For a normed space X , we let ball( X ) be the unitball of X , and B ( X ) (resp. K ( X )) be the algebra of all bounded linear (resp. M. ALAGHMANDAN, I. G. TODOROV, AND L. TUROWSKA compact) operators on X . If H is a Hilbert space and ξ, η ∈ H , we denote by ξ ⊗ η ∗ the rank one operator on H given by ( ξ ⊗ η ∗ )( ζ ) = ( ζ, η ) ξ , ζ ∈ H . By ω ξ,η we denote the vector functional on B ( H ) defined by ω ξ,η ( T ) = ( T ξ, η ).The pairing between elements of a normed space X and those of its dual X ∗ will be denoted by h· , ·i X , X ∗ ; when no risk of confusion arises, we writesimply h· , ·i . By M n ( X ) we denote the space of all n by n matrices withentries in X ; we set M n = M n ( C ). We let CB ( X ) be the (operator) spaceof all completely bounded maps on an operator space X .The algebraic tensor product of vector spaces X and Y will be denotedby X ⊙ Y ; if X and Y are Banach spaces, we let X ⊗ γ Y be their Banachprojective tensor product. If H and K are Hilbert spaces, we denote by H ⊗ K their Hilbertian tensor product. We let X ˆ ⊗Y denote the operatorprojective, and X ⊗ h Y the Haagerup, tensor product of the operator spaces X and Y . By X ⊗ eh Y we will denote the extended Haagerup tensor productof X and Y ; we refer the reader to [11] for its definition and properties. If X and Y are dual operator spaces, their weak* spacial tensor product will bedenoted by X ¯ ⊗Y , and their σ -Haagerup tensor product by X ⊗ σ h Y . Notethat, in the latter case, X ⊗ eh Y coincides with the weak* Haagerup tensorproduct of X and Y introduced in [4]. We often use the same symbol todenote both a bilinear map and its linearisation through a tensor product.Recall that a Banach algebra A equipped with an operator space structureis called completely contractive if k [ a i,j b k,l ] k M mn ( A ) ≤ k [ a i,j ] k M n ( A ) k [ b k,l ] k M m ( A ) for every [ a i,j ] ∈ M n ( A ) and [ b k,l ] ∈ M m ( A ) and n, m ∈ N . Thus, if A is acompletely contractive Banach algebra then the linearisation of the productextends to a completely contractive map m A : A ˆ ⊗A → A .Let A be a commutative regular semi-simple completely contractive Ba-nach algebra with Gelfand spectrum Ω; thus, A can be thought of as asubalgebra of the algebra C (Ω) of all continuous functions on Ω vanishingat infinity. A continuous function b : Ω → C is called a multiplier of A if b A ⊆ A ; in this case, we have a well-defined map m b on A , given by m b ( a ) = ba , which is automatically bounded. If the map m b is moreovercompletely bounded, b is called a completely bounded multiplier . We denoteby M A (resp. M cb A ) the space of all multipliers (resp. completely boundedmultipliers) of A . It is known that a (bounded) linear map T : A → A isof the form T = m b for some b ∈ M A if and only if T ( x ) y = xT ( y ) for all x, y ∈ A (see e.g. [23, Proposition 2.2.16]). Note that M A (resp. M cb A )is a closed subalgebra of B ( A ) (resp. CB ( A )). If b ∈ M cb A , we denote by k b k cbm the completely bounded norm of m b ; we often identify the functions b ∈ M cb A with the corresponding linear transformations m b . Note that, if a ∈ A , then k m ( n ) a [ a k,l ] k M n ( A ) = k [ aa k,l ] k M n ( A ) ≤ k a k A k [ a k,l ] k M n ( A )OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 5 for every [ a k,l ] ∈ M n ( A ) and every n ∈ N . Therefore, the mapping a m a from A into M cb A is a contraction.We next recall some basic facts from [13] and [6]. Let G be a locallycompact group. The Haar measure evaluated at a Borel set E ⊆ G will bedenoted by | E | , and integration with respect to it along the variable s willbe denoted by ds . As customary, a ∗ b denotes the convolution, wheneverdefined, of the functions a and b . For t ∈ G , we let λ t be the unitaryoperator on L ( G ), given by λ t f ( s ) = f ( t − s ), s ∈ G , f ∈ L ( G ). We let M ( G ) be the Banach *-algebra of all complex Borel measures on G and usethe symbol λ to denote the left regular *-representation of M ( G ) on L ( G );thus, ( λ ( µ ) f ) = Z G λ s f dµ ( s ) , µ ∈ M ( G ) , f ∈ L ( G ) , where the integral is understood in the weak sense. We identify L ( G ) witha *-subalgebra of M ( G ) in the canonical way.We let VN( G ) (resp. C ∗ r ( G ), C ∗ ( G )) be the von Neumann algebra (resp.the reduced C*-algebra, the full C*-algebra) of G . As usual, A ( G ) (resp. B ( G )) stands for the Fourier (resp. the Fourier-Stieltjes) algebra of G . Thus, C ∗ r ( G ) = { λ ( f ) : f ∈ L ( G ) } , VN( G ) = C ∗ r ( G ) w ∗ ,B ( G ) = { ( π ( · ) ξ, η ) : π : G → B ( H ) cont. unitary representation , ξ, η ∈ H } , and A ( G ) is the collection of the functions on G of the form s → ( λ s ξ, η ),where ξ, η ∈ L ( G ); see [13] for details. We denote by k · k A the normof A ( G ). Note that the dual of A ( G ) (resp. C ∗ ( G )) can be canonicallyidentified with VN( G ) (resp. B ( G )). More specifically, if φ ∈ A ( G ) and ξ, η ∈ L ( G ) are such that φ ( s ) = ( λ s ξ, η ), s ∈ G , then(1) h φ, T i = ( T ξ, η ) , T ∈ VN( G ) . We equip A ( G ) (resp. B ( G )) with the operator space structure arising fromthe latter identification. Note that both A ( G ) and B ( G ) are completelycontractive Banach algebras with respect to these operator space structures.For each ψ ∈ M A ( G ), the dual m ∗ ψ of the map m ψ acts on VN( G ); infact, m ∗ ψ ( λ t ) = ψ ( t ) λ t , t ∈ G , and m ∗ ψ ( λ ( f )) = λ ( ψf ), f ∈ L ( G ). Notethat a multiplier ψ ∈ M A ( G ) is completely bounded precisely when m ∗ ψ is completely bounded; in this case, k ψ k cbm = k m ∗ ψ k cb . We set ψ · T = m ∗ ψ ( T ). The elements of M cb A ( G ) are called Herz-Schur multipliers andwere introduced and originally studied in [6].Let H and K be separable Hilbert spaces and M ⊆ B ( H ) and N ⊆ B ( K )be von Neumann algebras. Every element u ∈ M⊗ eh N has a representation(2) u = ∞ X i =1 a i ⊗ b i , where ( a i ) i ∈ N ⊆ M and ( b i ) i ∈ N ⊆ N are sequences such that P ∞ i =1 a i a ∗ i and P ∞ i =1 b ∗ i b i are weak* convergent. In this case, the series (2) converges in M. ALAGHMANDAN, I. G. TODOROV, AND L. TUROWSKA the weak* topology of
M ⊗ eh N with respect to the completely isometricidentification [4](3) M ⊗ eh N ≡ ( M ∗ ⊗ h N ∗ ) ∗ , where M ∗ and N ∗ denote the preduals of M and N , respectively. Following[4], call (2) a w*-representation of u . Denoting by A (resp. B ) the row (resp.column) operator ( a i ) i ∈ N (resp. ( b i ) i ∈ N ), we write (2) as u = A ⊙ B . Everysuch u gives rise to a completely bounded weak* continuous M ′ , N ′ -modulemap Φ u : B ( K, H ) → B ( K, H ) given by(4) Φ u ( T ) = ∞ X i =1 a i T b i , T ∈ B ( K, H ) , and the map u → Φ u is a complete isometry from M ⊗ eh N onto the space CB w ∗ M ′ , N ′ ( B ( K, H )) of all weak* continuous completely bounded M ′ , N ′ -module maps on B ( K, H ) [4]. Note that the algebraic tensor product
M⊙N can be viewed in a natural way as a (weak* dense) subspace of
M ⊗ eh N .3. Multipliers of bivariate Fourier algebras
In this section, we introduce a natural bivariate version of Herz-Schurmultipliers and develop their basic properties. We set A h ( G ) = A ( G ) ⊗ h A ( G ) and VN eh ( G ) = VN( G ) ⊗ eh VN( G ) . According to (3), we have a completely isometric identification(5) A h ( G ) ∗ ≡ VN eh ( G );under this identification,(6) h φ ⊗ ψ, λ s ⊗ λ t i = φ ( s ) ψ ( t ) , φ, ψ ∈ A ( G ) , s, t ∈ G. We proceed with some certainly well-known considerations; because ofthe frequent lack of precise references, we provide the full details, whichalso serve our aim to set the appropriate context and notation for theirsubsequent applications. We first note that the natural injection ι G : A ( G ) → C ( G )is completely contractive. Indeed, let U = [ u i,j ] i,j ∈ M n ( A ( G )) and associateto U the map F U : VN( G ) → M n given by F U ( T ) = [ h u i,j , T i ] i,j . Then k ι ( n ) G ( U ) k M n ( C ( G )) = sup s ∈ G k [ u i,j ( s )] k M n = sup s ∈ G k [ h u i,j , λ s i ] k M n ≤ sup {k [ h u i,j , T i ] k M n : T ∈ ball(VN(( G )) } = sup {k F U ( T ) k M n : T ∈ ball(VN(( G )) } = k F U k ≤ k F U k cb = k U k M n ( A ( G )) . Thus, the map(7) ι h def = ι G ⊗ h ι G : A h ( G ) → C ( G ) ⊗ h C ( G ) OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 7 is completely contractive. On the other hand, there is a natural contractiveinjection(8) C ( G ) ⊗ h C ( G ) → C ( G ) ⊗ min C ( G ) ≡ C ( G × G ) , which allows us to view the elements of C ( G ) ⊗ h C ( G ) as continuousfunctions on G × G (vanishing at infinity). In fact, C ( G ) ⊗ h C ( G ) is a(Banach) algebra under pointwise addition and multiplication and, by theGrothendieck inequality, coincides up to renorming with the Varopoulos al-gebra C ( G ) ⊗ γ C ( G ). If v ∈ A h ( G ) then, in view of (7) and (8),(9) k ι h ( v ) k ∞ ≤ k v k h . The duality in the next lemma is the one arising from the identification(5).
Lemma 3.1. If s, t ∈ G and v ∈ A h ( G ) then (10) h v, λ s ⊗ λ t i = ι h ( v )( s, t ) . In particular, the map ι h is injective.Proof. Let s, t ∈ G , and suppose that v = φ ⊗ ψ for some φ, ψ ∈ A ( G ). Then h v, λ s ⊗ λ t i = h φ ⊗ ψ, λ s ⊗ λ t i = h φ, λ s ih ψ, λ t i = φ ( s ) ψ ( t ) = ι h ( φ ⊗ ψ )( s, t ) . It follows that (10) holds for all v ∈ A ( G ) ⊙ A ( G ). For every T ∈ VN eh ( G ),the map v → h v, T i is norm continuous. On the other hand, in view of (8), | ι h ( v )( s, t ) | ≤ k ι h ( v ) k ∞ ≤ k ι h ( v ) k h . Identity (10) now follows from the density of A ( G ) ⊙ A ( G ) in A h ( G ).Assuming ι h ( v ) = 0, we have that ι h ( v )( s, t ) = 0 for all s, t ∈ G . By(10), h v, λ s ⊗ λ t i = 0 for all s, t ∈ G . An application of Kaplansky’s DensityTheorem shows that the set { λ s ⊗ λ t : s, t ∈ G } spans a weak* dense subspaceof VN eh ( G ); it now follows that v = 0. (cid:3) Since the Haagerup norm is dominated by the operator projective one,the identity map on A ( G ) ⊙ A ( G ) extends to a complete contraction(11) ˆ ι : A ( G ) ˆ ⊗ A ( G ) → A h ( G ) . Identifying A ( G ) ˆ ⊗ A ( G ) with A ( G × G ) (see [10, Chapter 16]), we thusconsider ˆ ι as a complete contraction from A ( G × G ) into A h ( G ). Notethat ι h ◦ ˆ ι = ι G × G ; indeed, the latter identity is straightforward on thealgebraic tensor product A ( G ) ⊙ A ( G ), and hence holds by density andcontinuity. Since ι G × G is injective, we conclude that ˆ ι is injective. The dualˆ ι ∗ : VN eh ( G ) → VN( G × G ) of the map ˆ ι : A ( G × G ) → A h ( G ) is easily seento coincide with the canonical inclusion of VN eh ( G ) into VN( G × G ), and ishence (completely contractive and) injective [4, Corollary 3.8].In the sequel, we often suppress the notations ι G × G , ˆ ι and ι h and, by virtueof Lemma 3.1, consider the elements of A h ( G ) as (continuous) functions on G × G . M. ALAGHMANDAN, I. G. TODOROV, AND L. TUROWSKA
The next proposition contains the main facts that we will need about thealgebra A h ( G ). Recall that a normed algebra ( A , k · k A ) is said to have a (left) bounded approximate unit [36], if there exists a constant C > v ∈ A and every ǫ >
0, there exists u ∈ A such that k u k A ≤ C and k uv − v k A < ǫ . Proposition 3.2.
