Central and convolution Herz-Schur multipliers
Andrew McKee, Reyhaneh Pourshahami, Ivan G. Todorov, Lyudmila Turowska
aa r X i v : . [ m a t h . F A ] J a n CENTRAL AND CONVOLUTION HERZ–SCHURMULTIPLIERS
ANDREW MCKEE, REYHANEH POURSHAHAMI, IVAN G. TODOROV,AND LYUDMILA TUROWSKA
Abstract.
We obtain descriptions of central operator-valued Schurand Herz–Schur multipliers, akin to a classical characterisation due toGrothendieck, that reveals a close link between central (linear) multipli-ers and bilinear multipliers into the trace class. Restricting to dynamicalsystems where a locally compact group acts on itself by translation, weidentify their convolution multipliers as the right completely boundedmultipliers, in the sense of Junge–Neufang–Ruan, of a canonical quan-tum group associated with the underlying group. We provide charac-terisations of contractive idempotent operator-valued Schur and Herz–Schur multipliers. Exploiting the link between Herz–Schur multipliersand multipliers on transformation groupoids, we provide a combinato-rial characterisation of groupoid multipliers that are contractive andidempotent.
Contents
1. Introduction 22. Preliminaries 42.1. General background 42.2. Multipliers 62.3. Preliminary results 103. Central multipliers 123.1. Central Schur multipliers 123.2. Central Herz–Schur multipliers 183.3. Positive central multipliers 203.4. Connections with other types of multipliers 234. Convolution multipliers 254.1. Abelian case 254.2. General case 285. Idempotent multipliers 335.1. Central idempotent multipliers 335.2. Positive central idempotent multipliers 365.3. Idempotent convolution multipliers 38References 40
Date : 1 January 2021. Introduction
Schur multipliers originated in the work of Schur on the entry-wise (orHadamard) product of matrices in the early twentieth century. These arecomplex-valued functions, defined on the cartesian product X × Y of twomeasure spaces ( X, µ ) and (
Y, ν ) that give rise to completely bounded mapson the space K of all compact operators from L ( X, µ ) into L ( Y, ν ), actingby pointwise multiplication on the integral kernels of the operators from theHilbert-Schmidt class. A concrete description of these objects, which hasfound numerous applications thereafter, was given by Groth´endieck in hisResum´e [13]. Since then, Schur multipliers have played a significant role inoperator theory, the theory of Banach spaces, the theory of operator spaces,and have been linked to perturbation theory through the concept of doubleoperator integrals (see [6, 23] and the references therein).The theory of Herz–Schur, or completely bounded, multipliers of theFourier algebra of a locally compact group originated in the work of Herz [16],where they were viewed as a generalisation of Fourier–Stieltjes transforms.Similarly to Schur multipliers, Herz–Schur multipliers are complex-valuedfunctions, this time defined on a locally compact group G , that give riseto completely bounded maps on the reduced C ∗ -algebra C ∗ r ( G ) of G , actingby pointwise multiplication on its subalgebra L ( G ). An important develop-ment in the subject were the works of Gilbert and of Bo˙zejko and Fendler [4],showing that the Herz–Schur multipliers on the locally compact group G canbe isometrically identified with the space of all Schur multipliers on G × G of Toeplitz type. Haagerup [14] pioneered the use of Herz–Schur multipliersto study the approximation properties of operator algebras (see also [7]).Recently, several generalisations of Schur and Herz–Schur multipliers tothe ‘operator-valued’ case have appeared: B´edos and Conti [2] introducedmultipliers of a C ∗ -dynamical system based on a Hilbert module versionof the Fourier–Stieltjes algebra, and applied these techniques to study C ∗ -crossed products while, in [27], three of the present authors defined Schurand Herz–Schur multipliers with values in the space of all completely bound-ed maps on a C ∗ -algebra and obtained a version of the Bo˙zejko–Fendlercorrespondence. The use of multiplier techniques to study reduced crossedproducts, following Haagerup’s work, has been furthered by Skalski andthree of the present authors in [26], by the first author in [25], and by thefirst and the fourth authors in [28].In this paper we consider special cases of the multipliers defined in [27].We define central Schur and Herz–Schur multipliers in Definition 3.2 andDefinition 3.8, respectively. They are associated with completely boundedmaps on a C ∗ -algebra A that are multiplication operators by elements ofthe centre of the multiplier algebra of A , and are one of the most com-mon type of multipliers that arises in specific circumstances. A special caseof particular importance arises when A is abelian. Given a central Herz–Schur multiplier of the C ∗ - dynamical system ( A, G, α ), the corresponding
ENTRAL AND CONVOLUTION MULTIPLIERS 3 completely bounded map on the crossed product is an A -bimodule map.Such maps were considered by Dong and Ruan [8] in their study of theHilbert module Haagerup property of crossed products. Exploiting the factthat commutative (unital) C ∗ -algebras are simply algebras of continuousfunctions on compact topological spaces, we identify the central Schur andHerz–Schur multipliers with scalar-valued functions on three and two vari-ables, respectively. This allows us to identify a close link, that seems tohave remained unnoticed until now, between central multipliers and the bi-linear Schur multipliers into the trace class, introduced and characterisedby Coine, Le Merdy and Sukochev in [6] (see also [23]).A C ∗ -dynamical system of particular importance is ( C ( G ) , G, β ), where G is a locally compact group, C ( G ) is the C ∗ -algebra of all continuousfunctions on G vanishing at infinity, and β is the left translation action of G on C ( G ). The second main class of maps we are concerned with are theconvolution multipliers of ( C ( G ) , G, β ) introduced in [27]. We answer [27,Question 6.6], identifying the Herz–Schur multipliers of the latter dynamicalsystem with the right multipliers of a canonical quantum group associatedwith G ; in the case where G is abelian, we show that these multiplierscoincide with the elements of the Fourier–Stieltjes algebra B ( G × Γ), whereΓ is the dual group of G .Finally, we investigate when the special classes of multipliers consideredin this paper give rise to idempotent completely bounded maps. The gen-eral study of idempotent Herz–Schur multipliers goes back to Cohen [5], whocharacterised all idempotent elements of the measure algebra M ( G ). In [17],Host generalised Cohen’s characterisation by identifying the general form ofidempotents in B ( G ), for any locally compact group G , while Katavolos andPaulsen in [21] and Stan in [40] gave characterisations of contractive idem-potent Schur multipliers and contractive idempotent Herz-Schur multipliersrespectively, based on a combinatorial 3-of-4 property. In this paper, weuse the 3-of-4 property to obtain characterisations of various classes of cen-tral idempotent Schur multipliers and idempotent Herz–Schur multipliers ofdynamical systems.The paper is organised as follows. Section 2 contains background material,including a review of crossed products and multipliers as introduced in [27].The section also includes some preliminary results that will be needed later.In Section 3 we define central Schur A -multipliers, and present a character-isation of the central Schur C ( Z )-multipliers, followed by a similar char-acterisation of central Schur A -multipliers for an arbitrary C ∗ -algebra A .After introducing central Herz–Schur multipliers, we characterise the cen-tral Herz–Schur ( A, G, α )-multipliers, the central Herz–Schur ( C ( Z ) , G, α )-multipliers, as well as their canonical positive cones. Convolution multipliersare considered in Section 5, first in the abelian and then in the general case.Therein, we also investigate idempotent multipliers within the classes ofcentral and convolution multipliers from Section 3 and Section 4. A. MCKEE, R. POURSHAHAMI, I. G. TODOROV, AND L. TUROWSKA Preliminaries
Throughout this paper, we make standing separability assumptions: un-less otherwise stated, we consider only separable C ∗ -algebras, separableHilbert spaces and second-countable locally compact groups. These assump-tions allow us to consider multipliers defined on standard measure spaces.We however note that the results remain valid for the case of discrete spaceswith counting measure, in which case the separability assumptions can bedropped.2.1. General background.
Measure spaces.
We fix for the whole paper standard measure spaces(
X, µ ) and (
Y, ν ); this means that there exist locally compact, metrisable,complete, separable topologies on X and Y (called admissible topologies),with respect to which µ and ν are regular Borel σ -finite measures. Thedirect products X × Y and Y × X are equipped with the correspondingproduct measures. We use standard notation for the L p spaces over ( X, µ )and (
Y, ν ) ( p = 1 , , ∞ ); we will also consider (not necessarily countable)sets equipped with counting measure, in which case we write ℓ p ( X ) in placeof L p ( X ).Given a Banach space B , the space L p ( X, B ) ( p = 1 ,
2) is the space of(equivalence classes of) Bochner p -integrable functions from X to B withrespect to µ ; each of these spaces contains the algebraic tensor product C c ( X ) ⊙ B as a dense subspace. The identification L ( X, H ) ∼ = L ( X ) ⊗ H will be used frequently; here, and in the sequel, we denote by L ⊗ H
Hilber-tian tensor product of Hilbert spaces L and H . We refer to Williams [42, Ap-pendix B.I.4] for further details.Let B ( H , L ) be the space of all bounded linear operators from H into L ; wewrite as usual B ( H ) = B ( H , H ). For a weak ∗ -closed subspace M ⊆ B ( H , L )we let L ∞ ( X, M ) denote the space of (equivalence classes of) boundedfunctions f : X → M such that, for each x ∈ X and ξ ∈ L ( X, H ), η ∈ L ( X, L ), the functions x f ( x )( ξ ( x )) and x f ( x ) ∗ ( η ( x )) are weaklymeasurable as functions from X to H and from X to L , respectively. Weequip L ∞ ( X, M ) with the norm k f k := esssup x ∈ X k f ( x ) k and identify each f ∈ L ∞ ( X, M ) with the operator D f from L ( X, H ) to L ( X, L ) given by( D f ξ )( x ) = f ( x ) ξ ( x ). See Takesaki [41, Section IV.7] for details. We write L ∞ ( X, H ) for the space of (equivalence classes of) bounded weakly measur-able H -valued functions on X .Since we have a standing second-countability assumption for locally com-pact groups (except when we specify a discrete group) our groups are metris-able as topological spaces, and are hence standard measure spaces whenequipped with left Haar measure.2.1.2. Operator spaces.
Consider (concrete) operator spaces V ⊆ B ( H ) and W ⊆ B ( L ). The norm-closed spatial tensor product of V and W will bewritten V ⊗ W , while if V and W are weak*-closed, their weak*-spatial ENTRAL AND CONVOLUTION MULTIPLIERS 5 tensor product will be denoted V ⊗ W . The operator space projective tensorproduct V b ⊗ W satisfies the canonical completely isometric identifications( V b ⊗ W ) ∗ = CB( V, W ∗ ) = CB( W, V ∗ ) [9, Corollary 7.1.5]; if M and N arevon Neumann algebras, V = M ∗ and W = N ∗ , then ( V b ⊗ W ) ∗ = M ⊗ N ,up to a complete isometry [9, Theorem 7.2.4]. For u ∈ M n ( V ⊙ W ) let k u k h = inf {k a kk b k} , where the infimum is taken over all integers p , and allmatrices a ∈ M n,p ( V ) and b ∈ M p,n ( W ), such that u i,j = P k a i,k ⊗ b k,j ; theHaagerup tensor product V ⊗ h W is the completion of the operator space V ⊙ W in k · k h ; see [9, Chapter 9] for further details.For an index set I , we will write C ωI ( V ) for the operator space of fami-lies ( x i ) i ∈ I ⊆ V such that the sums P i ∈ J x ∗ i x i are uniformly bounded overall finite sets J ⊆ I ; equivalently, C ωI ( M ) = ℓ ( I ) c ⊗ M , where ℓ ( I ) c de-notes ℓ ( I ), equipped with the column operator space structure. Similarly, R ωI ( V ) denotes the operator space of families ( x i ) i ∈ I ⊆ V such that thesums P i ∈ J x i x ∗ i are uniformly bounded over all finite sets J ⊆ I ; equiv-alently, R ωI ( M ) = ℓ ( I ) r ⊗ M , where ℓ ( I ) r denotes ℓ ( I ), equipped withthe row operator space structure. Further details on the row and columnspaces can be found in [9] and [35]. If V and W are dual operator spacesthen their weak* Haagerup tensor product will be written V ⊗ w ∗ h W ; atypical element u ∈ V ⊗ w ∗ h W is u = P i ∈ I f i ⊗ g i , where I is some cardinal, f = ( f i ) i ∈ I ∈ R ωI ( V ) and g = ( g i ) i ∈ I ∈ C ωI ( W ); see [3] for further details.2.1.3. The trace and Hilbert–Schmidt classes.