The following statements hold true:(i) The space A h ( G ) is a regular semisimple Tauberian completely con-tractive Banach algebra with respect to the operation of pointwise multipli-cation, whose Gelfand spectrum can be identified with G × G .(ii) The map ι h is an algebra homomorphism.(iii) The algebra A h ( G ) has a bounded approximate identity if and only if G is amenable. Furthermore, if G is amenable then the bounded approximateidentity can be chosen to be compactly supported.Proof. (i), (ii) Let m : ( A ( G ) ⊙ A ( G )) × ( A ( G ) ⊙ A ( G )) → A ( G ) ⊙ A ( G )be the map given by m ( φ ⊗ ψ, φ ′ ⊗ ψ ′ ) = ( φφ ′ ) ⊗ ( ψψ ′ ) . By [8, Section 9.2], m linearises to a completely bounded bilinear map m h : A h ( G ) ˆ ⊗ A h ( G ) → A h ( G ) , turning A h ( G ) into a commutative completely contractive Banach algebra.Let w = φ ⊗ ψ and w ′ = φ ′ ⊗ ψ ′ for some φ, ψ, φ ′ , ψ ′ ∈ A ( G ); then ι h ( m ( w, w ′ ))( s, t ) = ι h (( φφ ′ ) ⊗ ( ψψ ′ ))( s, t ) = ( φφ ′ )( s )( ψψ ′ )( t )= ( φ ⊗ ψ )( s, t )( φ ′ ⊗ ψ ′ )( s, t ) = ι h ( w )( s, t ) ι h ( w ′ )( s, t ) . By the continuity of m , ι h ( m ( w, w ′ )) = ι h ( w ) ι h ( w ′ ) for all w, w ′ ∈ A h ( G )and ι h is a homomorphism. Therefore m coincides with the pointwise mul-tiplication. The fact that the Gelfand spectrum of A h ( G ) coincides with G × G follows from [34, Theorem 2]. Since ι G × G is injective and A ( G × G )is a regular Banach algebra, we conclude that A h ( G ) is regular, too. Notethat, since the elements λ s ⊗ λ t , s, t ∈ G , are characters of A h ( G ), the latteralgebra is also semi-simple.Note that the space X = A ( G ) ∩ C c ( G ) is dense in A ( G ); it follows thatthe space X ⊙ X is dense in A h ( G ). This implies that C c ( G × G ) ∩ A h ( G )is dense in A h ( G ), that is, A h ( G ) is Tauberian.(iii) Suppose that G is amenable. By Leptin’s Theorem, A ( G ) has abounded approximate identity say ( φ α ) α . Set w α = φ α ⊗ φ α . If ψ , ψ ∈ A ( G ) then, clearly, w α ( ψ ⊗ ψ ) → α ψ ⊗ ψ in A h ( G ). Now a straightfor-ward approximation argument shows that ( w α ) α is a (bounded) approximateidentity for A h ( G ).Conversely, suppose that ( w α ) α is a bounded approximate identity of A h ( G ). Let δ s denote the character on A ( G ) corresponding to an element s ∈ G . The map id ⊗ δ s : A h ( G ) → A ( G ) is a (completely) contractive OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 9 homomorphism. For an arbitrary 0 = v ∈ A ( G ), let s ∈ G so that v ( s ) = 0.Note that (id ⊗ δ s )( w α ) v = (id ⊗ δ s )( w α )(id ⊗ δ s )( v ⊗ v ( s ) − v )= (id ⊗ δ s )( w α ( v ⊗ v ( s ) − v )) → α (id ⊗ δ s )( v ⊗ v ( s ) − v ) = v. Thus, A ( G ) has a (left) bounded approximate unit. By [36, Theorem 1], A ( G ) has a bounded approximate identity. By Leptin’s Theorem, G isamenable. (cid:3) The following lemma will be needed shortly, but it may be interesting inits own right.
Lemma 3.3.
Let A be a commutative Banach algebra, B be a completelycontractive commutative Banach algebra, and θ : A → M cb B be a boundedhomomorphism. If A has a bounded approximate identity and the linear spanof { θ ( a ) b : a ∈ A , b ∈ B} is dense in B , then θ can be extended to a boundedmap θ : M A → M cb B . In particular, if A is a completely contractive Banachalgebra with a bounded approximate identity, then M A = M cb A .Proof. Fix a bounded approximate identity ( a α ) α of A . Let B = span { θ ( a )( b ) : a ∈ A , b ∈ B} . For a given c ∈ M A , define θ ( c ) on B by θ ( c ) m X k =1 θ ( a k )( b k ) ! := m X k =1 θ ( ca k )( b k ) , a k ∈ A , b k ∈ B , k = 1 , . . . , m. The mapping θ ( c ) is a well-defined linear map on B . In fact, if n X k =1 θ ( a (1) k ) b (1) k = m X l =1 θ ( a (2) k ) b (2) k , for some subsets { a (1) k , a (2) l : k = 1 , . . . , n, l = 1 , . . . , m } ⊆ A and { b (1) k , b (2) l : k = 1 , . . . , n, l = 1 , . . . , m } ⊆ B , then n X k =1 θ ( ca (1) k ) b (1) k = lim α n X k =1 θ ( ca α a (1) k ) b (1) k = lim α θ ( ca α ) n X k =1 θ ( a (1) k ) b (1) k ! (12) = lim α θ ( ca α ) m X l =1 θ ( a (2) l ) b (2) l ! = lim α m X l =1 θ ( ca α a (2) l ) b (2) l = m X l =1 θ ( ca (2) l ) b (2) l . We claim that θ ( c ) is a completely bounded map on B . Let " n i,j X k =1 θ ( a ( i,j ) k ) b ( i,j ) k i,j be an arbitrary element in the unit ball of M n ( B ). Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) θ ( c ) ( n ) " n i,j X k =1 θ ( a ( i,j ) k ) b ( i,j ) k i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" n i,j X k =1 θ ( ca ( i,j ) k ) b ( i,j ) k i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" n i,j X k =1 θ ( ca α a ( i,j ) k ) b ( i,j ) k i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) θ ( ca α ) ( n ) " n i,j X k =1 θ ( a ( i,j ) k ) b ( i,j ) k i,j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ sup α k θ ( ca α ) k cbm ≤ k θ kk c k M A sup α k a α k < ∞ . Since B is dense in B , the map θ ( c ) can be extended as a completelybounded map (denoted in the same way) on B . Furthermore, θ ( c ) is amultiplier. In fact, let b, b ′ ∈ B . Since B is dense in B , there is a sequence (cid:16)P n i k =1 θ ( a ( i ) k ) b ( i ) k (cid:17) i ∈ N in B converging to b . We have θ ( c )( bb ′ ) = lim i →∞ θ ( c ) n i X k =1 θ ( a ( i ) k ) b ( i ) k b ′ ! = lim i →∞ n i X k =1 θ ( ca ( i ) k ) b ( i ) k b ′ = lim i →∞ n i X k =1 θ ( ca ( i ) k ) b ( i ) k ! b ′ = lim i →∞ θ ( c ) n i X k =1 θ ( a ( i ) k ) b ( i ) k ! b ′ = θ ( c )( b ) b ′ ;thus, θ takes values in M cb B .To prove the last statement in the formulation of the Lemma, note thatif A is a completely contractive Banach algebra, A sits inside M cb A in anatural fashion. Since A possesses a (bounded) approximate identity, theset { ab : a, b ∈ A} is dense in A . By the first part of the proof, the identitymap can be extended to a map θ : M A → M cb A where for each b ∈ M A , θ ( b )( a ) = lim α ( ba α ) a = ba. OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 11
Therefore, the extension θ is still the identity map, and hence M A ⊆ M cb A .This completes the proof as the inclusion M cb A ⊆ M A holds by definition. (cid:3) Remark 3.4.
Lemma 3.3 was formulated in the generality that is neededlater, that is, for the case A is commutative and B is commutative andcompletely contractive. However, it holds more generally when A is anarbitrary Banach algebra and B is a completely bounded Banach algebra.We refer the reader to [8] for more details on completely bounded multipliersof completely bounded Banach algebras.We note that(13) A h ( G ) ⊆ M cb A h ( G ) ⊆ M A h ( G ) , where the first inclusion follows from the fact that A h ( G ) is a completelycontractive Banach algebra (see Proposition 3.2 (i)). The following corollaryis immediate from Lemma 3.3 and Proposition 3.2. Corollary 3.5. If G is an amenable locally compact group then M cb A h ( G )= M A h ( G ) . Proposition 3.6.
The following hold:(i) M cb A ( G ) ⊙ M cb A ( G ) ⊆ M cb A h ( G ) ;(ii) A h ( G ) ⊆ M cb A ( G ) ⊙ M cb A ( G ) k·k cbm .Moreover, if f, g ∈ M cb A ( G ) , then k f ⊗ g k cbm ≤ k f k cbm k g k cbm . Proof.
Let f, g ∈ M cb A ( G ). Then the map m f : A ( G ) → A ( G ), given by m f ( h ) = f h , is completely bounded. Thus, m f ⊗ id : A h ( G ) → A h ( G ) iscompletely bounded; however, it is easy to note that ( m f ⊗ id)( v ) = ( f ⊗ v , v ∈ A h ( G ). Thus, f ⊗ ∈ M cb A h ( G ). By symmetry, 1 ⊗ g ∈ M cb A h ( G )and hence f ⊗ g = ( f ⊗ ⊗ g ) ∈ M cb A h ( G ). The norm inequality isstraightforward from the fact that M cb A h ( G ) is a Banach algebra.Since A h ( G ) is a completely contractive Banach algebra, if v ∈ A h ( G )then k v k cbm ≤ k v k h . Now the fact that A ( G ) ⊆ M cb A ( G ) implies A h ( G ) = A ( G ) ⊙ A ( G ) k·k h ⊆ M cb A ( G ) ⊙ M cb A ( G ) k·k cbm . (cid:3) Since k φ k ∞ ≤ k φ k B ( G ) whenever φ ∈ B ( G ) (see [6, Corollary 1.8]), astraightforward argument shows that, if w = P ∞ i =1 φ i ⊗ ψ i is an elementof B ( G ) ⊗ γ B ( G ) (where we have assumed that P ∞ i =1 k φ i k B ( G ) < ∞ and P ∞ i =1 k ψ i k B ( G ) < ∞ ) then the series P ∞ i =1 φ i ( s ) ψ i ( t ) converges for all s, t ∈ G ; thus, w can be identified with a function on G × G . Proposition 3.6 nowimplies that B ( G ) ⊗ γ B ( G ) ⊆ M cb A ( G ) ⊙ M cb A ( G ) k·k cbm ⊆ M cb A h ( G ) . It is natural to ask whether M cb A h ( G ) can be obtained from the two copiesof M cb A ( G ) lying inside it. More specifically, we formulate the followingquestion. Question 3.7. (i) Is it true that B ( G ) ⊗ h B ( G ) ⊆ M cb A h ( G ) ?(ii) Is M cb A ( G ) ⊙ M cb A ( G ) dense in M cb A h ( G ) ? Given w ∈ M cb A h ( G ), let R w be the dual of m w ; clearly, R w is a com-pletely bounded weak* continuous map on VN eh ( G ) and k R w k cb = k w k cbm .The proof of the following proposition is similar to the proof of [6, Theo-rem 1.6] which characterises the completely bounded multipliers of Fourieralgebras of locally compact groups. We note that, if H is a finite group, then A ( H ) coincides, as a set, with the algebra of all complex valued functionson H , and hence the operator projective tensor product A h ( G ) ˆ ⊗ A ( H ) canbe identified in a natural fashion with a space of functions on G × G × H . Proposition 3.8.
Let u be a bounded continuous function on G × G . Thefollowing are equivalent:(i) u ∈ M cb A h ( G ) ;(ii) there exists C > such that for every finite group H , u ⊗ belongsto M ( A h ( G ) ˆ ⊗ A ( H )) and k u ⊗ k M ( A h ( G ) ˆ ⊗ A ( H )) ≤ C .Proof. Suppose that H is a finite group and let k , . . . , k n ∈ N be the dimen-sions of the (pairwise inequivalent) irreducible representations of H . ThenVN( H ) ∼ = L ni =1 M k i . Up to complete isometries, by Corollary 7.1.5 andequation (7.1.16) in [10], we have( A h ( G ) ˆ ⊗ A ( H )) ∗ = CB ( A h ( G ) , A ( H ) ∗ ) = CB ( A h ( G ) , n M i =1 M k i )= n M i =1 CB ( A h ( G ) , M k i ) = n M i =1 M k i ( A h ( G ) ∗ )(14) = n M i =1 M k i (VN eh ( G )) . (ii) ⇒ (i) Suppose that u is a bounded continuous function satisfying thecondition in (ii). For a fixed positive integer l , choose H so that for some i , k i = l . By restricting R u ⊗ to the i th component of ( A h ( G ) ˆ ⊗ A ( H )) ∗ in the decomposition (14), we get k R u ⊗ id k i k ≤ k m ∗ u ⊗ k B (VN eh ( G ) ⊗ VN( H )) = k u ⊗ k M ( A h ( G ) ˆ ⊗ A ( H )) ≤ C. It follows that R u is a completely bounded weakly ∗ continuous map onVN eh ( G ); consequently, u ∈ M cb A h ( G ).(i) ⇒ (ii) follows from the identification (14). (cid:3) In the sequel, for w ∈ M A h ( G ) and u ∈ VN eh ( G ), we often write w · u = R w ( u ). It is clear that(15) k w · u k eh ≤ k w k cbm k u k eh . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 13
Note that if w ∈ M cb A h ( G ) then(16) w · ( λ s ⊗ λ t ) = w ( s, t )( λ s ⊗ λ t ) , s, t ∈ G. Indeed, if v ∈ A h ( G ) then, by Lemma 3.1, h w · ( λ s ⊗ λ t ) , v i = h λ s ⊗ λ t , wv i = ( wv )( s, t ) = h w ( s, t )( λ s ⊗ λ t ) , v i , and (16) is proved. Since k λ s ⊗ λ t k eh = 1, we have that(17) | w ( s, t ) | ≤ k w k cbm , s, t ∈ G. In the next proposition, we equip M cb A h ( G ) with the operator spacestructure arising from its inclusion into CB ( A h ( G )). Proposition 3.9.
The map M cb A h ( G ) × VN eh ( G ) → VN eh ( G ) given by ( w, u ) → w · u turns VN eh ( G ) into a completely contractive operator M cb A h ( G ) -module. Moreover, the module action is weak* continuous withrespect to the second variable.Proof. The map θ : ( w, u ) → w · u is clearly bilinear. If v ∈ A h ( G ), w , w ∈ M cb A h ( G ), and u ∈ VN eh ( G ) then h ( w w ) · u, v i = h u, w w v i = h u, w w v i = h w · u, w v i = h w · ( w · u ) , v i , and hence the map ( w, u ) → w · u is a module action.For [ α p,q ] ∈ M n (VN eh ( G )) and [ β l,m ] ∈ M r ( A h ( G )), let hh [ α p,q ] , [ β l,m ] ii = [ h α p,q , β l,m i ] ∈ M nr . Suppose that [ u i,j ] ∈ M n (VN eh ( G )) and [ w p,q ] ∈ M k ( M cb A h ( G )). Then k θ ( n,k ) ([ w p,q ] , [ u i,j ]) k = sup {khh θ ( k,n ) ([ w p,q ] , [ u i,j ]) , [ v s,t ] iik : [ v s,t ] ∈ ball( M r ( A h ( G )) , r ∈ N } = sup {khh [ u i,j ] , [ w p,q v s,t ] iik : [ v s,t ] ∈ ball( M r ( A h ( G )) , r ∈ N }≤ k [ u i,j ] k eh sup {k [ w p,q v s,t ] k h : [ v s,t ] ∈ ball( M r ( A h ( G )) , r ∈ N } . For a fixed r ∈ N , let T [ w p,q ] : A h ( G ) → M k ( A h ( G )) be the operator givenby T [ w p,q ] ( v ) = [ w p,q v ], v ∈ A h ( G ). By the definition of the operator spacestructure of M cb A h ( G ), for each [ v s,t ] in the unit ball of M r ( A h ( G )) we have k [ T [ w p,q ] ( v s,t )] k h ≤ k [ w p,q ] k M k ( M cb A h ( G )) . This implies that k θ ( k,n ) ([ w p,q ] , [ u i,j ]) k eh ≤ k [ w p,q ] k M k ( M cb A h ( G )) k [ u i,j ] k eh . It remains to show that the module action is weak* continuous with re-spect to the second variable. To this end, let ( u i ) i ⊆ VN eh ( G ) be a netconverging in the weak* topology to u ∈ VN eh ( G ). If w ∈ M cb A h ( G ) and v ∈ A h ( G ) then h w · u i , v i = h u i , wv i → i h u, wv i = h w · u, v i , establishing the claim. (cid:3) Lemma 3.10.
Let ψ , ψ ∈ A ( G ) and w = ψ ⊗ ψ . If T , T ∈ VN( G ) then w · ( T ⊗ T ) = ( ψ · T ) ⊗ ( ψ · T ) .Proof. Whenever φ , φ ∈ A ( G ), we have h w · ( T ⊗ T ) , φ ⊗ φ i = h T ⊗ T , ψ φ ⊗ ψ φ i = h T , ψ φ ih T , ψ φ i = h ψ · T , φ ih ψ · T , φ i = h ( ψ · T ) ⊗ ( ψ · T ) , φ ⊗ φ i . Since both w · ( T ⊗ T ) and ( ψ · T ) ⊗ ( ψ · T ) are bounded functionals on A h ( G ) and A ( G ) ⊙ A ( G ) is dense in A h ( G ), we conclude that w · ( T ⊗ T ) =( ψ · T ) ⊗ ( ψ · T ). (cid:3) Recall that ˆ ι ∗ : VN eh ( G ) → VN( G × G ), the dual of ˆ ι : A ( G × G ) → A h ( G ),is completely contractive and injective (see [4, Corollary 3.8]). Lemma 3.11.
Let u ∈ VN eh ( G ) and w ∈ A ( G × G ) . Then ˆ ι ∗ (ˆ ι ( w ) · u ) = w · ˆ ι ∗ ( u ) .Proof. For every v ∈ A ( G × G ), we have h ˆ ι ∗ (ˆ ι ( w ) · u ) , v i = h ˆ ι ( w ) · u, ˆ ι ( v ) i = h u, ˆ ι ( w )ˆ ι ( v ) i = h u, ˆ ι ( wv ) i = h w · ˆ ι ∗ ( u ) , v i . (cid:3) We recall that, if
C >
0, a locally compact group G is called weaklyamenable with constant C [7], if there exists a net ( φ α ) α of compactly sup-ported elements of A ( G ) such that k φ α k cbm ≤ C for all α and φ α → Theorem 3.12.
Let G be a locally compact group and C > . The followingare equivalent:(i) G is weakly amenable with constant C ;(ii) there exists a net ( w α ) α of compactly supported elements of A ( G ) ⊙ A ( G ) such that k w α k cbm ≤ C for all α and w α v → v in A h ( G ) for every v ∈ A h ( G ) ;(iii) there exists a net ( w α ) α of compactly supported elements of M cb A h ( G ) such that k w α k cbm ≤ C for all α and w α v → v in A h ( G ) for every v ∈ A h ( G ) .Proof. (i) ⇒ (ii) Suppose G is weakly amenable. By [7, Proposition 1.1], thereexist C > φ α ) α ⊆ A ( G ) of compactly supported elements suchthat k φ α k cbm ≤ C for all α and φ α φ → φ in A ( G ) for all φ ∈ A ( G ). Set w α = φ α ⊗ φ α . If ψ , ψ ∈ A ( G ) then, clearly, w α ( ψ ⊗ ψ ) → α ψ ⊗ ψ in A h ( G ). If v ∈ A h ( G ) is arbitrary and ǫ >
0, fix v ∈ A ( G ) ⊙ A ( G ) so that k v − v k h < ǫ/ C . Let α be such that k w α v − v k h < ǫ/ α ≥ α .If α ≥ α then k w α v − v k h ≤ k w α v − w α v k h + k w α v − v k h + k v − v k h ≤ ǫ. OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 15 (ii) ⇒ (iii) follows from Proposition 3.6 and the fact that A ( G ) ⊆ M cb A ( G ).(iii) ⇒ (i) Suppose that ( w α ) α is a net of compactly supported elementsof M cb A h ( G ) such that k w α k cbm ≤ C for all α and w α v → v in A h ( G ) forevery v ∈ A h ( G ). By Proposition 3.2 (i), A h ( G ) is a regular Banach algebra;it follows that ( w α ) α ⊆ A h ( G ).Note that k R w α k cb ≤ C for each α and R w α ( T ) → T in the weak* topol-ogy of VN eh ( G ), for every T ∈ VN eh ( G ). Let Ψ : VN( G ) → VN eh ( G ) be themap given by Ψ( T ) = T ⊗ I . Clearly, Ψ is weak* continuous and completelycontractive; in fact, Ψ is the dual of the map id ⊗ δ e . Note, in addition,that the multiplication map m : T ⊗ S T S extends uniquely to a weak*continuous completely contractive map from VN σ h ( G ) onto VN( G ) (see [11,p. 133]). We denote again by m its restriction to a map from VN eh ( G )into VN( G ) (see [11, Theorem 5.7]). Clearly, ( m ◦ Ψ)( T ) = T for every T ∈ VN( G ). We thus have that ( m ◦ R w α ◦ Ψ) α is a net of weak* continuousmaps on VN( G ) whose completely bounded norm is uniformly bounded by C . Moreover, ( m ◦ R w α ◦ Ψ)( λ s ) = w α ( s, e ) λ s , for each s ∈ G . Let ψ α : G → C be the function given by ψ α ( s ) = w α ( s, e ).Assuming that supp( w α ) ⊆ K α × K α for some compact subset K α ⊆ G , let φ α ∈ A ( G ) be a compactly supported function taking the value 1 on K α andat e . Then φ α ⊗ φ α ∈ A h ( G ) and hence w α ( φ α ⊗ φ α ) ∈ A h ( G ). It followsthat ψ α = (id ⊗ δ e )( w α ( φ α ⊗ φ α )) ∈ A ( G ) . For each T ∈ VN( G ) and ψ ∈ A ( G ), we have h ψ α ψ − ψ, T i = h ψ, ψ α · T − T i = h ψ, ( m ◦ R w α ◦ Ψ)( T ) − T i = h ψ, m ◦ ( R w α − id) ◦ Ψ( T ) i = h m ∗ ( ψ ) , ( R w α − id) ◦ Ψ( T ) i → α . Therefore, ψ α ψ → ψ in the weak topology of A ( G ). Thus, ψ belongs to theweak closure of the convex hull of the set { ψ α ψ } α .Fix 0 = ψ ∈ A ( G ). Since the weak closure and the norm closure of aconvex set are equal, the previous paragraph implies the existence of a net( ψ ′ β ) β in A ( G ) (depending on ψ ) with sup β k ψ ′ β k cbm ≤ C and k ψ ′ β ψ − ψ k cbm ≤ k ψ ′ β ψ − ψ k A → β . Consequently, the normed algebra ( A ( G ) , k · k cbm ) has an approximate unit,bounded in k·k cbm by C . By [36, Theorem 1], ( A ( G ) , k·k cbm ) has a boundedapproximate identity, and the weak amenability of G follows. (cid:3) Remark 3.13.
By the proof of Theorem 3.12, if condition (iii) is satisfiedthen then the net ( w α ) α can be chosen of the form of φ α ⊗ φ α for a net ( φ α ) α of compactly supported elements of A ( G ). In the remainder of the section, we will be concerned with the extendedHaagerup tensor product A eh ( G ) := A ( G ) ⊗ eh A ( G ) and its connection with A h ( G ) and M cb A h ( G ). We will use some technical notions from [11] andwe refer the reader to the latter paper for details. SetVN σ h ( G ) := VN( G ) ⊗ σ h VN( G );we have the canonical identification [11] A eh ( G ) ∗ ≡ VN σ h ( G ) . Similarly to the elements of VN eh ( G ), every element w of A eh ( G ) has arepresentation w = φ ⊙ ψ := P ∞ i =1 φ i ⊗ ψ i , where φ = ( φ i ) i ∈ N (resp. ψ =( ψ i ) i ∈ N ) is a bounded row (resp. column) with entries in A ( G ). Recallingthe identification A eh ( G ) ≡ CB σm (VN( G ) × VN( G ) , C ) , where the latter space consists of all multiplicatively bounded separatelyweak* continuous bilinear functionals on VN( G ) × VN( G ) [11], with a given ω ∈ A eh ( G ), we associate the function w ω : G × G → C with w ω ( s, t ) = h ω, λ s ⊗ λ t i = ω ( λ s , λ t ) = ∞ X i =1 h φ i , λ s ih ψ i , λ t i = ∞ X i =1 φ i ( s ) ψ i ( t ) . By [11], A eh ( G ) is a completely contractive Banach algebra and the multi-plication is defined as the composition of the following maps: A eh ( G ) ˆ ⊗ A eh ( G ) Ψ −→ A eh ( G ) ⊗ nuc A eh ( G ) S e −→ ( A ( G ) ˆ ⊗ A ( G )) ⊗ eh ( A ( G ) ˆ ⊗ A ( G )) m A ⊗ eh m A −→ A eh ( G ) , where Ψ is the canonical complete contraction from the projective tensorproduct to the nuclear tensor product of two copies of the operator space A eh ( G ) (see [11, p. 139]), S e is the shuffle map (see [11, Theorem 6.1]) and m A is the multiplication in A ( G ). By [11, Theorem 6.1], S ∗ e = S σ , where S σ is the shuffle map(VN( G ) ¯ ⊗ VN( G )) ⊗ σ h (VN( G ) ¯ ⊗ VN( G )) → VN σ h ( G ) ¯ ⊗ VN σ h ( G )defined on the elementary tensors by S σ (( S ⊗ S ) ⊗ ( T ⊗ T )) = ( S ⊗ T ) ⊗ ( S ⊗ T ) . Note that m A ⊗ eh m A is defined as the restriction to the space A ( G × G ) ⊗ eh A ( G × G ) = ( A ( G ) ˆ ⊗ A ( G )) ⊗ eh ( A ( G ) ˆ ⊗ A ( G ))of the map( m ∗ A ⊗ h m ∗ A ) ∗ : (VN( G × G ) ⊗ h VN( G × G )) ∗ → (VN( G ) ⊗ h VN( G )) ∗ . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 17
Thus, for ω , ω ∈ A eh ( G ) we have h ω · ω , λ s ⊗ λ t i A eh ( G ) , VN σ h ( G ) = h ( m A ⊗ eh m A ) ◦ S e ◦ Ψ( ω ⊗ ω ) , λ s ⊗ λ t i A eh ( G ) , VN σ h ( G ) = h S e ◦ Ψ( ω ⊗ ω ) , m ∗ A ( λ s ) ⊗ m ∗ A ( λ t ) i A eh ( G × G ) , VN σ h ( G × G ) = h S e ◦ Ψ( ω ⊗ ω ) , λ s ⊗ λ s ⊗ λ t ⊗ λ t i A eh ( G × G ) , VN σ h ( G × G ) = h Ψ( ω ⊗ ω ) , ( λ s ⊗ λ t ) ⊗ ( λ s ⊗ λ t ) i A eh ( G ) ⊗ nuc A eh ( G ) , VN σ h ( G ) ¯ ⊗ VN σ h ( G ) = h ω , λ s ⊗ λ t i A eh ( G ) , VN σ h ( G ) h ω , λ s ⊗ λ t i A eh ( G ) , VN σ h ( G ) , giving(18) w ω · ω ( s, t ) = w ω ( s, t ) w ω ( s, t ) . This shows that the map ω → w ω from A eh ( G ) into the algebra of allseparately continuous functions on G × G is a homomorphism. Since theelementary tensors λ s ⊗ λ t span a weak* dense subspace of VN σ h ( G ) [11,Lemma 5.8], we have that the latter map is injective. This allows us toview A eh ( G ) as an algebra (with respect to pointwise multiplication) of(separately continuous) functions on G × G .The operator multiplication in VN( G ) can be extended uniquely to aweak* continuous completely contractive map m : VN( G ) ⊗ σ h VN( G ) → VN( G ) (see [11, Proposition 5.9]). Following M. Daws [8], we denote by m ∗ its predual; thus, m ∗ is a complete contraction from A ( G ) into A ( G ) ⊗ eh A ( G ). The following special case of [8, Theorem 9.2] combined with theremarks after its proof, will play a crucial role in the next section. Theorem 3.14 ([8]) . The range of m ∗ is in M cb A h ( G ) and m ∗ is a com-plete contraction when considered as a map from A ( G ) to M cb A h ( G ) . We note that m ∗ ( φ )( s, t ) = h m ∗ ( φ ) , λ s ⊗ λ t i = h φ, λ st i = φ ( st ) , for all φ ∈ A ( G ) and all s, t ∈ G .4. Spectral synthesis in A h ( G )By Proposition 3.2, A h ( G ) is a regular commutative semisimple Banachalgebra with Gelfand spectrum G × G , and thus the problem of spectralsynthesis for closed subsets of G × G is well-posed. In this section, we linkthis problem to the problem of spectral synthesis in A ( G ). We start byrecalling some definitions, which will be specialised to A h ( G ) and A ( G ) inthe sequel. Suppose that A is a regular commutative semisimple Banachalgebra with Gelfand spectrum Ω; we can thus identify A with a subalgebraof C (Ω). Given a subset J ⊆ A , we letnull( J ) = { x ∈ Ω : a ( x ) = 0 for all a ∈ J } be its null set. Given a closed subset E ⊆ Ω, let I A ( E ) = { a ∈ A : a ( x ) = 0 for all x ∈ E } , I c A ( E ) = { a ∈ I A ( E ) : a has compact support } , and J A ( E ) = { a ∈ A : a has compact support disjoint from E } . If J ⊆ A is a closed ideal, then null( J ) = E if and only if J A ( E ) ⊆ J ⊆ I A ( E ) (see e.g. [18]). The set E is called a set of spectral synthesis (resp. local spectral synthesis ) for A , if I A ( E ) = J A ( E ) (resp. I c A ( E ) = J A ( E )).Equivalently, E is a set of spectral synthesis if J A ( E ) ⊥ = I A ( E ) ⊥ where, fora subset J ⊆ A , we have set J ⊥ = { τ ∈ A ∗ : τ ( a ) = 0 , for all a ∈ J } to be the annihilator of J in A ∗ .For an element τ ∈ A ∗ , following [23, Definition 5.1.12], we setsupp A ( τ ) def = { x ∈ Ω : for all open V ⊆ X with x ∈ V there exists a ∈ A with supp( a ) ⊆ V such that h τ, a i 6 = 0 } . Clearly,(19) (supp A ( τ )) c = { x ∈ Ω : ∃ an open set V ⊆ Ω with x ∈ V such thatif a ∈ A and supp( a ) ⊆ V then h τ, a i = 0 } . It is easy to see that E is a set of spectral synthesis if and only if h τ, a i = 0for all a ∈ I A ( E ) and all τ ∈ A ∗ with supp A ( τ ) ⊆ E .For T ∈ VN( G ), we set supp G ( T ) def = supp A ( G ) ( T ) to be the support of T introduced by Eymard [13], and for u ∈ VN eh ( G ), we set supp h ( u ) def =supp A h ( G ) ( u ). In the latter case, there is another natural candidate for asupport of u , namely, the set supp G × G ( u ) def = supp G × G (ˆ ι ∗ ( u )) where ˆ ι : A ( G × G ) → A h ( G ) is the complete contraction defined in (11). In the nextlemma we show that these two concepts coincide. Lemma 4.1.