Let H and L denote Hilbertspaces. We write K ( H , L ) (resp. S ( H , L )) for the compact (resp. traceclass) operators from H to L and simplify K ( H ) := K ( H , H ), etc. The space S ( H , L ) is equipped with the norm k T k := tr( | T | ). Recall that, via traceduality, we have isometric identifications S ( H , L ) ∼ = K ( L , H ) ∗ and B ( L , H ) ∼ = S ( H , L ) ∗ . The space of Hilbert–Schmidt operators T : H → L , with the norm k T k :=(tr( T ∗ T )) / , will be denoted S ( H , L ). These spaces will often appear with H = L ( X, µ ) and L = L ( Y, ν ), in which case we will write S ( X, Y ), S ( X ), etc .2.1.4. Crossed products.
Let A be a C ∗ -algebra, viewed as a subalgebra of B ( H A ), where H A denotes the Hilbert space of the universal representationof A . Let G be a locally compact group with modular function ∆, equippedwith left Haar measure m G , and α : G → Aut( A ) be a group homomorphismwhich is continuous in the point-norm topology, i.e. for all a ∈ A the map s α s ( a ) is continuous from G to A ; we say ( A, G, α ) is a C ∗ -dynamicalsystem. The space L ( G, A ) is a Banach ∗ -algebra when equipped with theproduct × given by( f × g )( t ) := Z G f ( s ) α s (cid:0) g ( s − t ) (cid:1) ds, f, g ∈ L ( G, A ) , t ∈ G, A. MCKEE, R. POURSHAHAMI, I. G. TODOROV, AND L. TUROWSKA the involution ∗ defined by f ∗ ( s ) := ∆( s ) − α s (cid:0) f ( s − ) ∗ (cid:1) , f ∈ L ( G, A ) , s ∈ G, and the L -norm k f k := R G k f ( s ) k ds . These definitions also give a ∗ -algebra structure on C c ( G, A ), which is a dense ∗ -subalgebra of L ( G, A ).Given a faithful representation θ : A → B ( H θ ), we define new representa-tions of A and G on L ( G, H θ ) as follows: π θ : A → B ( L ( G, H θ )); (cid:0) π θ ( a ) ξ (cid:1) ( t ) := θ (cid:0) α t − ( a ) (cid:1)(cid:0) ξ ( t ) (cid:1) ,λ θ : G → B ( L ( G, H θ )); ( λ θt ξ )( s ) := ξ ( t − s ) , for all a ∈ A , s, t ∈ G , ξ ∈ L ( G, H θ ). Then λ θ is a (strongly continuous)unitary representation of G and π θ (cid:0) α t ( a ) (cid:1) = λ θt π θ ( a )( λ θt ) ∗ , a ∈ A, t ∈ G. The pair ( π θ , λ θ ) is thus a covariant representation of ( A, G, α ) and thereforegives rise to a ∗ -representation π θ ⋊ λ θ : L ( G, A ) → B ( L ( G, H θ )) given by( π θ ⋊ λ θ )( f ) := Z G π θ (cid:0) f ( s ) (cid:1) λ θs ds, f ∈ L ( G, A ) . The reduced crossed product A ⋊ α,r G of A by G is independent of thechoice of the faithful representation θ and is defined as the closure of ( π θ ⋊ λ θ )( L ( G, A )) in the operator norm of B ( L ( G, H θ )); if we want to empha-sise the representation θ of A was used, we will write A ⋊ α,θ G . In Section 4we will use the weak* closure A ⋊ w ∗ α,r G of A ⋊ α,r G . In what follows wewill often simplify our notation by omitting the superscript θ . More onreduced crossed products can be found in Pedersen [33, Chapter 7], andWilliams [42].2.2. Multipliers.
We will use some well-known results on classical Schurand Herz–Schur multipliers, as well as results from [27]. We recall somedefinitions and results required later.2.2.1.
Schur multipliers.
Let (
X, µ ) and (
Y, ν ) be standard measure spaces.We say E ⊆ X × Y is marginally null if there exist null sets M ⊆ X and N ⊆ Y such that E ⊆ ( M × Y ) ∪ ( X × N ). Two measurable sets E, F ⊆ X × Y are called marginally equivalent if their symmetric difference is marginallynull; we say that two functions ϕ, ψ : X × Y → C are marginally equivalent if they are equal up to a marginally null set. A measurable set E ⊆ X × Y iscalled ω -open if it is marginally equivalent to a set of the form ∪ k ∈ N I k × J k ,where I k ⊆ X and J k ⊆ Y are measurable, k ∈ N . The collection of ω -open subsets of X × Y is a pseudo-topology on X × Y — it is closed underfinite intersections and countable unions; see [10, Section 3]. A function h : X × Y → C is called ω -continuous [10] if h − ( U ) is ω -open for everyopen set U ⊆ C . ENTRAL AND CONVOLUTION MULTIPLIERS 7
Let H be a separable Hilbert space and A ⊆ B ( H ) be a separable C ∗ -algebra. With any k ∈ L ( Y × X, A ), one can associate an element T k ∈B ( L ( X, H ) , L ( Y, H )) with k T k k ≤ k k k , by letting( T k ξ )( y ) := Z X k ( y, x )( ξ ( x )) dx, ξ ∈ L ( X, H ) , y ∈ Y. The linear space of all such operators is denoted by S ( X, Y ; A ) and is normdense in K ( L ( X ) , L ( Y )) ⊗ A ; we equip it with the operator space structurearising from this inclusion. Note that if A = C then the map k → T k is anisometric identification of L ( Y × X ) and S ( X, Y ).If B is a(nother) C ∗ -algebra we write CB( A, B ) for the space of completelybounded maps from A to B and set CB( A ) = CB( A, A ). We say that ϕ : X × Y → CB(
A, B ) is pointwise-measurable if ( x, y ) ϕ ( x, y )( a ) ∈ B is weakly measurable for each a ∈ A . If ϕ : X × Y → CB( A ) is a bounded,pointwise-measurable function, we define ϕ · k ∈ L ( Y × X, A ) by( ϕ · k )( y, x ) := ϕ ( x, y ) (cid:0) k ( y, x ) (cid:1) , ( y, x ) ∈ Y × X. Let S ϕ denote the bounded linear map on S ( X, Y ; A ) given by S ϕ ( T k ) := T ϕ · k , k ∈ L ( Y × X, A ) . Definition 2.1.
A bounded, pointwise-measurable function ϕ : X × Y → CB( A ) is called a Schur A -multiplier if S ϕ is a completely bounded mapon S ( X, Y ; A ) . We denote the space of such functions by S ( X, Y ; A ) andendow it with the norm k ϕ k S ( X,Y ; A ) := k S ϕ k cb (we write k ϕ k S when X, Y and A are clear from context). This definition does not depend on the faithful ∗ -representation of A on aseparable Hilbert space [27, Proposition 2.3]. Theorem 2.2. [27, Theorem 2.6] Let A ⊆ B ( H ) be a separable C ∗ -algebraand ϕ : X × Y → CB( A ) a bounded, pointwise measurable function. Thefollowing are equivalent:i. ϕ is a Schur A -multiplier;ii. there exist a separable Hilbert space H ρ , a non-degenerate ∗ -representa-tion ρ : A → B ( H ρ ) , and V ∈ L ∞ ( X, B ( H , H ρ )) , W ∈ L ∞ ( Y, B ( H , H ρ )) such that ϕ ( x, y )( a ) = W ( y ) ∗ ρ ( a ) V ( x ) , a ∈ A for almost all ( x, y ) ∈ X × Y .Moreover, if these conditions hold then we may choose V and W so that k ϕ k S = esssup x ∈ X kV ( x ) k esssup y ∈ Y kW ( y ) k . Note that the definitions and theorems make sense in the case X , Y arediscrete spaces with counting measures, in which case we do not need toassume separability. A. MCKEE, R. POURSHAHAMI, I. G. TODOROV, AND L. TUROWSKA
When discussing Schur A -multipliers we shall always assume withoutmentioning that A is separable unless X and Y are discrete spaces withcounting measures in which case A can be arbitrary.In the case where A = C , Schur A -multipliers reduce to classical (mea-surable) Schur multipliers [34]. The elements P ∞ i =1 f i ⊗ g i of the projec-tive tensor product S ( Y, X ) = L ( X, µ ) b ⊗ L ( Y, µ ) (where we assume P ∞ i =1 k f i k < ∞ and P ∞ i =1 k g i k < ∞ ) can be identified with functions P ∞ i =1 f i ( x ) g i ( y ) on X × Y , well-defined up to a marginally null set [1]; un-der this identification, Schur multipliers coincide with the multipliers of S ( Y, X ).Given a ∈ L ∞ ( X, µ ), let M a be the operator on L ( X, µ ) defined by( M a ξ )( x ) := a ( x ) ξ ( x ) , x ∈ X. Let D X = { M a : a ∈ L ∞ ( X, µ ) } and define D Y analogously. By a well-known result of Haagerup [15] (see also [3]), there is a completely iso-metric weak*-homeomorphism between the algebra of weak*-continuous,completely bounded D Y , D X -bimodule maps on B ( L ( X ) , L ( Y )) and theweak* Haagerup tensor product D Y ⊗ w ∗ h D X [3]; this homeomorphism sends P ∞ k =1 b k ⊗ a k ∈ D Y ⊗ w ∗ h D X to the map T ∞ X k =1 b k T a k on B ( L ( X ) , L ( Y )). Note that D Y ⊗ w ∗ h D X can be viewed as a space of(equivalence classes of) functions, and each of these functions belongs to S ( X, Y ). Theorem 2.2 can be specialised as follows in the scalar-valuedcase.
Theorem 2.3.
Let ϕ ∈ L ∞ ( X × Y ) . The following are equivalent:i. ϕ ∈ S ( X, Y ) and k ϕ k S ≤ C ;ii. there exists sequences ( a k ) ∞ k =1 ⊆ L ∞ ( X, µ ) and ( b k ) ∞ k =1 ⊆ L ∞ ( Y, ν ) with esssup x ∈ X ∞ X k =1 | a k ( x ) | ≤ C and esssup y ∈ Y ∞ X k =1 | b k ( y ) | ≤ C such that ϕ ( x, y ) = ∞ X k =1 a k ( x ) b k ( y ) for almost all ( x, y ) ∈ X × Y ; iii. there exist a separable Hilbert space H and weakly measurable functions v : X → H , w : Y → H , such that esssup x ∈ X k v ( x ) k ≤ √ C, esssup y ∈ Y k w ( y ) k ≤ √ C and ϕ ( x, y ) = h v ( x ) , w ( y ) i , for almost all ( x, y ) ∈ X × Y ; iv. k T ϕ · k k ≤ C k T k k for all k ∈ L ( Y × X ) . ENTRAL AND CONVOLUTION MULTIPLIERS 9
We remark that if X and Y are discrete spaces with counting measuresthe theorem holds true with possibly uncountable families ( a k ) and ( b k ).2.2.2. Herz–Schur multipliers.