Let u ∈ VN eh ( G ) and w ∈ A h ( G ) . Then(i) supp h ( u ) = supp G × G ( u ) ;(ii) if supp h ( u ) = ∅ then u = 0 ;(iii) supp h ( w · u ) ⊆ supp( w ) ∩ supp h ( u ) .Proof. (i) Suppose that x ∈ supp G × G ( u ) and let V ⊆ G × G be an openneighbourhood of x . Then there exists w ∈ A ( G × G ) such that supp( w ) ⊆ V and h ˆ ι ∗ ( u ) , w i 6 = 0. Thus, h u, ˆ ι ( w ) i 6 = 0 and, since ˆ ι ( w ) is supported in V ,we have that x ∈ supp h ( u ).An argument similar to the one in the proof of [13, Proposition 4.4] showsthat supp h ( u ) is the set of all x ∈ G × G so that, if w ∈ A h ( G ) is such that w · u = 0, then w ( x ) = 0. Let x ∈ supp h ( u ) and w ∈ A ( G × G ) with w · ˆ ι ∗ ( u ) = 0. By Lemma 3.11, ˆ ι ( w ) · u = 0 and so ˆ ι ( w )( x ) = 0. By [13,Proposition 4.4], x ∈ supp G × G ( u ). OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 19 (ii) By (i) and the definition of supp G × G ( u ), the element ˆ ι ∗ ( u ) of VN( G × G ) has empty support. By [13, Proposition 4.6], ˆ ι ∗ ( u ) = 0 and, since ˆ ι ∗ isinjective, u = 0.(iii) Suppose there exists x ∈ supp h ( w · u ) not contained in supp( w ). Let V be an open neighbourhood of x such that V ∩ supp( w ) = ∅ . There exists v ∈ A h ( G ) such that supp( v ) ⊆ V and h w · u, v i 6 = 0. However, h w · u, v i = h u, vw i = 0 as vw = 0, a contradiction. Therefore, supp h ( w · u ) ⊆ supp( w ).Finally, suppose that x ∈ supp h ( w · u ) does not belong to supp h ( u ), andlet V ⊆ G × G be a neighbourhood of x with V ∩ supp h ( u ) = ∅ , satisfyingthe condition on the right hand side of (19). If v ∈ A h ( G ) is supported in V then so is wv . By (19), h w · u, v i = h u, wv i = 0 , showing that x supp h ( w · u ), a contradiction. (cid:3) For a subset E ⊆ G , let E ♯ = { ( s, t ) ∈ G × G : st ∈ E } . The proof of the following lemma is immediate and we omit it.
Lemma 4.2.
Let E be a subset of G . Then ( E ♯ ) = ( E ) ♯ and ( E c ) ♯ = ( E ♯ ) c . If w : G × G → C and s, t ∈ G , we let w s,t : G → C be the function givenby w s,t ( r ) = w ( sr, r − t ). For x ∈ G , we let w x : G × G → C be the functiongiven by w x ( s, t ) := w s,t ( x ) = w ( sx, x − t ), and ˆ w x : G → C be the functiongiven by ˆ w x ( t ) := w x ( e, t ) = w ( x, x − t ). Lemma 4.3.
Let w ∈ A h ( G ) . Then(i) for each r ∈ G , the function w r belongs to A h ( G ) and k w r k h = k w k h ;(ii) the map r → w r from G into A h ( G ) is continuous;(iii) for each r ∈ G , the function ˆ w r belongs to A ( G ) and k ˆ w r k A ≤ k w k h ;(iv) the map r → ˆ w r from G into A ( G ) is continuous.Proof. Fix, throughout the proof, w ∈ A h ( G ). For r ∈ G and φ ∈ A ( G ), let L r , R r : A ( G ) → A ( G ) be the operators given by L r ( φ )( t ) = φ ( r − t ) and R r ( φ )( s ) = φ ( sr ). It is well-known that L r and R r are complete isometries;thus, the operator R r ⊗ L r : A h ( G ) → A h ( G ) is a (complete) isometry.(i), (ii) Note that(20) ( R r ⊗ L r )( w ) = w r . Indeed, this identity is straightforward if w is an element of the algebraictensor product A ( G ) ⊙ A ( G ), and, by the density of A ( G ) ⊙ A ( G ) in A h ( G )and the fact that k · k h dominates the uniform norm, it holds for an arbitrary w ∈ A h ( G ).It is easy to see that the maps r R r ( φ ) and r L r ( φ ) from G into A ( G ) are continuous, for every φ ∈ A ( G ). By (20), the map r w r from G into A h ( G ) is continuous, and k w r k h = k w k h for every r ∈ G . (iii) Similarly to (20), one can show that ˆ w r = ( δ r ⊗ L r )( w ) where δ r denotes the evaluation at r . The map δ r ⊗ L r : A h ( G ) → A ( G ) is completelycontractive; it follows that ˆ w r ∈ A ( G ) and k ˆ w r k A ≤ k w k h , r ∈ G .(iv) Fix s ∈ G and ǫ >
0. Assume that w = φ ⊗ ψ , where φ, ψ ∈ A ( G )and ψ ( t ) = ( λ t ξ, η ), t ∈ G , for some ξ, η ∈ L ( G ).ˆ w r ( t ) − ˆ w s ( t ) = ( φ ( r ) − φ ( s )) ψ ( r − t ) + φ ( s )( ψ ( r − t ) − ψ ( s − t ))= ( φ ( r ) − φ ( s )) L r ( ψ )( t ) + φ ( s )( L r ( ψ )( t ) − L s ( ψ )( t )) , for all t ∈ G ; thus,(21) ˆ w r − ˆ w s = ( φ ( r ) − φ ( s )) L r ( ψ ) + φ ( s )( L r ( ψ ) − L s ( ψ )) . Let V s be a neighbourhood of s such that | φ ( r ) − φ ( s ) | < ǫ and k λ r η − λ s η k <ǫ for all r ∈ V s . By (21), k ˆ w r − ˆ w s k A ≤ | φ ( r ) − φ ( s ) |k ψ k A + | φ ( s ) |k ξ k k ( λ r − λ s ) η k < ǫ k ψ k A + ǫ | φ ( s ) |k ξ k , for all r ∈ V s . It follows that, if w = P ni =1 φ i ⊗ ψ i for φ i , ψ i ∈ A ( G ), i =1 , . . . , n , then there exists a neighbourhood V s of s , so that k ˆ w r − ˆ w s k A < ǫ for every r ∈ V s .Finally, if w ∈ A h ( G ) is arbitrary, let v ∈ A ( G ) ⊙ A ( G ) be such that k w − v k h < ǫ/
3. Let V s be a neighbourhood of s such that k ˆ v r − ˆ v s k A < ǫ for every r ∈ V s . Then, for every r ∈ V s , using (iii) we have k ˆ w r − ˆ w s k A ≤ k ˆ w r − ˆ v r k A + k ˆ v r − ˆ v s k A + k ˆ v s − ˆ w s k A ≤ k w − v k h + k ˆ v r − ˆ v s k A < ǫ. Thus, the map r ˆ w r is continuous. (cid:3) Lemma 4.4.
Let a ∈ A ( G ) be a compactly supported function and V a = aA ( G ) ⊙ aA ( G ) k·k h .(i) If v ∈ V a then the function s → v ( s, s − t ) is integrable for each t ∈ G .(ii) For v ∈ V a , the function Γ a ( v ) : G → C given by Γ a ( v )( t ) = Z G v ( s, s − t ) ds, t ∈ G, is compactly supported and belongs to A ( G ) .(iii) The map Γ a : V a → A ( G ) is bounded and k Γ a k ≤ | supp( a ) | .Proof. Set F = supp( a ). Note that, by the injectivity of the Haageruptensor product, there is a natural completely isometric identification V a ≡ aA ( G ) ⊗ h aA ( G ).(i) For v ∈ V a and t ∈ G , the function s → v ( s, s − t ) is continuous andsupported on the compact set F ∩ tF − ; it is hence integrable.(ii), (iii) Let v ∈ V a . Since v is supported on F × F , we have that ˆ v s = 0if s F . By Lemma 4.3 (iv), the function s → ˆ v s from G into A ( G ) is OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 21 continuous. It follows that the integralΓ ′ ( v ) = Z G ˆ v s ds is well-defined in the Bochner sense. By Lemma 4.3 (iii), k Γ ′ ( v ) k A ≤| F |k v k h , and hence Γ ′ is a bounded map on V a with norm not exceeding | F | . If t ∈ G thenΓ ′ ( v )( t ) = (cid:28)Z G ˆ v s ds, λ t (cid:29) = Z G h ˆ v s , λ t i ds = Z G v ( s, s − t ) ds = Γ a ( v )( t );thus, Γ ′ = Γ a . In particuar, Γ a takes values in A ( G ) and is bounded with k Γ a k ≤ | F | . In addition, Γ a ( v )( t ) = 0 whenever t F F , and so the functionΓ a ( v ) is compactly supported. (cid:3) Lemma 4.5.
Let E ⊆ G be a closed set and a ∈ A ( G ) be compactly sup-ported. Suppose that v ∈ V a has compact support disjoint from E ♯ . Then Γ a ( v ) ∈ J A ( G ) ( E ) .Proof. Suppose that v ∈ V a has compact support disjoint from E ♯ andset F = supp( v ). By Lemma 4.4, Γ a ( v ) is compactly supported. By thecontinuity of the group multiplication, the set ˜ F def = { xy : ( x, y ) ∈ F } iscompact; moreover, ˜ F ∩ E = ∅ . Therefore, there exists an open subset W ⊆ G such that W is compact and ˜ F ⊆ W ⊆ W ⊆ E c . Again by thecontinuity of the multiplication, W ♯ is an open subset of G × G . Clearly, F ⊆ ˜ F ♯ ⊆ W ♯ ⊆ W ♯ . Now let t ∈ W c ; then ( s, s − t ) ∈ ( W c ) ♯ for each s ∈ G . By Lemma 4.2,( s, s − t ) ∈ ( W ♯ ) c and so ( s, s − t ) / ∈ F , s ∈ G . Therefore,Γ a ( v )( t ) = Z G v ( s, s − t ) ds = 0 . It follows that Γ a ( v ) has compact support (within W ) disjoint from E , andhence Γ a ( v ) ∈ J A ( G ) ( E ). (cid:3) Theorem 4.6.
Let E ⊆ G be a closed set. If E ♯ is a set of spectral synthesisfor A h ( G ) then E is a set of local spectral synthesis for A ( G ) . Moreover,if A ( G ) has a (possibly unbounded) approximate identity then E is a set ofspectral synthesis.Proof. Let φ ∈ A ( G ) vanish on E . Then m ∗ ( φ ) vanishes on E ♯ . By Theo-rem 3.14, m ∗ ( φ ) ∈ M cb A h ( G ).Fix v ∈ A h ( G ) ∩ C c ( G × G ) and let K ⊆ G be a compact subset suchthat supp( v ) ⊆ K × K . The element m ∗ ( φ ) v of A h ( G ) vanishes on E ♯ ;since E ♯ is a set of spectral synthesis for A h ( G ), there exists a sequence( v n ) n ∈ N ⊆ A h ( G ), whose elements have compact support disjoint from E ♯ ,such that v n → n →∞ m ∗ ( φ ) v . Since A ( G ) is a regular Banach algebra, there exists a compactly sup-ported function a K ∈ A ( G ) which is equal to 1 on K . Setting a K × K = a K ⊗ a K , we have va K × K = v . After replacing v n by v n a K × K if necessary,we may assume that ( v n ) n ∈ N ⊆ a K × K A h ( G ).Note that a K × K A h ( G ) k·k h = ( a K A ( G )) ⊙ ( a K A ( G )) k·k h ;therefore ( v n ) n ∈ N ⊆ V a K and hence m ∗ ( f ) v ∈ V a K . By Lemma 4.4, for any t ∈ G we haveΓ a K ( m ∗ ( φ ) v )( t ) = Z G m ∗ ( φ )( s, s − t ) v ( s, s − t ) ds = φ ( t ) Z G v ( s, s − t ) ds = ( φ Γ a K ( v ))( t ) . Hence k φ Γ a K ( v ) − Γ a K ( v n ) k A ≤ k Γ a K kk m ∗ ( φ ) v − v n k h and therefore k φ Γ a K ( v ) − Γ a K ( v n ) k A → n →∞ a K ( v n ) ∈ J A ( G ) ( E ) for each n ∈ N . Fix T ∈ J A ( G ) ( E ) ⊥ .Then(22) h φ · T, Γ a K ( v ) i = h T, φ Γ a K ( v ) i = lim n →∞ h T, Γ a K ( v n ) i = 0 . Note that, by Lemma 4.4, Γ a K ( v ) is a compactly supported function from A ( G ).Let ( U α ) α be a neighbourhood basis at e consisting of relatively compactneighbourhoods uniformly contained in a compact neighbourhood of e andordered inversely by the inclusion relation. For each α , let b α be an elementin A ( G ) ∩ C c ( G ) so that supp( b α ) ⊆ U α and b α ( e ) = 1. Set e α = b α / k b α k .It is easy to check that ( e α ) α is a bounded approximate identity of L ( G )in A ( G ) ∩ C c ( G ), all of whose elements are supported in a fixed compactneighbourhood of the identity.Fix a compactly supported element b of A ( G ). Then ( e α ⊗ b ) α ⊆ A h ( G ) ∩ C c ( G × G ), and we may assume that, for a certain compact set F ⊆ G ,supp( e α ⊗ b ) ⊆ F × F , for all α . By Lemma 4.4, if a F ∈ A ( G ) is a compactlysupported function taking value 1 on F , then Γ a F ( e α ⊗ b ) = e α ∗ b ∈ A ( G ).Moreover, by the continuity of the map r → L r ( b ) from G into A ( G ), wehave that f ∗ b = Z G f ( r ) L r ( b ) dr, f ∈ L ( G ) , where the integral is understood in the Bochner sense and is A ( G )-valued.We thus have(23) k Γ a F ( e α ⊗ b ) − b k A = k e α ∗ b − b k A ≤ sup r ∈ U α k L r ( b ) − b k A → α . Now (22) implies that h φ · T, b i = 0, for all b ∈ C c ( G ) ∩ A ( G ). Since thecompactly supported functions in A ( G ) form a dense subset of A ( G ), we OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 23 conclude that φ · T = 0. If A ( G ) has an approximate identity, say ( a α ) α ,then h T, φ i = lim α h T, φa α i = lim α h φ · T, a α i = 0and hence E is a set of spectral synthesis.Assume now that φ has compact support and let a ∈ A ( G ) ∩ C c ( G ) sothat a | supp( φ ) ≡
1; hence, aφ = φ . Therefore, h T, φ i = h T, aφ i = h φ · T, a i = 0 . Since this holds for an arbitrary element φ of I cA ( G ) ( E ), we conclude that T ∈ I cA ( G ) ( E ) ⊥ , and thus E is a set of local spectral synthesis for A ( G ). (cid:3) Recall that a locally compact group G is called a Moore group [25] if eachcontinuous irreducible unitary representation of G is finite dimensional. Werefer the reader to [28] for background on this class of groups. In the sequel,we will use the fact that every Moore group is amenable and unimodular[28, p. 1486]. Note that if G is either virtually abelian (that is, contains anopen abelian subgroup of finite index) or compact then G is a Moore group(see [25, Theorems 1 and 2]).Our next aim is Theorem 4.11, the general structure of whose proof isinspired by that of the proof of [24, Theorem 4.11]. We proceed with somepreliminary facts.Let K be a compact subset of G . In what follows we will view the space C ( K ) of all continuous functions on K as a subspace of bounded Borelfunctions on G , equipped with the uniform norm; thus, the elements of C ( K )will be considered as functions f defined on the whole of G , and such that f ( s ) = 0 whenever s K . In the sequel, we will need to make a distinctionbetween the essential supremum norm and the supremum norm; thus, wewrite k · k ∞ for the former and k · k sup for the latter. Let ( f i ) i ∈ N , ( g i ) i ∈ N ⊆ C ( K ) be sequences with P ∞ i =1 k f i k < ∞ and P ∞ i =1 k g i k < ∞ , and let w = P ∞ i =1 f i ⊗ g i be the corresponding element of C ( K ) ⊗ γ C ( K ). For every s ∈ G , we have ∞ X i =1 | f i ( s ) | ≤ ∞ X i =1 k f i k < ∞ ;similarly, P ∞ i =1 | g i ( t ) | < ∞ , t ∈ G . By the Cauchy-Schwarz inequality,the series P ∞ i =1 f i ( s ) g i ( t ) is absolutely convergent for all s, t ∈ G . One canmoreover verify that its sum does not depend on the particular represen-tation of the element w ∈ C ( K ) ⊗ γ C ( K ). We thus view the elements of C ( K ) ⊗ γ C ( K ) as (bounded measurable) functions on G × G . Lemma 4.7.
Let G be a Moore group, K ⊆ G be a compact set and w ∈ C ( K ) ⊗ γ C ( K ) . Then w s,t ∈ L ( G ) and k w s,t k ≤ | K |k w k γ , for all s, t ∈ G . Proof.
Let ( f i ) i ∈ N , ( g i ) i ∈ N ⊆ C ( K ) be sequences with P ∞ i =1 k f i k < ∞ and P ∞ i =1 k g i k < ∞ such that w = P ∞ i =1 f i ⊗ g i in C ( K ) ⊗ γ C ( K ). We have Z G | w ( sr, r − t ) | dr ≤ ∞ X i =1 Z G | f i ( sr ) || g i ( r − t ) | dr ≤ ∞ X i =1 Z s − K | f i ( sr ) | dr ! ∞ X i =1 Z tK − | g i ( r − t ) | dr ! ≤ | K | ∞ X i =1 k f i k ! / ∞ X i =1 k g i k ! / . The claim now follows. (cid:3)
Let G be a Moore group. Fix a compact set K ⊆ G and a function w ∈ C ( K ) ⊗ γ C ( K ). By Lemma 4.7, for every function h ∈ L ∞ ( G ) and any s, t ∈ G , the function hw s,t is integrable; we set( h ◦ w )( s, t ) := Z G h ( r ) w s,t ( r ) dr. Note that, by Lemma 4.7,(24) | ( h ◦ w )( s, t ) | ≤ | K |k h k ∞ k w k γ , s, t ∈ G. As customary, by b G we denote the set of all (equivalence classes of) con-tinuous irreducible unitary representations of the group G . For π ∈ b G , let H π be the Hilbert space on which π acts. Setting d π = dim H π , let { e πi } d π i =1 be an orthonormal basis of H π . Denote by π i,j the corresponding coefficientfunctions of π , that is, the functions given by π i,j ( s ) = ( π ( s ) e πi , e πj ), s ∈ G .By Lemma 4.7, the integral(25) w π ( s, t ) := Z G w s,t ( r ) π ( r ) dr is well-defined as an element of B ( H π ) for all s, t ∈ G . Let˜ w π ( s, t ) := π ( s ) w π ( s, t ) , s, t ∈ G. For all i, j = 1 , . . . , d π , set w πi,j ( s, t ) := ( w π ( s, t ) e πi , e πj )and ˜ w πi,j ( s, t ) := ( ˜ w π ( s, t ) e πi , e πj );note that w πi,j ( s, t ) = ( π i,j ◦ w )( s, t ) . Lemma 4.8.
Let G be a Moore group, K ⊆ G be a compact set, π ∈ b G and w ∈ C ( K ) ⊗ γ C ( K ) . Then the functions w πi,j and ˜ w πi,j belong to M cb A h ( G ) for all i, j = 1 , . . . , d π . Moreover, ˜ w πi,j lies in the range of the map m ∗ fromTheorem 3.14. OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 25
Proof.
Fix i, j ∈ { , . . . , d π } and let K ⊆ G be a compact set. DefineΛ : C ( K ) × C ( K ) → A ( G ) by Λ( f, g ) = ( f π i,j ) ∗ g . Then k Λ( f, g ) k A ≤ k f π i,j k k g k ≤ | K |k f k sup k g k sup ;in other words, Λ is a bounded (bilinear) map and hence induces a map(denoted in the same fashion) Λ : C ( K ) ⊗ γ C ( K ) → A ( G ) so that k Λ( w ) k A ≤ | K |k w k γ , w ∈ C ( K ) ⊗ γ C ( K ) . Let Ψ = m ∗ ◦ Λ; by Theorem 3.14, the mapΨ : C ( K ) ⊗ γ C ( K ) → M cb A h ( G )is bounded with k Ψ k ≤ | K | .Let Φ : C ( K ) ⊗ γ C ( K ) → ℓ ∞ ( G × G ) be given by Φ( w ) = ˜ w πi,j . By thedefinition of ˜ w πi,j and Lemma 4.7, k Φ( w ) k ∞ ≤ | K |k w k γ . If f, g ∈ C ( K ), w = f ⊗ g and s, t ∈ G thenΦ( w )( s, t ) = Z G w ( sr, r − t )( π ( sr ) e πi , e πj ) dr = Z G w ( r, r − st ) π i,j ( r ) dr = Z G f ( r ) π i,j ( r ) g ( r − st ) dr = m ∗ ( f π i,j ∗ g )( s, t ) = Ψ( w )( s, t ) . By linearity,Φ( w )( s, t ) = Ψ( w )( s, t ) , s, t ∈ G, w ∈ C ( K ) ⊙ C ( K ) . Let w ∈ C ( K ) ⊗ γ C ( K ) and ( w k ) k ∈ N ⊆ C ( K ) ⊙ C ( K ) be a sequence with k w k − w k γ → k →∞
0. By (17),Ψ( w k )( s, t ) → Ψ( w )( s, t ) , s, t ∈ G. On the other hand, since Φ is bounded,Φ( w k )( s, t ) → Φ( w )( s, t ) , s, t ∈ G. It follows that Φ( w )( s, t ) = Ψ( w )( s, t ) for all s, t ∈ G ; since Ψ( w ) ∈ M cb A h ( G ), we conclude that ˜ w πi,j ∈ M cb A h ( G ).A calculation similar to the one in [24, (4.6)] implies that(26) w πi,j ( s, t ) = d π X l =1 π l,j ( s ) ˜ w πi,l ( s, t ) . Since π l,j ∈ B ( G ), Proposition 3.6 implies that π l,j ⊗ ∈ M cb A h ( G ). By(26), w πi,j ∈ M cb A h ( G ). (cid:3) Let w ∈ A h ( G ). By Lemma 4.3 (ii), the function from G into A h ( G ),mapping r to w r , is continuous. Therefore, if f ∈ L ( G ) then(27) f ⋆ w := Z G f ( r ) w r dr is a well-defined A h ( G )-valued integral in Bochner’s sense. Lemma 4.9.
Let G be a Moore group and w ∈ A h ( G ) . If ( e α ) α is a boundedapproximate identity of L ( G ) then e α ⋆ w → α w .Proof. Let ( U α ) α be a basis of neighbourhood of e , directed by inverse in-clusion, and ( f α ) α ⊆ L ( G ) be such that supp( f α ) ⊆ U α and k f α k = 1 forall α . As in the proof of Proposition 3.2, if X = A ( G ) ∩ C c ( G ) then X ⊙ X isdense in A h ( G ); in addition, X ⊙ X ⊆ A ( G × G ). For a given ǫ >
0, choose v ∈ X ⊙ X so that k w − v k h < ǫ/
3. By Lemma 4.3 (i), k w r − v r k h < ǫ/ , r ∈ G. Since v ∈ A ( G × G ), there is a neighbourhood V of e such that k v r − v k A < ǫ , r ∈ V. Let α be such that U α ⊆ V . For all α ≥ α we have k f α ⋆ w − w k h = (cid:13)(cid:13)(cid:13)(cid:13)Z G f α ( r ) w r dr − w (cid:13)(cid:13)(cid:13)(cid:13) h ≤ Z V | f α ( r ) |k w r − w k h dr ≤ Z V | f α ( r ) | ( k w r − v r k h + k v r − v k h + k v − w k h ) dr ≤ k w − v k h + Z V | f α ( r ) | k v r − v k A dr < ǫ. Thus, f α ⋆ w → α w . It is immediate that ( f α ) is a bounded approximateidentity of L ( G ). By Cohen’s factorisation theorem [19, 32.22] and [19,32.33(a)], e α ⋆ w → w for any bounded approximate identity ( e α ) α in L ( G ). (cid:3) Remark 4.10. If K ⊆ G is a compact set, w ∈ C ( K ) ⊗ γ C ( K ) and f ∈ L ∞ ( G ) is compactly supported then f ◦ w = f ⋆ w .Proof. For s, t ∈ G we have( f ⋆ w )( s, t ) = h f ⋆ w, λ s ⊗ λ t i = (cid:28)Z G f ( r ) w r dr, λ s ⊗ λ t (cid:29) = Z G f ( r ) h w r , λ s ⊗ λ t i dr = Z G f ( r ) w r ( s, t ) dr = Z G f ( r ) w s,t ( r ) dr = ( f ◦ w )( s, t ) . (cid:3) Theorem 4.11.
Let G be a Moore group and E ⊆ G be a closed set. If E is a set of spectral synthesis for A ( G ) then E ♯ is a set of spectral synthesisfor A h ( G ) .Proof. Let E ⊆ G be a set of spectral synthesis for A ( G ) and w ∈ I A h ( G ) ( E ♯ ).Since G is amenable, Theorem 3.12 implies that w can be approximated bycompactly supported functions in I A h ( G ) ( E ♯ ); we may thus assume that w is compactly supported itself. Let K ⊆ G be a compact set such that OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 27 supp( w ) ⊆ K × K . We have that w ∈ C ( G ) ⊗ h C ( G ); by Grothendieck’sinequality, w ∈ C ( G ) ⊗ γ C ( G ). Write w = P ∞ k =1 f k ⊗ g k , where ( f k ) k ∈ N and( g k ) k ∈ N are families of functions in C ( G ) such that P ∞ k =1 k f k k ∞ < ∞ and P ∞ k =1 k g k k ∞ < ∞ . Let ˜ f k = f k χ K (resp. ˜ g k = f k χ K ). Then ˜ f k , ˜ g k ∈ C ( K )for each k ∈ N ; moreover, P ∞ k =1 k ˜ f k k < ∞ and P ∞ k =1 k ˜ g k k < ∞ .Letting ˜ w m = P mk =1 ˜ f k ⊗ ˜ g k , m ∈ N , we thus have that the sequence ( ˜ w m ) m ∈ N converges in C ( K ) ⊗ γ C ( K ); let ˜ w be its limit. Since the uniform norm isdominated by the projective one, we easily see that w ( s, t ) = ˜ w ( s, t ) for all s, t ∈ K . Thus, w ∈ C ( K ) ⊗ γ C ( K ).Since w ∈ I A h ( G ) ( E ♯ ), we have that w r | E ♯ = 0 for all r ∈ G , and hencethe functions w πi,j and ˜ w πi,j vanish on E ♯ for all π ∈ b G and all 1 ≤ i, j ≤ d π .In the sequel, we fix u ∈ VN eh ( G ) with supp( u ) ⊆ E ♯ . We divide the restof the proof in three steps. Step 1. w πi,j · u = 0 for all π ∈ b G and all i, j = 1 , . . . , d π .Fix π ∈ b G and i, j ∈ { , . . . , d π } . By Lemma 4.8, there exists a ∈ A ( G )such that m ∗ ( a ) = ˜ w πi,j . Since ˜ w πi,j vanishes on E ♯ , we have that a ∈ I A ( G ) ( E ). Since E is a set of spectral synthesis for A ( G ), there exists asequence ( a n ) n ∈ N ⊆ A ( G ), whose elements have compact supports disjointfrom E , such that k a n − a k A → n →∞
0. Note that the element m ∗ ( a n ) of M cb A h ( G ) vanishes on a neighbourhood of E ♯ for each n ∈ N . By The-orem 3.14, if w ′′ ∈ A h ( G ) then m ∗ ( a n ) w ′′ ∈ A h ( G ). Moreover, if w ′′ iscompactly supported then m ∗ ( a n ) w ′′ is compactly supported and vanisheson a neighbourhood of E ♯ ; hence, m ∗ ( a n ) w ′′ ∈ J A h ( G ) ( E ♯ ). By Proposition3.2, every element w ′ of A h ( G ) is the limit of compactly supported elementsof A h ( G ). It follows that m ∗ ( a n ) w ′ ∈ J A h ( G ) ( E ♯ ) for every w ′ ∈ A h ( G ) andevery n ∈ N . Therefore, h ˜ w πi,j · u, w ′ i = h u, ˜ w πi,j w ′ i = h u, m ∗ ( a ) w ′ i = lim n →∞ h u, m ∗ ( a n ) w ′ i = 0 , for every w ′ ∈ A h ( G ). This shows that ˜ w πi,j · u = 0; by (26), w πi,j · u = 0 . Step 2.