Let G be a locally compact second countablegroup, vN( G ) (resp. C ∗ r ( G )) be its von Neumann algebra (resp. reduced C ∗ -algebra) and A ( G ) be the Fourier algebra of G [11]. Let A be a separable C ∗ -algebra. A bounded function F : G → CB( A ) will be called pointwise-measurable if, for every a ∈ A , the map s F ( s )( a ) is a weakly measurablefunction from G into A . Suppose that the function F : G → CB( A ) isbounded and pointwise-measurable, and define( F · f )( s ) := F ( s ) (cid:0) f ( s ) (cid:1) , f ∈ L ( G, A ) , s ∈ G. Since F is pointwise-measurable, F · f is weakly measurable, and k F · f k ≤ sup s ∈ G k F ( s ) kk f k ( f ∈ L ( G, A )); hence F · f ∈ L ( G, A ) for every f ∈ L ( G, A ). Definition 2.4.
A bounded, pointwise measurable function F : G → CB( A ) will be called a Herz–Schur (
A, G, α )-multiplier if the map S F on ( π ⋊ λ )( L ( G, A )) , given by S F (cid:0) ( π ⋊ λ )( f ) (cid:1) := ( π ⋊ λ )( F · f ) , is completely bounded. If F is a Herz–Schur ( A, G, α )-multiplier, we continue to denote by S F the corresponding extension to a completely bounded map on A ⋊ α,r G .Definition 2.4 is independent of the faithful representation of A [27, Re-mark 3.2(ii)]. We note that the set of all Herz–Schur ( A, G, α )-multipliersis an algebra with respect to the pointwise operations; we denote it by S ( A, G, α ) and endow it with the norm k F k HS := k S F k cb .The definition makes sense when G is an arbitrary discrete group. In thiscase we can drop the separability assumption on A .In what follows we shall always consider C ∗ -dynamical systems ( A, G, α )where either G is second countable and A is separable or G is discrete inwhich case A can be arbitrary.Given a function F : G → CB( A ), define N ( F ) : G × G → CB( A ) byletting N ( F )( s, t )( a ) := α t − (cid:0) F ( ts − ) (cid:0) α t ( a ) (cid:1)(cid:1) , s, t ∈ G, a ∈ A. Observe that if F is pointwise measurable then so is N ( F ). The follow-ing result [27, Theorem 3.5] relates Schur A -multipliers and Herz–Schur( A, G, α )-multipliers, generalising a classical transference result of Bo˙zejko–Fendler [4].
Theorem 2.5.
Let ( A, G, α ) be a C ∗ -dynamical system and F : G → CB( A ) a bounded, pointwise-measurable function. The following are equivalent:i. F is a Herz-Schur ( A, G, α ) -multiplier;ii. N ( F ) is a Schur A -multiplier. Moreover, if the above conditions hold then k F k HS = kN ( F ) k S . The Schur A -multipliers ϕ of the form ϕ = N ( F ) will be called α -invariant . We note that a different definition was given in [27] (see [27, Def-inition 3.14]), but by [27, Theorem 3.18], it agrees with the one adoptedhere.In the case where A = C and the action is trivial, Herz–Schur ( A, G, α )-multipliers coincide with the classical
Herz–Schur multipliers of G [7], thatis, with the functions u : G → C such that uA ( G ) ⊆ A ( G ) and the map m u : A ( G ) → A ( G ); m u ( v ) := uv, v ∈ A ( G ) , is completely bounded. Here we equip A ( G ) with the operator space struc-ture, arising from the identification A ( G ) ∗ = vN( G ) [11]. The space ofclassical Herz–Schur multipliers of G will be denoted by M cb A ( G ). We notethat if u ∈ M cb A ( G ) then the restriction S u := m ∗ u | C ∗ r ( G ) is a completelybounded map satisfying [7] S u : C ∗ r ( G ) → C ∗ r ( G ); S u (cid:0) λ ( f ) (cid:1) = λ ( uf ) , f ∈ L ( G ) . Preliminary results.
In this subsection, we give several technical re-sults on Schur and Herz–Schur multipliers that will be needed in the sequel.The equivalence between (i) and (iii) in the next proposition was given, inthe scalar-valued case, in [21, Theorem 7].
Proposition 2.6.
Let H be a separable Hilbert space, A ⊆ B ( H ) a separa-ble C ∗ -algebra and ϕ : X × Y → CB( A ) a bounded, pointwise-measurablefunction. The following are equivalent:i. ϕ ( x, y ) = 0 for almost all ( x, y ) ∈ X × Y ;ii. S ϕ = 0 .If ϕ is a Schur A -multiplier of the form ϕ ( x, y )( a ) = W ( y ) ∗ ρ ( a ) V ( x ) , a ∈ A ,as in Theorem 2.2, then these conditions are equivalent to:iii. ϕ ( x, y ) = 0 for marginally almost all ( x, y ) ∈ X × Y .Proof. (i) = ⇒ (ii) Let T k ∈ S ( X, Y ; A ). If ϕ ( x, y ) = 0 for almost all( x, y ) ∈ X × Y then ϕ · k = 0 almost everywhere, for every k ∈ L ( Y × X, A ),and hence S ϕ ( T k ) = T ϕ · k = 0 for every k ∈ L ( Y × X, A ).(ii) = ⇒ (i) Suppose S ϕ = 0 and let k ∈ L ( Y × X, A ). We have S ϕ ( T k ) = T ϕ · k = 0, so we conclude that ϕ · k = 0 almost everywhere by [27, Lemma2.1]. We claim that ϕ ( x, y ) = 0 for almost all ( x, y ) ∈ X × Y . Indeed, let { e i } i ∈ N be a dense subset of H , ξ ∈ L ( X ) and η ∈ L ( Y ); then h S ϕ ( T k )( ξ ⊗ e i ) , η ⊗ e j i = Z Y h S ϕ ( T k )( ξ ⊗ e i )( y ) , ( η ⊗ e j )( y ) i dy (1) = Z Y (cid:28)Z X ( ϕ · k )( y, x )( ξ ⊗ e i )( x ) dx, ( η ⊗ e j )( y ) (cid:29) dy = Z Y Z X (cid:10) ϕ ( x, y ) (cid:0) k ( y, x ) (cid:1) e i , e j (cid:11) ξ ( x ) η ( y ) dx dy. ENTRAL AND CONVOLUTION MULTIPLIERS 11
Fix a ∈ A , choose w ∈ L ( Y × X ), and let k ( y, x ) = w ( y, x ) a . Then (1)implies Z Y Z X h ϕ ( x, y )( a ) e i , e j i w ( y, x ) ξ ( x ) η ( y ) dx dy = 0 . Since ϕ ( x, y )( a ) is a bounded operator, we conclude that h ϕ ( x, y )( a ) e i , e j i =0 almost everywhere for all i, j ∈ N . Hence ϕ ( x, y ) = 0 almost everywhereby the separability of A and the continuity of ϕ ( x, y ).Now suppose that ϕ is a Schur A -multiplier.(iii) = ⇒ (i) is trivial.(i) = ⇒ (iii) Assume that the set R := { ( x, y ) ∈ X × Y : ϕ ( x, y ) = 0 } is null. Let A and H be countable dense subsets of A and H respectively;then R c = { ( x, y ) : ϕ ( x, y ) = 0 } = \ a ∈ A , ξ,η ∈H { ( x, y ) : h ϕ ( x, y )( a ) ξ, η i = 0 } = \ a ∈ A , ξ,η ∈H { ( x, y ) : h ρ ( a ) V ( x ) ξ, W ( y ) η i = 0 } . It is easily seen that a function of the form ( x, y )
7→ h α ( x ) , β ( y ) i , where α ∈ L ∞ ( X, H ρ ) and β ∈ L ∞ ( Y, H ρ ), is ω -continuous; thus, the set { ( x, y ) : h α ( x ) , β ( y ) i 6 = 0 } is ω -open. It follows that the set [ a ∈ A , ξ,η ∈H { ( x, y ) : h ρ ( a ) V ( x ) ξ, W ( y ) η i 6 = 0 } is ω -open. Hence there are families A n ⊆ X, B n ⊆ Y of measurable setssuch that R is marginally equivalent to ∪ ∞ n =1 A n × B n . Since ( µ × ν )( R ) = 0we have µ ( A n ) ν ( B n ) = 0 for each n . Let N := [ ν ( B n ) =0 A n and N := [ µ ( A n ) =0 B n . We have that µ ( N ) = 0, ν ( N ) = 0 and R that is marginally equivalent toa subset of N × Y ∪ X × N ; thus, R is marginally null. (cid:3) The next lemma contains a completely isometric version of the main trans-ference result of [27, Section 3].
Lemma 2.7.
Let ( A, G, α ) be a C ∗ -dynamical system. The map N is acompletely isometric algebra homomorphism from the space of Herz–Schur ( A, G, α ) -multipliers to the space of Schur A -multipliers on G × G .Proof. Fix n ∈ N and Herz–Schur ( A, G, α )-multipliers F i,j , 1 ≤ i, j ≤ n .Since ( S F i,j ) i,j is an element of CB( A ⋊ α,r G, M n ( A ⋊ α,r G )) there exist arepresentation ρ : A ⋊ α,r G → B ( H ρ ) and operators V, W : L ( G, H ) → H ρ such that ( S F i,j ) i,j = W ∗ ρ ( · ) V and k V kk W k = k ( S F i,j ) i,j k cb . Take a ∈ A and r ∈ G . Arguing as in the proof of [27, Theorem 3.8] we obtainrepresentations ρ A and ρ G , of A and G respectively, such that (cid:0) π ( F i,j ( t )( a )) λ r (cid:1) i,j = (cid:0) S F i,j ( π ( a ) λ r ) (cid:1) i,j = W ∗ ρ A ( a ) ρ G ( r ) V. Define V ( s ) := ρ G ( s − ) V λ s and W ( t ) := ρ G ( t − ) W λ t , so that sup s ∈ G kV ( s ) k sup t ∈ G kW ( t ) k = k V kk W k = k ( S F i,j ) i,j k cb . Calcula-tions as in the proof of [27, Theorem 3.8] show that( N ( F i,j )( s, t )( a )) i,j = W ( t ) ∗ ρ A ( a ) V ( s ) , almost everywhere, so k ( S N ( F i,j ) ) i,j k cb ≤ sup s ∈ G kV ( s ) k sup t ∈ G kW ( t ) k = k V kk W k = k ( S F i,j ) i,j k cb . In the converse direction, note that ( S F i,j ) i,j is the restriction of ( S N ( F i,j ) ) i,j to M n ( A ⋊ α,r G ), so k ( S F i,j ) i,j k cb ≤ k ( S N ( F i,j ) ) i,j k cb . Thus F
7→ N ( F ) is acomplete isometry. The homomorphism claim is trivial. (cid:3) Central multipliers
Let (
X, µ ) and (
Y, ν ) be standard measure spaces. We denote for brevityby B (resp. K ) the space B ( L ( X, µ ) , L ( Y, ν )) (resp. K ( L ( X, µ ) , L ( Y, ν ))).Throughout this section A denotes a separable C ∗ -algebra, acting non-degenerately on a separable Hilbert space H . The multiplier algebra of A will be written M ( A ) and identified with the idealiser of A in B ( H ): M ( A ) = { c ∈ B ( H ) : ca, ac ∈ A for all a ∈ A } . As usual, we denote by Z ( B ) the centre of the C ∗ -algebra B .The following is immediate, and will be used several times in the sequel. Remark 3.1.
Let B ⊆ A be a C ∗ -subalgebra, and ϕ : X × Y → CB( A ) bea Schur A -multiplier. Suppose that ϕ ( x, y ) leaves B invariant for almost all( x, y ), and let ϕ B : X × Y → CB( B ) be the map given by ϕ B ( x, y )( b ) := ϕ ( x, y )( b ) ( b ∈ B , ( x, y ) ∈ X × Y ). Then ϕ B is a Schur B -multiplier and k ϕ B k S ≤ k ϕ k S .3.1. Central Schur multipliers.Definition 3.2.
A Schur A -multiplier ϕ ∈ S ( X, Y ; A ) will be called central if there exists a family ( a x,y ) ( x,y ) ∈ X × Y ⊆ Z ( M ( A )) such that (2) ϕ ( x, y )( a ) = a x,y a, a ∈ A. Remark 3.3.
Let ϕ ∈ S ( X, Y ; A ) be a central Schur A -multiplier.i. The family ( a x,y ) ( x,y ) ∈ X × Y associated to ϕ in Definition 3.2 is uniqueup to a set of zero product measure.ii. If ( a x,y ) ( x,y ) ∈ X × Y is associated to ϕ as in Definition 3.2 then the map X × Y → Z ( M ( A )), ( x, y ) a x,y , is weakly measurable. ENTRAL AND CONVOLUTION MULTIPLIERS 13
Let A be a commutative C ∗ -algebra, and assume that A = C ( Z ), where Z is a locally compact Hausdorff space. The standing separability assump-tion implies that Z is second-countable, and hence metrisable. Since C ( Z )is separable it has a faithful state, so the associated Radon measure m on Z has full support.Let C ( Z, B ) be the space of all continuous functions from Z into a normedspace B vanishing at infinity. We write K = K ( L ( X ) , L ( Y )) and note that,up to a canonical ∗ -isomorphism,(3) K ⊗ C ( Z ) = C ( Z, K ) . The algebraic tensor product L ( Y × X ) ⊙ C ( Z ) can thus be viewed as a(dense) subspace of any of the spaces K ⊗ C ( Z ) and C ( Z, K ).Let ϕ ∈ S ( X, Y ; C ( Z )) be a central Schur C ( Z )-multiplier, associatedwith a family ( a x,y ) ( x,y ) ∈ X × Y ⊆ C b ( Z ) as in Definition 3.2; we view ϕ as ascalar-valued function on X × Y × Z by letting ϕ ( x, y, z ) = a x,y ( z ) , x ∈ X, y ∈ Y, z ∈ Z. By definition, ϕ is a bounded, measurable function on X × Y × Z which iscontinuous in the Z -variable. On the other hand, suppose ϕ : X × Y × Z → C is a bounded measurable function, continuous in the Z -variable. Then( x, y ) ϕ ( x, y, · ) a ( · ) ∈ C ( Z ) is weakly measurable for each a ∈ C ( Z ).Indeed, the function ( x, y ) δ z ( ϕ ( x, y )( a )) = ϕ ( x, y, z ) a ( z ) is measurablefor each z ∈ Z (here δ z stands for the point mass measure at z ∈ Z ). Asany m ∈ M ( Z ) = C ( Z ) ∗ is the weak* limit of linear combinations of pointmass measures, we conclude that the function ( x, y ) m ( ϕ ( x, y )( a )) ismeasurable for all m ∈ M ( Z ). We thus identify the central Schur C ( Z )-multipliers with bounded measurable functions ϕ : X × Y × Z → C , contin-uous in the Z -variable. For each z ∈ Z , let ϕ z : X × Y → C be given by ϕ z ( x, y ) = ϕ ( x, y, z ); clearly, ϕ z is a measurable function for each z ∈ Z .We recall some terminology from [6] that will be used in the sequel. Let ϕ ∈ L ∞ ( X × Y × Z ) and associate with it a bounded bilinear mapΛ ϕ : S ( Y, Z ) × S ( X, Y ) → S ( X, Z ); Λ ϕ ( T h , T k ) := T ϕ ( h ∗ k ) , where k ∈ L ( Y × X ) , h ∈ L ( Z × Y ) and ϕ ( h ∗ k )( z, x ) := Z Y ϕ ( x, y, z ) h ( z, y ) k ( y, x ) dy, ( x, z ) ∈ X × Z. By [6, Corollary 10] the norm k Λ ϕ k of Λ ϕ as a bilinear map, where thespaces S ( Y, Z ) and S ( X, Y ) are equipped with their Hilbert-Schmidt norm,is equal to k ϕ k ∞ . We say that ϕ is an operator S -multiplier if Λ ϕ maps S ( Y, Z ) × S ( X, Y ) into S ( X, Z ). The following characterisation of oper-ator S -multipliers was obtained in [6]: Theorem 3.4.
Let ϕ : X × Y × Z → C be a bounded measurable function.The following are equivalent:i. the function ϕ is an operator S -multiplier; ii. there exist a Hilbert space L and weakly measurable functions v : X × Z → L , w : Y × Z → L , satisfying esssup ( x,z ) ∈ X × Z k v ( x, z ) k < ∞ , esssup ( y,z ) ∈ Y × Z k w ( y, z ) k < ∞ , such that (4) ϕ ( x, y, z ) = h v ( x, z ) , w ( y, z ) i , almost all ( x, y, z ) ∈ X × Y × Z .Moreover, if these conditions hold then k ϕ k S = esssup ( x,z ) ∈ X × Z k v ( x, z ) k esssup ( y,z ) ∈ Y × Z k w ( y, z ) k . In Theorem 3.6, we relate operator S -multipliers to central multipliers.We first include a lemma. If E is an operator space then we identify C ( Z ) ⊙E with a dense subspace of the minimal tensor product C ( Z ) ⊗ E (and equipit with the operator space structure arising from this inclusion), and itselements — with continuous functions from Z into E . If E is in addition anoperator system, we equip the algebraic tensor product C ( Z ) ⊙ E with theoperator system structure arising from its inclusion in C ( Z ) ⊗ E . Lemma 3.5.
Let Z be a locally compact Hausdorff space and E be an oper-ator space. Let Φ z : E → E be a linear map, z ∈ Z , and Φ : C ( Z ) ⊙ E → C ( Z ) ⊗ E a linear map defined by Φ( a ⊗ T )( z ) = a ( z )Φ z ( T ) , z ∈ Z. The following are equivalent:i. Φ is completely bounded;ii. Φ z is completely bounded for every z ∈ Z and sup z ∈ Z k Φ z k cb < ∞ .Moreover, if these conditions are fulfilled then k Φ k cb = sup z ∈ Z k Φ z k cb .Assume that E is an operator system. The following are equivalent:i’. Φ is completely positive;ii’. Φ z is completely positive for every z ∈ Z .Proof. (i) = ⇒ (ii) Fix z ∈ Z and note that, if a ∈ C ( Z ) has norm one and a ( z ) = 1 then Φ z ( T ) = ( δ z ⊗ id)(Φ( a ⊗ T )) , T ∈ E . It follows that Φ z is completely bounded and(5) sup z ∈ Z k Φ z k cb ≤ k Φ k cb . (ii) = ⇒ (i) We identify M n ( C ( Z ) ⊙ E ) with a subspace of C ( Z, M n ( E ))in the canonical way. Let ( h i,j ) i,j ∈ M n ( C ( Z ) ⊙ E ). The claim is immediatefrom the fact thatΦ ( n ) (( h i,j ) i,j ) ( z ) = (Φ( h i,j )( z )) i,j = (Φ z ( h i,j ( z ))) i,j . It remains to note the reverse inequality in (5); it follows by the fact that,if a ∈ C ( Z ) has norm one and a ( z ) = 1 then k Φ ( n ) z ( T ) k ≤ k Φ ( n ) ( a ⊗ T ) k ,for every T ∈ M n ( E ). ENTRAL AND CONVOLUTION MULTIPLIERS 15
Now assume that E is an operator system.(i’) = ⇒ (ii’) follows as the implication (i) = ⇒ (ii), by choosing the func-tion a to be in addition positive.(ii’) = ⇒ (i’) follows similarly to the implication (ii) = ⇒ (i), by taking intoaccount that a matrix ( h i,j ) i,j belongs to the positive cone of M n ( C ( Z ) ⊙ E )if and only if ( h i,j ( z )) i,j ∈ M + n for every z ∈ Z . (cid:3) Theorem 3.6.