If supp h ( u ) ⊆ K × K then ( f ⋆ w ) · u = 0 for all f ∈ L ( G ).Let U be an open relatively compact subset of G such that K ⊆ U . Let a ∈ A ( G ) ∩ C c ( G ) so that a | U ≡
1. Let F ⊆ G be a compact set such that F − = F and supp( a ) ⊆ F . For a compactly supported element f ∈ L ( G ),using Lemma 4.1, for all v ∈ A h ( G ) we have h u, ( f ⋆ w ) v i = h ( a ⊗ a ) · u, ( f ⋆ w ) v i = h u, ( a ⊗ a )( f ⋆ w ) v i = h u, χ F × F ( a ⊗ a )( f ⋆ w ) v i . (28) On the other hand, using Remark 4.10 we have χ F × F ( f ⋆ w )( s, t ) = Z G χ F ( s ) χ F ( t ) f ( r ) w ( sr, r − t ) dr = χ F ( s ) χ F ( t ) Z F − F f ( r ) w ( sr, r − t ) dr = χ F × F (( χ F − F f ) ⋆ w )( s, t ) . Now (28) implies h u, ( f ⋆ w ) v i = h u, ( a ⊗ a ) χ F × F (( χ F − F f ) ⋆ w ) v i = h u, ( a ⊗ a )(( χ F − F f ) ⋆ w ) v i = h ( a ⊗ a ) · u, (( f χ F − F ) ⋆ w ) v i = h u, (( f χ F − F ) ⋆ w ) v i . Thus,(29) ( f ⋆ w ) · u = ( f χ F − F ⋆ w ) · u. If π ∈ b G , i, j ∈ { , . . . , d π } and v ∈ A h ( G ) then( a ⊗ a ) w πi,j ( s, t ) = a ( s ) a ( t ) Z G π i,j ( r ) w s,t ( r ) dr = a ( s ) a ( t ) χ F ( s ) χ F ( t ) Z G π i,j ( r ) w ( sr, r − t ) dr = a ( s ) a ( t ) Z G π i,j ( r ) χ F − F ( r ) w ( sr, r − t ) dr = ( a ⊗ a )( π i,j χ F − F ◦ w )( s, t ) . By Remark 4.10 and Step 1 we now have(( π i,j χ F − F ) ⋆ w ) · u = ( π i,j χ F − F ◦ w ) · u = ( π i,j χ F − F ◦ w ) · (( a ⊗ a ) · u )= ( a ⊗ a )( π i,j χ F − F ◦ w ) · u = ( a ⊗ a ) w πi,j · u = w πi,j · (( a ⊗ a ) · u )) = w πi,j · u = 0 . Since G is unimodular, by [9, 13.6.5], f χ F − F can be approximated in L ( G )by finite linear combinations of the form(30) m X k =1 c k χ F − F f k where, for every k , the function f k has the form π i,j for some π ∈ b G andsome i, j ∈ { , . . . , d π } . Hence, by (29),( f ⋆ w ) · u = ( f χ F − F ⋆ w ) · u = 0 . Step 3. h u, w i = 0.Let ( ε β ) β be a bounded approximate identity of A ( G ), such that supp( ε β ) ⊆ K β for some compact set K β ⊆ G . We can assume, without loss of OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 29 generality, that K ⊆ K β for each β . Assume first that u is compactlysupported and let ( e α ) α be a bounded approximate identity for L ( G ). UsingStep 2 and Lemma 4.9, we have h u, w i = lim β h u, ( ε β ⊗ ε β ) w i = lim β h ( ε β ⊗ ε β ) · u, w i = lim β lim α h ( e α ⋆ w ) · u, ε β ⊗ ε β i = 0 . If u is arbitrary then h u, w i = lim β h u, ( ε β ⊗ ε β ) w i = lim β h ( ε β ⊗ ε β ) · u, w i = 0 . We have thus shown that h u, w i = 0 whenever w ∈ I A h ( G ) ( E ♯ ) and u ∈ VN eh ( G ) is supported in E ♯ . This shows that E ♯ is a set of spectral synthesisfor A h ( G ). (cid:3) Remark.
We note that, in the proof of Theorem 4.11, the fact that G is aMoore group was essentially used in the finiteness of the sum (26). Corollary 4.12.
Let G be a Moore group. A closed set E ⊆ G is a set ofspectral synthesis for A ( G ) if and only if E ♯ is a set of spectral synthesis for A h ( G ) .Proof. Immediate from Theorems 4.6 and 4.11. (cid:3)
The subset ˜∆ = { ( s, s − ) : s ∈ G } of G × G is usually referred to as the antidiagonal of G . It is known that ifthe group G is compact, the antidiagonal is not a set of spectral synthesis for A ( G × G ) unless the connected component of the neutral element is abelian(see [15, Theorem 2.5]). On the other hand, the antidiagonal coincideswith { e } ♯ ; since { e } is a set of spectral synthesis for A ( G ), Theorem 4.11implies that ˜∆ is a set of spectral synthesis for A h ( G ) if G is a Moore group.In Section 6, we will refine this statement and give a characterisation ofall elements in the dual of A h ( G ) supported in the antidiagonal for moregeneral groups.5. The case of virtually abelian groups
It is easy to see that the flip of variables preserves spectral synthesis inthe algebra A ( G × G ). The question of whether the same holds true forthe algebra A h ( G ) is the motivation behind the present section. Recallthat a locally compact group is called virtually abelian, if it has an openabelian subgroup of finite index. We assume, in this section, that G isa virtually abelian group. We first give a general result on the extendedHaagerup tensor product; in the case of the Haagerup tensor product, itwas established in [21]. Proposition 5.1.
Let M be a unital C*-algebra. The following are equiv-alent:(i) M is subhomogeneous;(ii) the linear map f : M ⊙ M → M ⊗ eh M , given on elementary tensorsby f ( a ⊗ b ) = b ⊗ a , extends to a completely bounded map on M ⊗ eh M .Proof. (i) ⇒ (ii) Suppose that M ⊆ Z ⊗ M n ( C ) for some n and some commu-tative von Neumann algebra Z . Assume that Z coincides with the multi-plication masa of L ∞ ( X, µ ), acting on the Hilbert space L ( X, µ ), for somesuitably chosen measure space (
X, µ ). Denote by γ the involution on Z ,that is, γ ( f ) = f ∗ , f ∈ Z . Let τ n be the matrix transpose acting on M n ( C ).If a ∈ M then a ∗ = ( γ ⊗ τ n )( a ). We note first that γ ⊗ τ n is completelybounded. Indeed, if m ∈ N then making the identification M m ( Z ⊗ M n ) ≡Z ⊗ M m ( M n ), we have that the map ( γ ⊗ τ n ) ( m ) corresponds to γ ⊗ τ ( m ) n .Since τ n is completely bounded with k τ n k cb = k τ ( n ) n k = n , we have that k ( γ ⊗ τ n ) ( m ) k ≤ n for every m .Let x ∈ M ⊗ eh M and write x = A ⊙ B , where A is the row operator( a α ) α ∈ A , while B is the column operator ( b α ) α ∈ A . We have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ A a ∗ α a α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ( γ ⊗ τ ( ∞ ) n )( A )( γ ⊗ τ ( ∞ ) n )( A ∗ ) (cid:13)(cid:13)(cid:13) ≤ k γ ⊗ τ n k k A k = k γ ⊗ τ n k k AA ∗ k ≤ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ A a α a ∗ α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . A similar argument shows that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ A b α b ∗ α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ A b ∗ α b α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , and (ii) is established.(ii) ⇒ (i) We have that M ⊗ h M ⊆ M ⊗ eh M completely isometrically,and that the algebraic tensor product M ⊙ M is dense in
M ⊗ h M . Itfollows that the map f leaves M ⊗ h M invariant. By [21, Theorem 4], M issubhomogeneous. (cid:3) Theorem 5.2.
Let G be a locally compact group. The following are equiv-alent:(i) G is virtually abelian;(ii) the linear map σ : A ( G ) ⊙ A ( G ) → A ( G ) ⊗ h A ( G ) , given on elementarytensors by σ ( φ ⊗ ψ ) = ψ ⊗ φ , extends to a completely bounded map on A h ( G ) .Proof. (i) ⇒ (ii) By [25], the unitary representations of G have uniformlybounded dimension, and hence C ∗ r ( G ) is subhomogeneous. This easily im-plies that VN( G ) is subhomogeneous and, by Proposition 5.1, the flip ex-tends to a completely bounded map f on VN eh ( G ). Note that f is weak*continuous; indeed, suppose that ( u i ) i is a bounded net that converges to OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 31 an element u ∈ VN eh ( G ) in the weak* topology. For φ, ψ ∈ A ( G ) we thenhave h f ( u i ) , φ ⊗ ψ i = h u i , ψ ⊗ φ i → h u, ψ ⊗ φ i = h f ( u ) , φ ⊗ ψ i ;by the uniform boundedness of ( u i ) i and the density of the algebraic tensorproduct A ( G ) ⊙ A ( G ) in A h ( G ), we have that f ( u i ) → f ( u ) in the weak*topology. It follows that the map f has a completely bounded predual;a straightforward argument shows that this predual coincides with σ on A ( G ) ⊙ A ( G ).(ii) ⇒ (i) The dual of the map σ is easily seen to coincide with the flip onthe algebraic tensor product VN( G ) ⊙ VN( G ). By Proposition 5.1, VN( G )is subhomogeneous. By the proof of [14, Proposition 1.5], G is virtuallyabelian. (cid:3) Corollary 5.3.
Let G be a virtually abelian locally compact group and let E be a closed subset of G × G that satisfies spectral synthesis in A h ( G ) . Thenthe set ˜ E := { ( s, t ) : ( t, s ) ∈ E } satisfies spectral synthesis in A h ( G ) .Proof. By Theorem 5.2, the flip σ extends to a completely bounded mapon A h ( G ). It is easy to see that, if w ∈ A h ( G ) then σ ( w )( s, t ) = w ( t, s ), s, t ∈ G . Thus, σ carries I A h ( G ) ( E ) (resp. J A h ( G ) ( E )) onto I A h ( G ) ( ˜ E ) (resp. J A h ( G ) ( ˜ E )). The claim is now clear. (cid:3) It is well-known that the linear map S : A ( G ) → A ( G ) given by S ( u )( s ) = u ( s − ), is an isometry, and that it is completely bounded if and only if G isvirtually abelian [14, Proposition 1.5]. The adjoint S ∗ : VN( G ) → VN( G )of S is given by S ∗ ( λ s ) = λ s − ; clearly, S ∗ is weak* continuous.Suppose that G is a virtually abelian group. Let N : A ( G ) → M A h ( G )be the map given by(31) N ( a )( s, t ) = a ( st − ) , s, t ∈ G, that is, N ( a ) = (id ⊗ S ) ◦ m ∗ ( a ). If v ∈ A h ( G ) and a ∈ A ( G ) then, byTheorem 3.14, k N ( a ) v k h = k (id ⊗ S ) ( m ∗ ( a )( v )) k h ≤ k S k cb k a k A k v k h ;thus, N is bounded. One can modify the proof of Theorem 4.11 where thefunction w π defined in (25) is replaced by the function( s, t ) → Z G w ( sr, tr ) π ( r ) dr, a similar change is implemented in (27), and where the map m ∗ is replacedby the map N . The modified proof shows that if E is a set of spectralsynthesis for A ( G ) then E ♭ = { ( s, t ) ∈ G × G : st − ∈ E } is a set of spectral synthesis for A h ( G ). Similarly, the proof of Theorem 4.6 can be modified by using the map ˆΓgiven by ˆΓ( v )( t ) = Z G v ( ts, s ) ds (note that ˆΓ( a ⊗ b ) = S ( b ∗ S ( a ))). Thus instead of (23), one can show that k ˆΓ( w ⊗ e α ) − w k A → α
0. We also have ˆΓ( N ( f ) v ) = f ˆΓ( v ). Working with N in the place of m ∗ , a modification of the proof of Theorem 4.6 showsthat if E ♭ is a set of spectral synthesis for A h ( G ) then E is a set of spectralsynthesis for A ( G ).Let E ∗ = { ( s, t ) ∈ G × G : ts − ∈ E } . Combining the observations in the last two paragraphs with Corollary 5.3,we obtain the following corollary.
Corollary 5.4.
Let G be a virtually abelian group and E ⊆ G be a closedset. The following conditions are equivalent:(i) E is a set of spectral synthesis for A ( G ) ;(ii) E ♯ is a set of spectral synthesis for A h ( G ) ;(iii) E ∗ is a set of spectral synthesis for A h ( G ) . We summarise some implications of Corollary 5.4 and [29, Proposition 3.1]in the next remark.
Remark 5.5.
Let G be a virtually abelian compact group. Then the follow-ing are equivalent:(i) E is a set of spectral synthesis for A ( G ) ;(ii) E ♯ is a set of spectral synthesis for C ( G ) ⊗ γ C ( G ) ;(iii) E ∗ is a set of spectral synthesis for A ( G × G ) ;(iv) E ∗ is a set of spectral synthesis for A h ( G ) ;(v) E ♯ is a set of spectral synthesis for A h ( G ) .
6. VN( G ) ′ -bimodule maps and supports This section is centred around the correspondence between the elementsof the extended Haagerup tensor product VN eh ( G ) = VN( G ) ⊗ eh VN( G )and the completely bounded weak* continuous VN( G ) ′ -module maps on B ( L ( G )). We assume throughout the section that G is a second countablelocally compact group, and, in what follows, relate the support of an ele-ment u ∈ VN eh ( G ) to certain invariant subspaces of the completely boundedmap corresponding to u (see Theorem 6.6 and Corollary 6.7). Further, wecharacterise the elements u ∈ VN eh ( G ) supported on the antidiagonal asthose, for which the corresponding completely bounded map leaves the mul-tiplication masa of L ∞ ( G ) invariant (Theorem 6.10). It is well-known (see[26] and [27]) that the latter class consists precisely of the maps of the formΘ( µ ) with µ ∈ M ( G ), whereΘ( µ )( T ) = Z G λ s T λ ∗ s dµ ( s ) , T ∈ B ( L ( G )) . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 33
Note that, since λ ( L ( G )) is a (weak* dense) subspace of VN( G ), thealgebraic tensor product λ ( L ( G )) ⊙ λ ( L ( G )) sits naturally inside VN eh ( G ).We refer the reader to (4) for the definition of the map Φ u associated withan element u of VN eh ( G ). Lemma 6.1.