Let ϕ : X × Y × Z → C be a bounded measurable function,continuous in the Z -variable. The following are equivalent:i. ϕ is a central Schur C ( Z ) -multiplier;ii. the function ϕ z is a Schur multiplier for every z ∈ Z , and the map D ϕ : C ( Z, K ) → C ( Z, K ) given by D ϕ ( h )( z ) = S ϕ z ( h ( z )) , z ∈ Z, is completely bounded;iii. the function ϕ z is a Schur multiplier for every z ∈ Z , and sup z ∈ Z k ϕ z k S < ∞ ; iv. the function ϕ is an operator S -multiplier.If these conditions hold then k ϕ k S = sup z ∈ Z k ϕ z k S .Proof. (i) ⇐⇒ (ii) Let ϕ be a central Schur C ( Z )-multiplier. We fix ameasure m ∈ M ( Z ) so that the representation of C ( Z ) on L ( Z, m ), givenby a M a , where( M a ξ )( z ) := a ( z ) ξ ( z ) , a ∈ C ( Z ) , ξ ∈ L ( Z, m ) , z ∈ Z, is faithful. By [27, Proposition 2.3], we may identify C ( Z ) with its imagein B ( L ( Z )), so we abuse notation by writing a in place of M a . We recallthat the map S ϕ extends to a completely bounded map on K ⊗ C ( Z ). Weobserve that, when the identification (3) is made, we have that the map S ϕ (which is defined as a transformation on K ⊗ C ( Z )) is identified with D ϕ .Indeed, if k ∈ L ( Y × X ) and a ∈ C ( Z ) then S ϕ ( k ⊗ a )( z ) = ( ϕ ( · , · , z ) · k ) a ( z ) = D ϕ ( k ⊗ a )( z ) , z ∈ Z. The equivalence now follows.(ii) ⇐⇒ (iii) is immediate from Lemma 3.5.(i) = ⇒ (iv) Define a map ψ : f ψ f , on L ( Z ) by letting ψ f ( x, y ) := Z Z ϕ ( x, y, z ) f ( z ) dz, ( x, y ) ∈ X × Y. We will show that ψ f belongs to L ∞ ( X ) ⊗ w ∗ h L ∞ ( Y ) and has norm at most k ϕ k S . Take f ∈ C c ( Z ), k ∈ L ( Y × X ), and a ∈ C ( Z ) with k a k = 1and a ( z ) = 1 for all z ∈ supp( f ). Writing f = f f , f , f ∈ L ( Z ), k f k = k f k k f k , for ξ ∈ L ( X ) and η ∈ L ( Y ), we have (cid:12)(cid:12)(cid:10) S ψ f ( T k ) ξ, η (cid:11)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z X × Y (cid:18)Z Z ϕ ( x, y, z ) f ( z ) dz (cid:19) k ( y, x ) ξ ( x ) η ( y ) dx dy (cid:12)(cid:12)(cid:12)(cid:12) = |h S ϕ ( T k ⊗ a )( ξ ⊗ f ) , η ⊗ f i|≤ k ϕ k S k T k ⊗ a kk ξ k k f k k η k k f k ≤ k ϕ k S k T k kk f k k ξ k k η k . Thus the map S ψ f is bounded in the operator norm, implying that ψ f isa Schur multiplier with k ψ f k S ≤ k ϕ k S k f k . It follows from the density of C c ( Z ) in L ( Z ) that ψ is a bounded map, with k ψ k ≤ k ϕ k S ; we view ψ as taking values in L ∞ ( X ) ⊗ w ∗ h L ∞ ( Y ) using the standard identification ofthis tensor product with the Schur multipliers on X × Y .By standard operator space identifications (see [6] and [23]), we have ψ ∈ B ( L ( Z ) , L ∞ ( X ) ⊗ w ∗ h L ∞ ( Y )) ∼ = L ∞ ( Z ) ⊗ ( L ∞ ( X ) ⊗ w ∗ h L ∞ ( Y )) , where ϕ ∈ L ∞ ( X × Y × Z ) is the corresponding element in L ∞ ( Z ) ⊗ ( L ∞ ( X ) ⊗ w ∗ h L ∞ ( Y )). Condition (iv) now follows by [6, Theorem 19] andTheorem 3.4.(iv) = ⇒ (i) Let v and w be the functions arising as in Theorem 3.4, and M ⊆ X × Y × Z be a set with ( µ × ν × m )( M c ) = 0, such that (4) holds for all( x, y, z ) ∈ M . Set M x,y = { z : ( x, y, z ) ∈ M } and N = { ( x, y ) : m ( M cx,y ) =0 } ; it is clear that ( µ × ν )( N c ) = 0. Write W ( y ) : L ( Z ) → L ⊗ L ( Z ) and V ( x ) : L ( Z ) → L ⊗ L ( Z ) for the maps, given by (cid:0) V ( x ) ξ (cid:1) ( z ) := v ( x, z ) ξ ( z ) and (cid:0) W ( y ) ξ (cid:1) ( z ) := w ( y, z ) ξ ( z ) , ξ ∈ L ( Z );we have esssup x ∈ X kV ( x ) k = esssup ( x,z ) ∈ X × Z k v ( x, z ) k < ∞ , esssup y ∈ Y kW ( y ) k = esssup ( y,z ) ∈ Y × Z k w ( y, z ) k < ∞ . For a ∈ C ( Z ), ξ, η ∈ L ( Z ) and ( x, y ) ∈ N , we have hW ( y ) ∗ ( I ⊗ M a ) V ( x ) ξ, η i = h ( I ⊗ M a ) V ( x ) ξ, W ( y ) η i = Z Z a ( z ) h v ( x, z ) , w ( y, z ) i ξ ( z ) η ( z ) dm ( z )= Z Z a ( z ) ϕ ( x, y, z ) ξ ( z ) η ( z ) dm ( z ) . It follows that, if ( x, y ) ∈ N then W ( y ) ∗ ( I ⊗ M a ) V ( x ) = M ϕ x,y a , a ∈ C ( Z )(here ϕ x,y is the function on Z given by ϕ x,y ( z ) = ϕ ( x, y, z )). By [27,Theorem 2.6], ϕ is a Schur C ( Z )-multiplier which is clearly central, and k ϕ k S ≤ esssup x ∈ X kV ( x ) k esssup y ∈ Y kW ( y ) k = esssup z ∈ Z k ϕ z k S . ENTRAL AND CONVOLUTION MULTIPLIERS 17
Finally, from the proof of (i) = ⇒ (ii) = ⇒ (iii), equation (5), and theestimate in (iv) = ⇒ (i) we have k ϕ k S = sup z ∈ Z k ϕ z k S . (cid:3) In the next result we assume that A acts non-degenerately on a separableHilbert space H , and we identify the elements of the centre Z ( M ( A )) of A with completely bounded maps on A acting by operator multiplication. Corollary 3.7.
Let ϕ : X × Y → Z ( M ( A )) be a pointwise measurablefunction, and assume that Z ( A ) A = A . The following are equivalent:i. ϕ is a central Schur A -multiplier;ii. there exist an index set I and operators V ∈ C ωI ( L ∞ ( X, Z ( A ) ′′ )) and W ∈ C ωI ( L ∞ ( Y, Z ( A ) ′′ )) , such that ϕ ( x, y ) = X i ∈ I W i ( y ) ∗ V i ( x ) , for almost all ( x, y ) ∈ X × Y .Moreover, if ϕ : X × Y → Z ( M ( A )) is weakly measurable then the aboveconditions are equivalent to:iii. ϕ is a central Schur B -multiplier for any C ∗ -algebra B ⊆ B ( H ) with Z ( A ) ⊆ Z ( B ) .If the conditions hold we may choose V, W such that k ϕ k S = k V k C ωI ( L ∞ ( X,Z ( A ) ′′ )) k W k C ωI ( L ∞ ( Y,Z ( A ) ′′ )) , where k ϕ k S is the norm of the Schur multiplier in either (i) or (iii).Proof. Since Z ( A ) A = A , the algebra Z ( A ) is non-degenerate and hence Z ( A ) ′′ = Z ( A ) w , where the closure in the weak operator topology.(i) = ⇒ (ii) By Remark 3.1, ϕ is a Schur Z ( A )-multiplier. Followingthe proof of Theorem 3.6, and using the identification Z ( A ) ∼ = C ( Z ) and Z ( A ) ′′ ∼ = L ∞ ( Z, m ), for some measure space (
Z, m ), we identify ϕ with anelement of L ∞ ( Z, m ) ⊗ ( L ∞ ( X ) ⊗ w ∗ h L ∞ ( Y )). Using [6], we see that thereexist an index set I and two families ( V i ) i ∈ I , ( W i ) i ∈ I , where V i : X → Z ( A ) ′′ and W i : Y → Z ( A ) ′′ are measurable functions satisfyingesssup x ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I V i ( x ) ∗ V i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ∞ and esssup y ∈ Y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I W i ( y ) ∗ W i ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ∞ , such that ϕ ( x, y ) = P i ∈ I W i ( y ) ∗ V i ( x ) almost everywhere on X × Y (theseries converges weakly) and(6) k ϕ k S ( X,Y ; Z ( A )) = esssup x ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I V i ( x ) ∗ V i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) esssup y ∈ Y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I W i ( y ) ∗ W i ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (ii) = ⇒ (i) For a ∈ A , we have(7) ϕ ( x, y )( a ) = X i ∈ I W i ( y ) ∗ V i ( x ) a = X i ∈ I W i ( y ) ∗ aV i ( x ) = W ∗ ( y ) ρ ( a ) V ( x ) , where V ( x ) := ( V i ( x )) i ∈ I , W ( y ) := ( W i ( y ) ∗ ) i ∈ I and ρ ( a ) := id ℓ ( I ) ⊗ a .By [27, Theorem 2.6] ϕ is a Schur A -multiplier, and it is clearly central.(ii) = ⇒ (iii) The assumption implies that ( x, y ) ϕ ( x, y )( b ) ∈ B isweakly measurable for all b ∈ B , so it makes sense to speak of ϕ being a Schur B -multiplier. Now the same proof as that of the implication (ii) = ⇒ (i) canbe applied.(iii) = ⇒ (i) is trivial.For the norm equality observe that k ϕ k S ( X,Y ; B ) ≥ k ϕ k S ( X,Y ; Z ( A )) while,by (7), we have k ϕ k S ( X,Y ; B ) ≤ esssup x ∈ X kV ( x ) k esssup y ∈ Y kW ( y ) k = esssup x ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I V i ( x ) ∗ V i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) esssup x ∈ X (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I V i ( x ) ∗ V i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k V k C ωI ( L ∞ ( X,Z ( A ) ′′ )) k W k C ωI ( L ∞ ( Y,Z ( A ) ′′ )) . The equality follows by combining this with (6). (cid:3)
We remark that the results of this subsection and the rest of the sectionremain true when X and Y are discrete spaces with counting measures, Z is an arbitrary (not necessarily second countable) locally compact Hausdorffspace and A is an arbitrary (not necessarily separable ) C ∗ -algebra.3.2. Central Herz–Schur multipliers.
In this subsection, similarly toTheorem 3.6, we characterise central Herz–Schur multipliers, a natural in-variant version of central Schur multipliers, which we now introduce.
Definition 3.8.
Let ( A, G, α ) be a C ∗ -dynamical system. A Herz–Schur ( A, G, α ) -multiplier F will be called central if there exists a family ( a r ) r ∈ G ⊆ Z ( M ( A )) such that F ( r )( a ) = a r a, a ∈ A, r ∈ G. Proposition 3.9.
Let A be a C ∗ -algebra such that Z ( A ) A = A , ( A, G, α ) bea C ∗ -dynamical system, ( a r ) r ∈ G be a family in Z ( M ( A )) and suppose thatthe map F : G → CB( A ) , given by F ( r )( a ) = a r a , is pointwise measurable.The following are equivalent:i. F is a central Herz–Schur ( Z ( A ) , G, α ) -multiplier;ii. F is a central Herz–Schur ( A, G, α ) -multiplier;iii. there exist V, W ∈ C ωI ( L ∞ ( G, Z ( A ) ′′ )) such that α t − ( a ts − ) = X i ∈ I W i ( t ) ∗ V i ( s ) , for almost all ( s, t ) ∈ G × G. Moreover, V and W may be chosen so that k F k HS = k V k C ωI ( L ∞ ( X,Z ( A ) ′′ )) k W k C ωI ( L ∞ ( Y,Z ( A ) ′′ )) where k F k HS refers to the norm of F in either (i) or (ii). ENTRAL AND CONVOLUTION MULTIPLIERS 19
Proof. (i) = ⇒ (ii) By [27, Theorem 3.8] N ( F ) is a Schur Z ( A )-multiplier; itis clearly central. Using the assumption Z ( A ) A = A we observe that Z ( A )acts non-degenerately on any Hilbert space where A acts non-degenerately,so by Corollary 3.7 we have that N ( F ) is a central Schur A -multiplier.Applying again [27, Theorem 3.8], we obtain that F is a central Herz–Schur( A, G, α )-multiplier.(ii) = ⇒ (i) Immediate from [27, Theorem 3.8] and Remark 3.1.(i) = ⇒ (iii) By [27, Theorem 3.8] N ( F ) is a central Schur Z ( A )-multiplier,and for a ∈ A and s, t ∈ G , N ( F )( s, t )( a ) = α t − ( a ts − ) a, a ∈ A. By Corollary 3.7(ii), there exist
V, W ∈ C ωI ( L ∞ ( G, Z ( A ) ′′ )) such that α t − ( a ts − ) a = X i ∈ I W i ( t ) ∗ aV i ( s ) = X i ∈ I W i ( t ) ∗ V i ( s ) a almost everywhere.Since this holds for every a ∈ A and A ⊆ B ( H ) is separable and non-degenerate, we conclude that α t − ( a ts − ) = X i ∈ I W i ( t ) ∗ V i ( s ) , for almost all ( s, t ) ∈ G × G .(iii) = ⇒ (i) For a ∈ A and almost all s, t ∈ G we have N ( F )( s, t )( a ) = α t − ( a ts − ) a = X i ∈ I W i ( t ) ∗ aV i ( s ) = W ( t ) ∗ ρ ( a ) V ( s ) , where ρ ( a ) := id ℓ ( I ) ⊗ a , V ( s ) := ( V i ( s )) i ∈ I and W ( t ) := ( W i ( t )) i ∈ I . There-fore F is a Herz–Schur ( Z ( A ) , G, α )-multiplier by [27, Theorem 3.8].Since N is an isometry, the norm equality follows from the norm equalityin Theorem 3.7. (cid:3) A central Herz–Schur ( C ( Z ) , G, α )-multiplier F : G → CB( C ( Z )), as-sociated with a family ( a r ) r ∈ G ⊆ C b ( Z ), may be identified with a boundedmeasurable function, continuous in the Z -variable, given by F : G × Z → C ; F ( r, z ) = a r ( z ) , r ∈ G, z ∈ Z ;conversely, if F : G × Z → C is a bounded measurable function, continuous inthe Z -variable, then the associated function F : G → CB( C ( Z )) is boundedand pointwise-measurable. In the sequel, if Z is a locally compact Hausdorffspace and ( C ( Z ) , G, α ) is a C ∗ -dynamical system, we let ( z, t ) → zt be themapping from Z × G into Z that satisfies the condition f ( zt ) = α t ( f )( z ), z ∈ Z , t ∈ G . The mapping is jointly continuous and satisfies z ( st ) = ( zs ) t for all z ∈ Z and s, t ∈ G . Corollary 3.10.
Let ( C ( Z ) , G, α ) be a C ∗ -dynamical system, and F : G × Z → C a bounded measurable function, continuous in the Z -variable. Thefollowing are equivalent:i. F is a central Herz–Schur ( C ( Z ) , G, α ) -multiplier; ii. there exists a Hilbert space L and weakly measurable bounded functions v, w : G × Z → L such that F ( ts − , zt − ) = h v ( s, z ) , w ( t, z ) i almost all ( s, t, z ) ∈ G × G × Z .Moreover, k F k HS = esssup ( s,x ) ∈ G × Z k v ( s, x ) k esssup ( t,y ) ∈ G × Z k w ( t, y ) k .Proof. Immediate from Proposition 3.9 by taking L := ℓ ( I ), v ( s, x ) i := (cid:0) V i ( s ) (cid:1) ( x ) and w ( t, y ) i := (cid:0) W i ( t ) (cid:1) ( y ) , s, t ∈ G, x, y ∈ Z. (cid:3) Positive central multipliers.
Positive Schur A -multipliers, in thecase of sets equipped with the counting measure, were studied in [26] (see [26,Definition 2.3] and [26, Theorem 2.6]). Here we extend this by consideringarbitrary standard measure spaces and identifying corresponding versions ofthe previous results. Definition 3.11.
Let A be a C ∗ -algebra. A Schur A -multiplier ϕ : X × X → CB( A ) is called positive if S ϕ is completely positive. Before giving a completely positive version of Theorem 3.6, we includea lemma. Since L ∞ ( X ) ⊗ w ∗ h L ∞ ( X ) = ( L ( X ) ⊗ h L ( X )) ∗ , every Schurmultiplier ϕ on X × X gives rise to a canonical bilinear map F ϕ : L ( X ) × L ( X ) → C . As usual, we write F ( n,n ) ϕ for the corresponding amplification,a bilinear map from M n ( L ( X )) × M n ( L ( X )) into M n . Lemma 3.12.
Let ( X, µ ) be a standard measure space and ϕ ∈ L ∞ ( X ) ⊗ w ∗ h L ∞ ( X ) be a positive Schur multiplier. If T = ( f i,j ) ni,j =1 ∈ M n ( L ( X )) and T ∗ = ( f j,i ) ni,j =1 then F ( n,n ) ϕ ( T, T ∗ ) ∈ M + n .Proof. Note that, if ϕ is a positive Schur multiplier, by virtue of [15], onemay write ϕ = P ∞ i =1 a i ⊗ a i , where ( a i ) ∞ i =1 is a bounded row operator withentries in L ∞ ( X ). It thus suffices to prove the statement in the case where ϕ = a ⊗ a , for some a ∈ L ∞ ( X ). However, then we have F ( n,n ) ϕ ( T, T ∗ ) = n X k =1 h f i,k , a ih f j,k , a i ! ni,j =1 = n X k =1 (cid:16) h f i,k , a ih f j,k , a i (cid:17) ni,j =1 , and the conclusion follows. (cid:3) Theorem 3.13.
Let ϕ : X × X × Z → C be a bounded measurable function,continuous in the Z -variable. The following are equivalent:i. ϕ is a positive central Schur C ( Z ) -multiplier;ii. there exists a Hilbert space L and an essentially bounded, weakly mea-surable function v : X × Z → L such that ϕ ( x, y, z ) = h v ( x, z ) , v ( y, z ) i for almost all ( x, y, z ) ∈ X × X × Z ;iii. for each z ∈ Z the function ϕ z is a positive Schur multiplier, and sup z ∈ Z k ϕ z k S < ∞ . ENTRAL AND CONVOLUTION MULTIPLIERS 21
Moreover, if the space X is discrete and µ is counting measure the aboveconditions are equivalent to:iv. for any x , . . . , x n ∈ X and z ∈ Z the matrix ( ϕ ( x i , x j , z )) i,j is positivein M n .Proof. (i) = ⇒ (ii) Suppose that ϕ is a positive central Schur C ( Z )-multipli-er. We have seen in the proof of Theorem 3.6 that ϕ ∈ L ∞ ( Z ) ⊗ ( L ∞ ( X ) ⊗ w ∗ h L ∞ ( X )). With ϕ we associate the completely bounded bilinear map Φ ϕ : L ( X ) × L ( X ) → L ∞ ( Z ) given byΦ ϕ (( f, g ))( h ) = h ϕ, h ⊗ ( f ⊗ g ) i , f, g ∈ L ( X ) , h ∈ L ( Z ) . We obtainΦ ϕ (( f, g ))( h ) = Z Z Z X × X × Z ϕ ( x, y, z ) h ( z ) f ( x ) g ( y ) dx dy dz = Z Z (cid:18)Z X × X ϕ z ( x, y ) f ( x ) g ( y ) dxdy (cid:19) h ( z ) dz (8)and Φ ϕ (( f, g ))( z ) = Z X × X ϕ z ( x, y ) f ( x ) g ( y ) dxdy a.e. . Set Φ ϕ z (( f, g )) = Φ ϕ (( f, g ))( z ) , z ∈ Z. By Lemma 3.5, ϕ z is a positive Schur multiplier and, by Lemma 3.12,Φ ( n,n ) ϕ z ((( f i,j ) , ( f ∗ i,j ))) ∈ M + n for any ( f i,j ) ∈ M n ( L ( X )). By [39, Theorem4.4, Remark 4.5(iii)], there exists a family ( ψ i ) i ∈ Λ ⊆ CB( L ( X ) , L ∞ ( Z ))such that k P i ∈ I | ψ i ( a ) | k ∞ ≤ C k a k , a ∈ L ( X ), for some constant C > ϕ (( a, b )) = X i ∈ Λ ψ i ( a ) ψ i ( b ∗ ) ∗ , a, b ∈ L ( X ) . Identifying each ψ i with an element ψ i of L ∞ ( X × Z ) via ψ i ( f )( h ) = Z X Z Z ψ i ( x, z ) f ( x ) h ( z ) dx dz, f ∈ L ( X ) , h ∈ L ( Z ) , letting L = ℓ (Λ) and v ( x, z ) := ( ψ i ( x, z )) i ∈ Λ gives (ii).(ii) = ⇒ (i) Define V ( x ) : L ( Z ) → L ⊗ L ( Z ); (cid:0) V ( x ) ξ (cid:1) ( z ) := v ( x, z ) ξ ( z ) , ξ ∈ L ( Z ) . Then ϕ ( x, y )( a ) = V ( y ) ∗ (cid:0) id ⊗ M a (cid:1) V ( x ) , a ∈ C ( Z )for almost all ( x, y ) (see the proof of Theorem 3.6 (iv) ⇐⇒ (i)). Therefore ϕ is a central Schur C ( Z )-multiplier, and (as in the proof of [27, Theorem2.6]) writing ρ for the representation a id ⊗ M a of C ( Z ) on L ⊗ L ( Z )we have S ϕ ( T ) = V ∗ (id ⊗ ρ )( T ) V , T ∈ K ( L ( X )) ⊗ C ( Z ) . Hence S ϕ is completely positive. (i) ⇐⇒ (iii) follows from the following two facts: (a) since ϕ is a Schur C ( Z )-multiplier, we have that S ϕ ( K )( z ) = S ϕ z ( K ( z )), z ∈ Z for any K ∈ C ( Z, K ), and (b) an element K ∈ C ( Z, K ) is positive if and only if K ( z ) ≥ K for all z ∈ Z .Now assume that µ is counting measure on the discrete space X . Observethat (iv) is equivalent to ( ϕ ( x i , x j )) being a positive element of M n ( C ( Z )).(i) = ⇒ (iv) Let x , . . . , x n ∈ X . By [26, Theorem 2.6], the matrix( ϕ ( x i , x j )( a )) ∈ M n ( C ( Z )) is positive when a ∈ C ( Z ) is positive. Fora fixed z ∈ Z , let a ∈ C ( Z ) be such that a ( z ) = 1. It follows that( ϕ ( x i , x j , z )) i,j ∈ M + n .(iv) = ⇒ (i) For a positive ( a i,j ) ∈ M n ( C ( Z )), the matrix ( ϕ ( x i , x j )( a i,j ))is the Schur product of ( ϕ ( x i , x j )) and ( a i,j ) in M n ( C ( Z )). Since (iv)ensures the positivity of ( ϕ ( x i , x j )), and the Schur product of two positivematrices over a commutative C ∗ -algebra is positive, (i) follows from [26,Theorem 2.6]. (cid:3) In the next corollary we assume A acts nondegenerately on a separableHilbert space H . Corollary 3.14.
Let ϕ : X × X → Z ( M ( A )) ⊆ CB( A ) be a pointwisemeasurable function, and assume that Z ( A ) A = A . The following are equiv-alent:i. ϕ is a positive central Schur A -multiplier;ii. there exist an index set I and V ∈ C ωI ( L ∞ ( X, Z ( A ) ′′ )) such that ϕ ( x, y ) = X i ∈ I V i ( y ) ∗ V i ( x ) , for almost all ( x, y ) ∈ X × Y .Moreover, if ϕ : X × X → Z ( M ( A )) is weakly measurable then the aboveconditions are equivalent to:iii. ϕ is a positive central Schur B -multiplier for any C ∗ -algebra B ⊆ B ( H ) with Z ( A ) ⊆ Z ( B ) .Proof. Follows from Theorem 3.13 in the same way as Corollary 3.7 followsfrom Theorem 3.6. (cid:3)
We recall the following definition from [26].
Definition 3.15.
A Herz–Schur ( A, G, α ) -multiplier F : G → CB( A ) iscalled completely positive if S F is completely positive on A ⋊ α,r G . Theorem 3.16.