Let u ∈ VN eh ( G ) . Then there exists a net ( u α ) α ⊆ λ ( L ( G )) ⊙ λ ( L ( G )) such that(i) k u α k eh ≤ k u k eh for all α ,(ii) u α → u in the weak* topology of VN eh ( G ) , and(iii) Φ u α ( x ) → α Φ u ( x ) in the weak* topology of B ( L ( G )) , for every x ∈B ( L ( G )) .Proof. Suppose that u = P ∞ i =1 a i ⊗ b i is a w*-representation of u with theproperty that k u k eh = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 a i a ∗ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 b ∗ i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Let F ( ℓ ) be the algebra of all operators of finite rank on ℓ . Then thealgebraic tensor product F ( ℓ ) ⊙ λ ( L ( G )) is a weak* dense *-subalgebra ofthe von Neumann algebra B ( ℓ ) ¯ ⊗ VN( G ). Realise B ( ℓ ) ¯ ⊗ VN( G ) as a spaceof matrices (of infinite size) with entries in VN( G ), and note that, since P ∞ i =1 a i a ∗ i (resp. P ∞ i =1 b ∗ i b i ) is weak* convergent, ( a i ) ∞ i =1 (resp. ( b i ) ∞ i =1 ) canbe viewed as an element A (resp. B ) of B ( ℓ ) ¯ ⊗ VN( G ) supported by the firstrow (resp. by the first column). By the Kaplansky Density Theorem, thereexist nets ( ˜ A α ) α and ( ˜ B α ) α in F ( ℓ ) ⊙ λ ( L ( G )) such that k ˜ A α k ≤ k A k , k ˜ B α k ≤ k B k for all α , and˜ A α → α A, ˜ B α → α B in the strong* topology. Let A α (resp. B α ) be the compression of ˜ A α (resp.˜ B α ) to the first row (resp. column). Then k A α k ≤ k A k , k B α k ≤ k B k for all α , and A α → α A, B α → α B in the strong* topology. Let u α = A α ⊙ B α ; then u α ∈ λ ( L ( G )) ⊙ λ ( L ( G ))and(32) k u α k eh ≤ k A α kk B α k ≤ k A kk B k = k u k eh for all α . If x ∈ B ( L ( G )) thenΦ u α ( x ) = A α (1 ⊗ x ) B α → α A (1 ⊗ x ) B in the weak operator topology. By (32), k Φ u α ( x ) k ≤ k u α k eh k x k ≤ k u k eh k x k for all α , and hence Φ u α ( x ) → Φ u ( x ) in the weak* topology.It remains to show that u α → u in the weak* topology of VN eh ( G ). Let φ, ψ ∈ A ( G ), viewed as (weak* continuous) functionals on VN( G ), and let ξ, ξ ′ , η, η ′ ∈ L ( G ) be such that φ ( s ) = ( λ s ξ, η ) and ψ ( s ) = ( λ s ξ ′ , η ′ ), s ∈ G .Write A α = ( a αi ) ∞ i =1 and B α = ( b αi ) ∞ i =1 . Then, as pointed out in [11], h A ⊙ B, φ ⊗ ψ i = ∞ X i =1 h a i , φ ih b i , ψ i and h A α ⊙ B α , φ ⊗ ψ i = ∞ X i =1 h a αi , φ ih b αi , ψ i . Thus, using (1), we obtain |h A α ⊙ B α , φ ⊗ ψ i − h A ⊙ B, φ ⊗ ψ i|≤ ∞ X i =1 (cid:12)(cid:12) ( a αi ξ, η )( b αi ξ ′ , η ′ ) − ( a i ξ, η )( b i ξ ′ , η ′ ) (cid:12)(cid:12) ≤ ∞ X i =1 (cid:12)(cid:12) ( a αi ξ, η )( b αi ξ ′ , η ′ ) − ( a αi ξ, η )( b i ξ ′ , η ′ ) (cid:12)(cid:12) + ∞ X i =1 (cid:12)(cid:12) ( a αi ξ, η )( b i ξ ′ , η ′ ) − ( a i ξ, η )( b i ξ ′ , η ′ ) (cid:12)(cid:12) = ∞ X i =1 | ( a αi ξ, η ) || ( b αi ξ ′ , η ′ ) − ( b i ξ ′ , η ′ ) | + ∞ X i =1 | ( b i ξ ′ , η ′ ) || ( a αi ξ, η ) − ( a i ξ, η ) |≤ k η kk η ′ k ∞ X i =1 k a αi ξ kk b αi ξ ′ − b i ξ ′ k + k η kk η ′ k ∞ X i =1 k b i ξ ′ kk a αi ξ − a i ξ k≤ k η kk η ′ k ∞ X i =1 k a αi ξ k ! / ∞ X i =1 k b αi ξ ′ − b i ξ ′ k ! / + k η kk η ′ k ∞ X i =1 k b i ξ ′ k ! / ∞ X i =1 k a αi ξ − a i ξ k ! / = k η kk η ′ kk A α ξ kk B α ξ ′ − Bξ ′ k + k η kk η ′ kk Bξ ′ kk A α ξ − Aξ k . It follows that h A α ⊙ B α , φ ⊗ ψ i − h A ⊙ B, φ ⊗ ψ i → α . Since A ( G ) ⊙ A ( G ) is dense in A h ( G ) and the family ( A α ⊙ B α ) α of functionalson A h ( G ) is uniformly bounded, we conclude that h A α ⊙ B α , w i − h A ⊙ B, w i → α w ∈ A h ( G ), that is, u α → u in the weak* topology of VN eh ( G ). (cid:3) In the sequel, we write K = K ( L ( G )). Lemma 6.2.
Let ( u α ) α ⊆ VN eh ( G ) be a uniformly bounded net, convergingin the weak* topology to an element u ∈ VN eh ( G ) . Then (Φ u α ( T ) ξ, η ) → α (Φ u ( T ) ξ, η ) , for all T ∈ K and all ξ, η ∈ L ( G ) . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 35
Proof.
Fix ξ, η ∈ L ( G ), and assume first that T = f ⊗ g ∗ where f, g ∈ L ( G ). Let φ (resp. ψ ) be the restriction of the vector functional ω f,η (resp. ω ξ,g ) to VN( G ), viewed as an element of A ( G ). Suppose that u = P ∞ i =1 a i ⊗ b i is a w*-representation of u . Then(Φ u ( T ) ξ, η ) = (Φ u ( f ⊗ g ∗ ) ξ, η ) = ∞ X i =1 ( a i ( f ⊗ g ∗ ) b i ξ, η )= ∞ X i =1 ((( a i f ) ⊗ ( b ∗ i g ) ∗ ) ξ, η ) = ∞ X i =1 ( ξ, b ∗ i g )( a i f, η )(33) = ∞ X i =1 ( b i ξ, g )( a i f, η ) = ∞ X i =1 h a i , φ ih b i , ψ i = h u, φ ⊗ ψ i , where the last equality follows from [11, (5.9)]. Since u α → α u in the weak*topology, we conclude that(Φ u α ( T ) ξ, η ) → α (Φ u ( T ) ξ, η ) , whenever T has rank one. By linearity, the convergence holds for any finiterank operator T .Assume that T ∈ K is arbitrary and let ǫ >
0. Suppose that
C > k u k eh ≤ C and k u α k eh ≤ C for all α , and choose a finite rankoperator T with k T − T k < ǫ C k ξ kk η k . Let α be such that | (Φ u α ( T ) ξ, η ) − (Φ u ( T ) ξ, η ) | < ǫ for every α ≥ α . If α ≥ α then | (Φ u α ( T ) ξ, η ) − (Φ u ( T ) ξ, η ) |≤ | (Φ u α ( T ) ξ, η ) − (Φ u α ( T ) ξ, η ) | + | (Φ u α ( T ) ξ, η ) − (Φ u ( T ) ξ, η ) | + | (Φ u ( T ) ξ, η ) − (Φ u ( T ) ξ, η ) | ≤ C k ξ kk η kk T − T k + ǫ/ < ǫ. (cid:3) In the next statement, we use Proposition 3.6 to identify M cb A ( G ) ⊙ M cb A ( G ) with a subspace of M cb A h ( G ). Proposition 6.3.
Let µ, ν ∈ M ( G ) and u = λ ( µ ) ⊗ λ ( ν ) ∈ VN eh ( G ) . Forall w ∈ M cb A ( G ) ⊙ M cb A ( G ) k·k cbm , all T ∈ B ( L ( G )) and all ξ, η ∈ L ( G ) ,the function ( s, t ) → w ( s, t )( λ s Xλ t ξ, η ) is | µ | × | ν | -integrable and (34) (Φ w · u ( T ) ξ, η ) = Z G × G w ( t, s )( λ t T λ s ξ, η ) dµ ( t ) dν ( s ) . Proof.
Fix T ∈ B ( L ( G )) and ξ, η ∈ L ( G ). For w ∈ M cb A h ( G ), set ϕ w ( t, s ) = w ( t, s )( λ t T λ s ξ, η ) , t, s ∈ G. Using (17), we have Z G × G | ϕ w | d ( | µ | × | ν | ) ≤ k w k cbm Z G × G | ( T λ t ξ, λ s − η ) | d | µ | ( s ) d | ν | ( t ) ≤ k w k cbm k T k Z G × G k λ t ξ kk λ s − η k d | µ | ( s ) d | ν | ( t )= k w k cbm k T k Z G k λ t ξ k d | ν | ( t ) Z G k λ s − η k d | µ | ( s ) ≤ k w k cbm k T kk ξ kk η k| µ | ( G ) | ν | ( G ) . Since µ and ν are complex measures, they have finite total variation, andhence ϕ w ∈ L ( G × G, | µ | × | ν | ).Let φ, ψ ∈ M cb A ( G ) and w = φ ⊗ ψ . For every ζ ∈ L ( G ) we have Z G ( T λ ( ψν ) ξ )( r ) ζ ( r ) dr = ( T λ ( ψν ) ξ, ζ ) = ( λ ( ψν ) ξ, T ∗ ζ )= Z G ( ψ ( s ) λ s ξ, T ∗ ζ ) dν ( s )= Z G ( ψ ( s ) T λ s ξ, ζ ) dν ( s )= Z G Z G ψ ( s )( T λ s ξ )( r ) ζ ( r ) drdν ( s )= Z G (cid:18)Z G ψ ( s )( T λ s ξ )( r ) dν ( s ) (cid:19) ζ ( r ) dr. It follows that(35) (
T λ ( ψν ) ξ )( r ) = Z G ψ ( s )( T λ s ξ )( r ) dν ( s ) , for almost all r ∈ G. By (35) and Lemma 3.10,(Φ w · u ( T ) ξ, η ) = (Φ λ ( φµ ) ⊗ λ ( ψν ) ( T ) ξ, η ) = ( T λ ( ψν ) ξ, λ ( φµ ) ∗ η )= Z G Z G × G ψ ( s )( T λ s ξ )( r ) φ ( t )( λ t − η )( r ) dν ( s ) dµ ( t ) dr = Z G × G ϕ w ( t, s ) dµ ( t ) dν ( s ) . By linearity, (34) holds for every w ∈ M cb A ( G ) ⊙ M cb A ( G ).Now suppose that w is in the closure of M cb A ( G ) ⊙ M cb A ( G ), and let( w k ) k ∈ N be a sequence in M cb A ( G ) ⊙ M cb A ( G ) such that k w k − w k cbm → k →∞
0. By Proposition 3.9, w k · u → w · u in the norm of VN eh ( G ); thus,Φ w k · u → Φ w · u in the completely bounded norm, and hence(36) (Φ w k · u ( T ) ξ, η ) → k →∞ (Φ w · u ( T ) ξ, η ) . On the other hand, by (17), w k → w pointwise. By the first paragraph ofthe proof, the function ( t, s ) → ( λ t T λ s ξ, η ) is | µ | × | ν | -integrable. It follows OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 37 that the functions ϕ w k , k ∈ N , are dominated pointwise by an integrablefunction. Now the Lebesgue Dominated Convergence Theorem implies that Z G × G w k ( t, s )( λ t T λ s ξ, η ) dµ ( t ) dν ( s ) → Z G × G w ( t, s )( λ t T λ s ξ, η ) dµ ( t ) dν ( s )and this, together with (36) and the second paragraph of the proof, showsthat (34) holds for the function w . (cid:3) If f ∈ L ∞ ( G ), let M f be the operator on L ( G ) of multiplication by f ,and set D = { M f : f ∈ L ∞ ( G ) } . Recall that, for µ ∈ M ( G ), we let Θ( µ ) bethe completely bounded linear map on B ( L ( G )) given byΘ( µ )( T ) = Z G λ s T λ ∗ s dµ ( s ) , T ∈ B ( L ( G )) , where the intergral is understood in the weak sense. It is well-known thatΘ maps M ( G ) onto the space of all completely bounded weak* continu-ous VN( G ) ′ -bimodule maps that leave D invariant (see [26] and [27, Theo-rem 3.2]).We recall some notions from [2] and [12]. A measurable set κ ⊆ G × G iscalled marginally null if there exists a null set M ⊆ G such that κ ⊆ ( M × G ) ∪ ( G × M ). Two measurable sets κ and κ of G × G are called ω -equivalent (written κ ∼ = κ ) if their symmetric difference is marginally null; we say that κ is marginally contained in κ if κ \ κ is marginally null. The set κ iscalled ω -open if it is ω -equivalent to a countable union of sets of the form α × β , where α, β ⊆ G are measurable. It is called ω -closed if its complementis ω -open. For measurable α ⊆ G , let P ( α ) ∈ D be the projection givenby multiplication by the characteristic function of α . An operator T ∈B ( L ( G )) is said to be supported by κ if P ( β ) T P ( α ) = 0 whenever α, β ⊆ G are measurable sets with ( α × β ) ∩ κ ∼ = ∅ . The ω -support of a subset U ⊆B ( L ( G )) is the smallest (with respect to marginal containment) ω -closedset κ such that every element of U is supported by κ . It was shown in [2]and [12] that, given any ω -closed set κ , there exist a largest weak* closed D -bimodule M max ( κ ) (namely, the space of all operators on L ( G ) supportedby κ ) and a smallest weak* closed D -bimodule M min ( κ ) with support κ .We recall that for E ⊆ G , we write E ∗ = { ( s, t ) ∈ G × G : ts − ∈ E } and E ♯ = { ( s, t ) ∈ G × G : st ∈ E } . Proposition 6.4.
Let G be a second countable locally compact group, E ⊆ G be a compact set and V ⊆ G be a compact neighbourhood of e . Then M max ( E ∗ ) ⊆ M max (( EV ) ∗ ) ∩ K w ∗ . Proof.