Let ( A, G, α ) be a C ∗ -dynamical system such that Z ( A ) A = A , and F : G → Z ( M ( A )) be a pointwise measurable function. The follow-ing are equivalent:i. F is a completely positive central Herz–Schur ( Z ( A ) , G, α ) -multiplier;ii. F is a completely positive central Herz–Schur ( A, G, α ) -multiplier;iii. N ( F ) is a positive central Schur Z ( A ) -multiplier;iv. N ( F ) is a positive central Schur A -multiplier. ENTRAL AND CONVOLUTION MULTIPLIERS 23
Proof. (ii) = ⇒ (iv) Assume that F : G → Z ( M ( A )) is a positive centralHerz–Schur ( A, G, α )-multiplier. By the proof of [27, Theorem 3.8], using theStinespring dilation theorem in place of the Haagerup–Paulsen–Wittstocktheorem, we have S F ( T ) = V ∗ ρ ( T ) V , T ∈ A ⋊ α,r G . The representation ρ ◦ ( π ⋊ λ ) of the full crossed product A ⋊ α G has the form ρ A ⋊ ρ G , where( ρ A , ρ G ) is a covariant pair. Let V ( s ) := ρ G ( s − ) V λ s ; as in [27, page 408],we have N ( F )( s, t )( a ) = V ( t ) ∗ ρ A ( a ) V ( s ), so S N ( F ) = V ∗ ( ρ A ⊗ id)( · ) V iscompletely positive. Therefore N ( F ) is a positive Schur A -multiplier, andit is clearly central.(iv) = ⇒ (ii) As in the proof of [27, Theorem 3.8] we have that S F = S N ( F ) | A ⋊ α,r G , so S F is completely positive.(iv) = ⇒ (iii) Follows from Remark 3.1.(iii) = ⇒ (iv) Let N ( F ) be a positive central Schur Z ( A )-multiplier.Following the proof of the implication (i) = ⇒ (ii) of Corollary 3.7 andapplying [39, Remark 4.5(iii)], we see that there exists an index set I and an essentially bounded function V ∈ C ωI ( L ∞ ( G, Z ( A ) ′′ )) such that N ( F )( s, t ) = P i ∈ I V i ( t ) ∗ V i ( s ) almost everywhere on G × G (the series con-verges weakly). Hence for a ∈ A and s, t ∈ G we have N ( F )( s, t )( a ) = X i ∈ I V i ( t ) ∗ V i ( s ) a = X i ∈ I V i ( t ) ∗ aV i ( s ) = V ( t ) ∗ ρ ( a ) V ( s ) , where V ( r ) := ( V i ( r )) i ∈ I and ρ ( a ) = id ⊗ a . As in the proof of the implication(ii) = ⇒ (i) of [27, Theorem 2.6], it follows that S N ( F ) = V ∗ (id ⊗ ρ )( · ) V iscompletely positive, so N ( F ) is a positive central Schur A -multiplier.(i) ⇐⇒ (iii) This is a special case of (ii) ⇐⇒ (iv). (cid:3) Using Theorem 3.13, similarly to Corollary 3.10, one can obtain the fol-lowing description of completely positive central Herz–Schur ( C ( Z ) , G, α )-multipliers. Corollary 3.17.
Let ( C ( Z ) , G, α ) be a C ∗ -dynamical system, and F : G × Z → C a measurable function, continuous in the Z -variable. The followingare equivalent:i. F is a completely positive central Herz–Schur ( C ( Z ) , G, α ) -multiplier;ii. there exists a Hilbert space L and a weakly measurable function v : G × Z → L such that F ( ts − , xt − ) = h v ( s, x ) , v ( t, x ) i almost everywhereon G × G × Z . Connections with other types of multipliers.
Let Z be a locallycompact Hausdorff space, equipped with an action of a locally compactgroup G ; thus, we are given a map Z × G → Z , ( x, s ) → xs , jointly contin-uous and such that x ( st ) = ( xs ) t for all x ∈ Z and all s, t ∈ G . We considerthe crossed product C ( Z ) ⋊ α,r G , where α is the corresponding action of G on C ( Z ). The set G = Z × G is a groupoid, where the set G of composablepairs is given by G = { [( x , t ) , ( x , t )] : x = x t } , and if [( x , t ) , ( x , t )] ∈ G , the product ( x , t ) · ( x , t ) is defined to be( x , t t ), while the inverse ( x, t ) − of ( x, t ) is defined to be ( xt, t − ). Thedomain and range maps are given by d (( x, t )) := ( x, t ) − · ( x, t ) = ( xt, e ) , r (( x, t )) := ( x, t ) · ( x, t ) − = ( x, e ) . The unit space G of the groupoid, which is by definition equal to the com-mon image of the maps d and r , can therefore be canonically identified with X . We refer to [36] for background on groupoids (see also [27, Section 5.2]).Let ψ : Z × G → C be a bounded continuous function. Let F ψ ( s ) ∈ CB ( C ( Z )) be given by F ψ ( s )( f )( x ) := ψ ( x, s ) f ( x ), f ∈ C ( Z ), s ∈ G .In [27, Section 5] it was shown that such a function ψ is a Herz–Schur( C ( Z ) , G, α )-multiplier if and only if ψ is a completely bounded multiplierof the Fourier algebra of G in the sense of Renault [37]. In the terminology ofthis paper such functions ψ are central Herz–Schur ( C ( Z ) , G, α )-multipliers.The following is therefore immediate from [27, Proposition 5.3] and Corol-lary 3.10. Corollary 3.18.
Let ( C ( Z ) , G, α ) be a C ∗ -dynamical system, and write G for the underlying groupoid. Let ψ : Z × G → C be a bounded continuousfunction and write F ψ ( r )( f )( x ) := ψ ( x, r ) f ( x ) , f ∈ C ( Z ) . The followingare equivalent:i. F ψ is a central Herz–Schur ( C ( Z ) , G, α ) -multiplier;ii. ψ is a completely bounded multiplier of the Fourier algebra of G ;iii. there exist a Hilbert space L and essentially bounded functions v, w : G × Z → L such that ψ ( xt − , ts − ) = h v ( s, x ) , w ( t, x ) i , s, t ∈ G, almost all x ∈ X. If the conditions hold then we can choose v and w such that k ψ k HS = esssup ( s,x ) ∈ G × Z k v ( s, x ) k esssup ( t,x ) ∈ G × Z k w ( t, x ) k . We next link central multipliers to the multipliers studied by Dong–Ruanin [8]. Let (
A, G, α ) be a C ∗ -dynamical system with A unital and G discrete.Dong–Ruan define a function h : G → A to be a multiplier with respect to α if there is an A -bimodule map Φ on A ⋊ α,r G such that Φ( λ r ) = λ r π ( h ( r )).The A -bimodule requirement forces h ( r ) ∈ Z ( A ) for all r ∈ G . HenceΦ = S F for the central ( A, G, α ) multiplier given by F ( r )( a ) = h ( r ) a .In [8, Section 6], the authors use the fact that classical (positive) Schurmultipliers on a discrete group G give rise to (positive) central Herz–Schurmultipliers of ( ℓ ∞ ( G ) , G, β ) (here β denotes the left translation action). Thisconnection is also utilised by Ozawa [32]. We formalise this connection inthe next proposition. Proposition 3.19.
Let G be a discrete group. Consider a function ϕ : G × G → C and a family a = ( a r ) r ∈ G ⊆ C b ( G ) . Define a ϕr ( p ) := ϕ ( r − p − , p − ) and ϕ a ( s, t ) := a ts − ( t − ) . ENTRAL AND CONVOLUTION MULTIPLIERS 25
The assignments ϕ a ϕ and a ϕ a are mutual inverses, and give a one-to-one correspondence between the classical Schur multipliers and the centralHerz–Schur ( C ( G ) , G, β ) -multipliers. This bijection is an isometric algebraisomorphism which preserves positivity.Proof. It is easy to check that ϕ a ϕ = ϕ and a ϕ a = a and that these assign-ments are linear and multiplicative.Now suppose that a = ( a r ) r ∈ G is a central Herz–Schur ( C ( G ) , G, β )-multiplier. By Corollary 3.10, there exists a Hilbert space L and weaklymeasurable functions v, w : G × G → L , such that ϕ a ( s, t ) = a ts − ( t − ) = h v ( s, e ) , w ( t, e ) i , s, t ∈ G. It follows from [4] that ϕ a is a Schur multiplier and k ϕ a k S ≤ k a k HS .Conversely, suppose ϕ : G × G → C is a Schur multiplier, and take v, w : G → H are such that ϕ ( s, t ) = h v ( s ) , w ( t ) i and k ϕ k S = sup s ∈ G k v ( s ) k sup t ∈ G k w ( t ) k . Then, for s, t, x ∈ G , a ϕts − ( xt − ) = ϕ ( st − tx − , tx − ) = ϕ ( sx − , tx − ) = (cid:10) v ( sx − ) , w ( tx − ) (cid:11) , so, by Corollary 3.10, a ϕ = ( a ϕr ) r ∈ G is a central Herz–Schur ( C ( G ) , G, α )-multiplier with k a ϕ k HS ≤ k ϕ k S .If a is a positive central multiplier (resp. ϕ is a positive Schur multiplier)then applying Corollary 3.17, taking v = w in the above calculations, shows ϕ a (resp. a ϕ ) is also positive. (cid:3) Convolution multipliers
In this section, we give a characterisation of Herz–Schur convolution mul-tipliers first studied in [27, Section 6]. We will use the notion of a Herz–Schur θ -multiplier of a C ∗ -dynamical system ( A, G, α ), introduced in [27, Defini-tion 3.3]. Let θ : A → B ( H θ ) be a faithful representation of (the separable C ∗ -algebra) A on the separable Hilbert space H θ , and let ( π θ , λ θ ) be the reg-ular covariant pair associated to this representation (see Subsection 2.1.4).A function F : G → CB( A ) will be called a Herz–Schur θ -multiplier of ( A, G, α ) if the map π θ ( a ) λ θr π θ (cid:0) F ( r )( a ) (cid:1) λ θr extends to a completely bounded, weak*-continuous map on A ⋊ w ∗ α,θ G . Asbefore we assume that G is either second countable or discrete.4.1. Abelian case.
Let G be an abelian locally compact group equippedwith a Haar measure and Γ be its dual group. We denote by λ Γ the leftregular representation on L (Γ). We shall identify each element s ∈ G witha character on Γ, and use β to denote the natural action of G on C ∗ r (Γ) byletting β s (cid:0) λ Γ ( f ) (cid:1) := λ Γ ( sf ) , s ∈ G, f ∈ L (Γ);thus, ( C ∗ r (Γ) , G, β ) is a C ∗ -dynamical system. Given a bounded measurable function ψ : G × Γ → C and t ∈ G (resp. x ∈ Γ), let the function ψ t : Γ → C (resp. ψ x : G → C ) be given by ψ t ( y ) := ψ ( t, y ) (resp. ψ x ( s ) := ψ ( s, x )). We call ψ admissible if ψ t ∈ B (Γ)for every t ∈ G and sup t k ψ t k B (Γ) < ∞ . Assuming that ψ is admissible, let F ψ ( t ) : C ∗ r (Γ) → C ∗ r (Γ) be the map given by F ψ ( t )( λ Γ ( g )) = λ Γ ( ψ t g ) , g ∈ L (Γ) . We define the
Herz–Schur convolution multipliers of G to be the elementsof the set S conv ( G ) := { ψ : G × Γ → C : ψ is admissible and F ψ isa Herz–Schur ( C ∗ r (Γ) , G, β )-multiplier } , and write S idconv ( G ) = { ψ : G × Γ → C : ψ is admissible and F ψ isa Herz–Schur id-multiplier of ( C ∗ r (Γ) , G, β ) } . Here we write id for the canonical representation of C ∗ r (Γ) on L (Γ). Clearly,the space S conv ( G ) is an algebra with respect to the operations of pointwiseaddition and multiplication, and S idconv ( G ) is a subalgebra of S conv ( G ). For ψ ∈ S conv ( G ), let k ψ k HS = k F ψ k HS , and use S ψ to denote the map S F ψ .We identify an elementary tensor u ⊗ h , where u ∈ B ( G ) and h ∈ B (Γ),with the function ( s, x ) → u ( s ) h ( x ), s ∈ G , x ∈ Γ. Let F ( B ( G ) , B (Γ)) be thecomplex vector space of all separately continuous functions ψ : G × Γ → C such that, for every s ∈ G (resp. x ∈ Γ), the function ψ s : Γ → C (resp. ψ x : G → C ) belongs to B (Γ) (resp. B ( G )). By [27, Section 6], we have thefollowing inclusions: B ( G ) ⊙ B (Γ) ⊆ S idconv ( G ) ⊆ F ( B ( G ) , B (Γ)) . We now answer [27, Question 6.6] by identifying S idconv ( G ). Theorem 4.1.