Let U ⊆ G be an open set such that e ∈ U and U = V . The set EU is open, and hence ( EU ) ∗ is ω -open. Write ( EU ) ∗ ∼ = ∪ ∞ i =1 α i × β i , where α i , β i ⊆ G are measuarble subsets. Let ǫ >
0. By [12, Lemma 3.4], thereexist l ǫ ∈ N and a measurable subset L ǫ ⊆ G such that | L cǫ | < ǫ and(37) ( EU ) ∗ ∩ ( L ǫ × L ǫ ) ⊆ ∪ l ǫ i =1 α i × β i . It is easy to see that there exist (finite) families { σ p } Mp =1 and { τ q } Nq =1 ofpairwise disjoint measurable subsets of G and a subset R ⊆ { , . . . , M } ×{ , . . . , N } such that(38) ∪ l ǫ i =1 α i × β i = ∪ ( p,q ) ∈ R σ p × τ q . For each p, q ∈ { , . . . , M } × { , . . . , N } , let Π p,q : B ( L ( G )) → B ( L ( G )) bethe idempotent given byΠ p,q ( T ) = P ( τ q ) T P ( σ p ) , T ∈ B ( L ( G )) . By (37) and (38), T = X ( p,q ) ∈ R Π p,q ( T ) , for every T ∈ M max ( E ∗ ∩ ( L ǫ × L ǫ )) . However, Π p,q ( T ) ∈ M max ( σ p × τ q ), while M max ( σ p × τ q ) ⊆ M max ( σ p × τ q ) ∩ K w ∗ ⊆ M max (( EV ) ∗ ) ∩ K w ∗ , whenever ( p, q ) ∈ R . It follows that M max ( E ∗ ∩ ( L ǫ × L ǫ )) ⊆ M max (( EV ) ∗ ) ∩ K w ∗ . On the other hand, M max ( E ∗ ∩ ( L ǫ × L ǫ )) = P ( L ǫ ) M max ( E ∗ ) P ( L ǫ )and the claim follows after passing to a limit as ǫ → (cid:3) Lemma 6.5. If ( κ m ) m ∈ N is a decreasing sequence of ω -closed sets, then ∩ m ∈ N M max ( κ m ) = M max ( ∩ m ∈ N κ m ) .Proof. Set κ = ∩ m ∈ N κ m . Since κ ⊆ κ m , we have that M max ( κ ) ⊆ M max ( κ m ), m ∈ N ; thus, M max ( κ ) ⊆ ∩ m ∈ N M max ( κ m ).Assuming that κ cm ∼ = ∪ k ∈ N α mk × β mk , where α mk and β mk are measurablesubsets of G , we have that κ c ∼ = ∪ k,m ∈ N α mk × β mk . Suppose that α and β aremeasurable subsets of G such that κ ∩ ( α × β ) ∼ = ∅ and T ∈ ∩ m ∈ N M max ( κ m ).By deleting null sets from α and β if necessary, we can assume that α × β ⊆ κ c . Using [12, Lemma 3.4], one can see that there exists an increasingsequence ( K n ) n ∈ N of compact subsets of G such that G \ ( ∪ n ∈ N K n ) is null,and l n ∈ N such that( α ∩ K n ) × ( β ∩ K n ) ⊆ ∪ l n k,m =1 α mk × β mk . Using a decomposition analogous to (38), we conclude that P ( β ∩ K n ) T P ( α ∩ K n ) = 0. Since this holds for all n ∈ N , we conclude after passing to a limitthat P ( β ) T P ( α ) = 0. This shows that T ∈ M max ( κ ) and the proof iscomplete. (cid:3) For subsets
E, K ⊆ G , write Ω E,K = ∪ s ∈ K sEs − . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 39
Theorem 6.6.
Let G be a second countable locally compact group and E and K be compact subsets of G . If u ∈ VN eh ( G ) , supp h ( u ) ⊆ E ♯ ∩ ( K × K ) ,and w K ∈ A ( G × G ) is supported in K × K then Φ w K · u maps M max ( E ∗ ) into M max ((Ω E,K E ) ∗ ) .Proof. Suppose that supp h ( u ) ⊆ E ♯ ∩ ( K × K ). Let U and V be com-pact symmetric neighbourhoods of the neutral element e , and let U be anopen symmetric neighbourhood of e and W an open set such that U ⊆ W ⊆ U . Let w ∈ A ( G × G ) be a function supported in ( EU ) ♯ such that w | ( EU ) ♯ ∩ ( K × K ) ≡ T ∈ M max (( EV ) ∗ ) ∩ K . We claim that(39) Φ w K · u ( T ) ∈ M max ((Ω EV,K EU ) ∗ ) . Suppose that
L, M ⊆ G are measurable sets with ( L × M ) ∩ (Ω EV,K EU ) ∗ ∼ = ∅ .By deleting null sets from L and M if necessary, we may assume that, infact, ( L × M ) ∩ (Ω EV,K EU ) ∗ = ∅ , that is,(40) ( M L − ) ∩ (Ω EV,K EU ) = ∅ . Let ξ ∈ L ( G ) (resp. η ∈ L ( G )) be a function that vanishes almosteverywhere on L c (resp. M c ). By Lemma 6.1, there exists a uniformlybounded net ( u α ) α ⊆ λ ( L ( G )) ⊙ λ ( L ( G )) such that u α → α u in the weak*topology of VN eh ( G ). Write ˜ u α for the function on G × G corresponding tosome representative of u α , so that u α = ( λ ⊗ λ )(˜ u α ). Recall that w K is anelement of A ( G × G ) and hence of A h ( G ); by Proposition 3.6, w K belongsto the k · k cbm -closure of M cb A ( G ) ⊙ M cb A ( G ). Using Lemmas 3.11 and 6.2as well as Propositions 3.9 and 6.3, we have(Φ w K · u ( T ) ξ, η ) = (Φ w K w · u ( T ) ξ, η ) = lim α (Φ w K w · u α ( T ) ξ, η )= lim α Z G × G w K ( s, t ) w ( s, t )( λ s T λ t ξ, η )˜ u α ( s, t ) dsdt. Set h ( s, t ) = ( λ s T λ t ξ, η ) , s, t ∈ G. We claim that the integrand w K wh ˜ u α is identically zero. We consider thefollowing cases: Case 1. ( s, t ) K × K . In this case, w K ( s, t ) = 0. Case 2. ( s, t ) ( EU ) ♯ . In this case, w ( s, t ) = 0. Case 3. ( s, t ) ∈ ( K × K ) ∩ ( EU ) ♯ . We claim that, in this case h ( s, t ) = 0.To see this, note that supp( λ t ξ ) ⊆ tL and supp( λ s − η ) ⊆ s − M . Since T is supported by ( EV ) ∗ , it suffices to see that ( tL × s − M ) ∩ ( EV ) ∗ = ∅ .Assume, by way of contradiction, that there exist x, y ∈ G with( x, y ) ∈ ( tL × s − M ) ∩ ( EV ) ∗ . Write x = tx and y = s − y for some x ∈ L and y ∈ M . Then s − y x − t − = yx − ∈ EV, and so y x − ∈ sEV t. On the other hand, st ∈ EU and hence t ∈ s − EU . Thus, y x − ∈ sEV s − EU ⊆ Ω EV,K
EU.
This contradicts (40).It now follows that (Φ w K · u ( T ) ξ, η ) = 0 whenever ξ = P ( L ) ξ and η = P ( M ) η ; this implies that Φ w K · u ( T ) ∈ M max ((Ω EV,K EU ) ∗ ).Let ( U k ) k ∈ N be a sequence of compact neighbourhoods of e such that ∩ k ∈ N U k = { e } . Note that(41) ∩ k ∈ N Ω EV,K EU k ⊆ Ω EV,K E = Ω EV,K E. To see (41), assume that t ∈ ∩ k ∈ N Ω EV,K EU k and, for every k , write t = s k t k ,where s k ∈ Ω V,K E and t k ∈ U k . Then t k → k →∞ e and hence s k → t . Theinclusion in (41) is thus proved; the equality follows from the fact thatΩ EV,K E is compact.By Lemma 6.5, Φ w K · u ( T ) ∈ M max ((Ω EV,K E ) ∗ ). We have thus shown thatΦ w K · u ( M max (( EV ) ∗ ) ∩ K ) ⊆ M max ((Ω EV,K E ) ∗ ) . By Proposition 6.4 and the weak* continuity of Φ w K · u we have that(42) Φ w K · u ( M max ( E ∗ )) ⊆ M max ((Ω EV,K E ) ∗ ) . Let ( V k ) k ∈ N be a sequence of compact neighbourhoods of e such that ∩ k ∈ N V k = { e } . We claim that(43) ∩ k ∈ N Ω EV k ,K E = Ω E,K E. To show (43), assume that t ∈ ∩ k ∈ N Ω EV k ,K E . For each k ∈ N , there exist s k ∈ K , r k , p k ∈ E and t k ∈ V k such that s k r k t k s − k p k = t . Then t k → e and by the compactness of E and K we have that t ∈ Ω E,K E ; (43) is henceproved. It now follows from (42) and Lemma 6.5 thatΦ w K · u ( M max ( E ∗ )) ⊆ M max ((Ω E,K E ) ∗ ) . (cid:3) In the following corollaries, if H is a closed subgroup of G , we writeVN( H ) for the von Neumann subalgebra of VN( G ) generated by λ s , s ∈ H . Corollary 6.7.
Let G be a second countable locally compact group, H ⊆ G be a compact normal subgroup and K ⊆ G be a compact subset. If supp h ( u ) ⊆ H ♯ ∩ ( K × K ) then Φ u leaves the von Neumann algebra M H generated by VN( H ) and D invariant.Proof. Under the stated assumptions, Ω
H,K ⊆ H and hence Ω H,K H ⊆ HH ⊆ H . Suppose that supp h ( u ) ⊆ H ♯ ∩ ( K × K ) and let w ˜ K ∈ A h ( G ) besuch that w ˜ K = 1 on K × K and supp( w ˜ K ) ⊆ ˜ K for some compact set ˜ K containing a neighbourhood of K × K . Then w ˜ K .u = u and, by Theorem 6.6,Φ u ( M max ( H ∗ )) = Φ w ˜ K · u ( M max ( H ∗ )) ⊆ M max ( H ∗ ) . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 41
The claim now follows from the fact that M max ( H ∗ ) = M H (see [1]). (cid:3) Corollary 6.8.
Let G be a second countable compact group and H ⊆ G be aclosed normal subgroup. If supp( u ) ⊆ H ♯ then Φ u leaves the von Neumannalgebra M H generated by VN( H ) and D invariant.Proof. Immediate from Corollary 6.7. (cid:3)
Proposition 6.9.
Let u ∈ VN eh ( G ) and µ ∈ M ( G ) be such that Φ u = Θ( µ ) .Then h u, v i = Z G v ( s, s − ) dµ ( s ) , for all v ∈ A h ( G ) . Proof.
Let ξ, η, f, g ∈ L ( G ) and φ ( s ) = ( λ s f, η ) and ψ ( s ) = ( λ s ξ, g ) ( s ∈ G ).Then by (33) we have h u, φ ⊗ ψ i = (Φ u ( f ⊗ g ∗ ) ξ, η ) = h Θ( µ )( f ⊗ g ∗ ) ξ, η i = (cid:18)(cid:18)Z G λ s ( f ⊗ g ∗ ) λ ∗ s dµ ( s ) (cid:19) ξ, η (cid:19) = (cid:18)(cid:18)Z G ( λ s f ) ⊗ ( λ s g ) ∗ dµ ( s ) (cid:19) ξ, η (cid:19) = Z G ( λ s f, η )( ξ, λ s g ) dµ ( s )= Z G φ ( s ) ψ ( s − ) dµ ( s )= Z G ( φ ⊗ ψ )( s, s − ) dµ ( s ) . On the other hand, if v ∈ A h ( G ) then, by (17), (cid:12)(cid:12)(cid:12)(cid:12)Z G v ( s, s − ) dµ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k µ kk v k ∞ ≤ k µ kk v k h . Thus, there exists a (unique) u ′ ∈ VN eh ( G ) such that(44) h u ′ , v i = Z G v ( s, s − ) dµ ( s ) , v ∈ A h ( G ) . As A ( G ) ⊙ A ( G ) is dense in A h ( G ) and h u, v i = h u ′ , v i for every v ∈ A ( G ) ⊙ A ( G ), we obtain the statement. (cid:3) We recall that ˜∆ = { ( s, s − ) : s ∈ G } is the antidiagonal of G . Theorem 6.10.
Let G be a second countable weakly amenable locally com-pact group and u ∈ VN eh ( G ) . The following conditions are equivalent:(i) supp h ( u ) ⊆ ˜∆ ;(ii) there exists µ ∈ M ( G ) such that Φ u = Θ( µ ) . Proof. (i) ⇒ (ii) Suppose that supp h ( u ) ⊆ ˜∆. For a compact set K ⊆ G and w ∈ A ( G × G ) with support in K × K , we have, by Lemma 4.1, thatsupp h ( w · u ) ⊆ ˜∆ ∩ K × K . Hence, by Corollary 6.7,Φ w · u ( D ) ⊆ D ;by [27, Theorem 3.2], there exists a (unique) measure µ K,w ∈ M ( G ) suchthat(45) Φ w · u = Θ( µ K,w ) . Since G is weakly amenable, by Theorem 3.12, there exist a constant C >
0, compact sets K α ⊆ G and a net ( w α ) α of elements in A ( G ) ⊙ A ( G )supported in K α × K α such that k w α k cbm ≤ C for all α and w α v → v in A h ( G ) for all v ∈ A h ( G ). Set µ α = µ K α ,w α ; then, by Proposition 3.9, k µ α k = k Θ( µ α ) k cb = k w α · u k eh ≤ C k u k eh for all α . Thus, the net ( µ α ) α has a weak* cluster point; we assume with-out loss of generality that µ α → α µ in the weak* topology of M ( G ). Let f, g, ξ, η ∈ L ( G ). Then the functions s → ( λ s − ξ, g ) and s → ( f, λ s − η )belong to C ( G ) and(Θ( µ α )( f ⊗ g ∗ ) ξ, η ) = Z G ( λ s ( f ⊗ g ∗ ) λ ∗ s ξ, η ) dµ α ( s )= Z G (( f ⊗ g ∗ )( λ s − ξ ) , λ s − η ) dµ α ( s )= Z G ( λ s − ξ, g )( f, λ s − η ) dµ α ( s ) → α Z G ( λ s − ξ, g )( f, λ s − η ) dµ ( s )= (Θ( µ )( f ⊗ g ∗ ) ξ, η ) . It follows that(46) (Θ( µ α )( T ) ξ, η ) → α (Θ( µ )( T ) ξ, η )whenever T is an operator of finite rank.On the other hand, w α v → v for every v ∈ A h ( G ) and hence w α · u → u in the weak* topology of VN eh ( G ). By (15) and Lemma 6.2,(Φ w α · u ( T ) ξ, η ) → α (Φ u ( T ) ξ, η ) , for all T ∈ K . It now follows from (46) and (45) thatΦ u ( T ) = Θ( µ )( T ) , for every finite rank operator T . Since Φ u and Θ( µ ) are weak* continuous,we conclude that Φ u = Θ( µ ).(ii) ⇒ (i) Suppose that v ∈ A h ( G ) vanishes on ˜∆. By Proposition 6.9, h u, v i = Z G v ( s, s − ) dµ ( s ) = 0 . OMPLETELY BOUNDED BIMODULE MAPS AND SPECTRAL SYNTHESIS 43
It follows that supp( u ) ⊆ ˜∆. (cid:3) Remark
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Department of Mathematical Sciences, Chalmers University of Technol-ogy and the University of Gothenburg, Gothenburg SE-412 96, Sweden
E-mail address : [email protected] Pure Mathematics Research Centre, Queen’s University Belfast, BelfastBT7 1NN, United Kingdom
E-mail address : [email protected] Department of Mathematical Sciences, Chalmers University of Technol-ogy and the University of Gothenburg, Gothenburg SE-412 96, Sweden
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