Let G be a locally compact abelian groupand ψ : G × Γ → C be an admissible function. The following are equiva-lent:i. ψ ∈ S idconv ( G ) ;ii. ψ ∈ B ( G × Γ) .The identification is an isometric algebra homomorphism.Proof. (i) = ⇒ (ii) Let ψ ∈ S idconv ( G ) and let F ψ : G → CB( C ∗ r (Γ)) be thecorresponding Herz-Schur multiplier of ( C ∗ r (Γ) , G, β ). By [27, Theorem 3.8], N ( F ψ )( s, t ) is a Schur C ∗ r (Γ)-multiplier and hence there exists a Hilbertspace H ρ , operators V , W ∈ L ∞ ( G, B ( L (Γ) , H ρ )), a continuous unitaryrepresentation ρ : Γ → B ( H ρ ) and a subset N ⊆ G × G with ( m G × m G )( N ) =0, such that N ( F ψ )( s, t ) (cid:0) λ Γ ( f ) (cid:1) = W ( t ) ∗ ρ ( f ) V ( s ) , f ∈ L (Γ) , ENTRAL AND CONVOLUTION MULTIPLIERS 27 for all ( s, t ) N , and(9) k ψ k S = esssup s ∈ G kV ( s ) k esssup t ∈ G kW ( t ) k . As N ( F ψ )( s, t )( λ Γ ( f )) = β t − ( F ψ ( ts − )( β t ( λ Γ ( f )))) = λ Γ ( ψ ts − f ) , we obtain λ Γ ( ψ ts − f ) = W ( t ) ∗ ρ ( f ) V ( s ) , f ∈ L (Γ) , for all ( s, t ) / ∈ N . As ψ ts − ∈ B (Γ), we have that ψ ts − is a completelybounded multiplier of A (Γ), and the map S ψ ts − can be extended to a weak*-continuous linear operator on vN(Γ); we have ψ ( ts − , x ) λ Γ x = W ( t ) ∗ ρ ( x ) V ( s ) , x ∈ Γ , ( s, t ) / ∈ N. Thus, for ξ ∈ L (Γ) with k ξ k = 1, we have ψ ( ts − , xy − ) h ξ, ξ i = (cid:10) λ Γ x − W ( t ) ∗ ρ ( x ) ρ ( y ) ∗ V ( s ) λ Γ y ξ, ξ (cid:11) = (cid:10) ρ ( y ) ∗ V ( s ) λ Γ y ξ, ρ ( x ) ∗ W ( t ) λ Γ x ξ (cid:11) . Letting v ( s, y ) := ρ ( y ) ∗ V ( s ) λ Γ y ξ and w ( t, x ) := ρ ( x ) ∗ W ( t ) λ Γ x ξ , we obtain ψ (cid:0) ( t, x )( s, y ) − (cid:1) = h v ( s, y ) , w ( t, x ) i , ( s, t ) / ∈ N. By [4], ψ is equal almost everywhere to a completely bounded multiplier of A ( G × Γ), and hence to an element u ∈ B ( G × Γ) [20, Theorem 5.1.8]. Tosee that ψ ( t, x ) = u ( t, x ) for all ( t, x ), for each t ∈ G we let N t = { x ∈ Γ : ψ ( t, x ) = u ( t, x ) } . By Fubini’s Theorem, the set { t ∈ G : m Γ ( N ct ) > } has measure zero, thatis, for almost all t ∈ G , we have that ψ ( t, x ) = u ( t, x ) almost everywhere.As ψ is separately continuous, the last equality holds for all x ∈ Γ. Usingagain the separate continuity of ψ we obtain that ψ ( t, x ) = u ( t, x ) for all( t, x ). Furthermore, by (9), k ψ k B ( G × Γ) ≤ esssup ( s,y ) ∈ G × Γ k ρ ( y ) ∗ V ( s ) λ Γ y ξ k esssup ( t,x ) ∈ G × Γ k ρ ( x ) ∗ W ( t ) λ Γ x ξ k≤ esssup s ∈ G kV ( s ) k esssup t ∈ G kW ( t ) k = k ψ k S . (ii) = ⇒ (i) Assume that ψ ∈ B ( G × Γ). By [4], there exist a Hilbert space H and continuous v, w : G × Γ → H such that ψ ( ts − , xy − ) = h v ( s, y ) , w ( t, x ) i , s, t ∈ G, x, y ∈ Γ , and k ψ k B ( G × Γ) = sup ( s,y ) k v ( s, y ) k sup ( t,x ) k w ( t, x ) k . Choose an orthonormal basis { e i } i ∈ I in H and let v i ( s, y ) := h v ( s, y ) , e i i and w i ( t, x ) := h e i , w ( t, x ) i . Then ψ ( ts − , xy − ) = X i ∈ I v i ( s, y ) w i ( t, x ) , s, t ∈ G, x, y ∈ Γ . Let S be the completely bounded operator on B ( L ( G × Γ)), given by S ( T ) := P i ∈ I M w i T M v i . Clearly,(10) k S k cb = k ψ k B ( G × Γ) . To complete the proof, it suffices to show that the restriction of the operator S to C ∗ r (Γ) ⋊ w ∗ β, id G is given by(11) S (cid:0) π id ( λ Γ x ) λ id s (cid:1) = π id (cid:0) ψ ( s, x ) λ Γ x (cid:1) λ id s . First note that(12) ( π id ( λ Γ x ) ξ )( t ) = β t − ( λ Γ x ) ξ ( t ) = t ( x ) λ Γ x ξ ( t ) , ξ ∈ L ( G, L (Γ)) . Writing v i ( t )( · ) and w i ( t )( · ) for v i ( t, · ) and w i ( t, · ), respectively, for t ∈ G and y ∈ Γ, and fixing ξ, η ∈ L ( G, L (Γ)), we have D S ( π id ( λ Γ x ) λ id s ) ξ, η E = X i ∈ I D M w i π id ( λ Γ x ) λ id s M v i ξ, η E = X i ∈ I Z (cid:16) M w i ( t ) t ( x ) λ Γ x M v i ( s − t ) ξ ( s − t ) (cid:17) ( y ) η ( t, y ) dtdy = X i ∈ I Z w i ( t, y ) v i ( s − t, x − y ) t ( x ) ξ ( s − t, x − y ) η ( t, y ) dtdy = Z ψ ( tt − s, yy − x ) t ( x ) ξ ( s − t, x − y ) η ( t, y ) dtdy = Z ψ ( s, x ) (cid:0) π id ( λ Γ x ) λ id s ξ (cid:1) ( t, y ) η ( t, y ) dtdy. Together with (12), this establishes (11). In addition, k ψ k S = (cid:13)(cid:13)(cid:13)(cid:13) S (cid:12)(cid:12) C ∗ r (Γ) ⋊ w ∗ β, id G (cid:13)(cid:13)(cid:13)(cid:13) cb ≤ k S k cb = k ψ k B ( G × Γ) , which together with (10) gives the desired equality.To see that the identification is multiplicative, observe that if ψ, χ ∈ S idconv ( G ) then S F ψ S F χ = S F ψχ . (cid:3) In Theorem 4.4 below we will show that the identification in Theorem 4.1is in fact a complete isometry.4.2.
General case.
Now let G be an arbitrary locally compact group. Inorder to define convolution multipliers, we replace C ∗ r (Γ) with the quantumgroup dual of C ∗ r ( G ), namely C ( G ), equipped with its natural action of G . Similarly we replace B (Γ) by M ( G ), the Banach algebra of all complex-valued Radon measures on G with the convolution multiplication, given by( µ ∗ ν )( f ) := Z G Z G f ( st ) dµ ( s ) dν ( t ) , f ∈ C ( G ) , µ, ν ∈ M ( G ) . ENTRAL AND CONVOLUTION MULTIPLIERS 29
We identify L ( G ) with the norm-closed ideal in M ( G ) consisting of abso-lutely continuous measures with respect to left Haar measure. We have that L ( G ) is an M ( G )-bimodule in the natural way. Using the identification L ( G ) ∗ = L ∞ ( G ), we arrive at an M ( G )-bimodule structure on L ∞ ( G ),given by h µ · f, h i = h f, h ∗ µ i and h f · µ, h i = h f, µ ∗ h i , for h ∈ L ( G ), f ∈ L ∞ ( G ), µ ∈ M ( G ). In particular,( µ · f )( s ) = Z G f ( st ) dµ ( t ) and ( f · µ )( t ) = Z G f ( st ) dµ ( s ) . Let ρ be the right regular representation of G on L ( G ); thus,( ρ s ξ )( t ) = ∆( s ) / ξ ( ts ) . For µ ∈ M ( G ), define a bounded linear operator θ ( µ )( a ), a ∈ B ( L ( G )), by θ ( µ )( a ) := Z G ρ s aρ ∗ s dµ ( s ) . By [30, Theorem 3.2] (see also [29, Theorem 4.5]), the map θ is a weak ∗ -weak ∗ continuous completely isometric homomorphism from M ( G ) to thespace CB σ ( B ( L ( G ))) of all completely bounded weak* continuous linearmaps on B ( L ( G )) and k θ ( µ ) k cb = k θ ( µ ) k = k µ k . We have θ ( µ )( f ) = µ · f ∈ L ∞ ( G ) , f ∈ L ∞ ( G ) . Moreover, θ ( µ ) is a vN( G )-bimodule map.For each t ∈ G , let β t : L ∞ ( G ) → L ∞ ( G ) be given by β t ( f ) := λ Gt f λ Gt − = f t , where f t ( x ) = f ( t − x ). Then(13) β t ◦ θ ( µ ) = θ ( µ ) ◦ β t , t ∈ G. For Λ = { µ t } t ∈ G ⊆ M ( G ), define F Λ : G → CB( C ( G )) by F Λ ( t )( f ) := θ ( µ t )( f ) , t ∈ G, f ∈ C ( G ) . Definition 4.2.
A family
Λ = { µ t } t ∈ G ⊆ M ( G ) is called a convolutionmultiplier if F Λ is a Herz–Schur ( C ( G ) , G, β ) -multiplier. If Λ = { µ t } t ∈ G is a convolution multiplier, we set k Λ k HS = k F Λ k HS .Let id denote the representation of C ( G ) on L ( G ) by multiplication op-erators and S idconv ( G ) be the collection of families Λ = { µ t } t ∈ G ⊆ M ( G ) suchthat F Λ is a Herz–Schur id-multiplier of ( C ( G ) , G, β ), endowed with the al-gebra structure coming from pointwise operations on the maps F Λ . When G is abelian, the identifications C ( G ) ≡ C ∗ r (Γ) and M ( G ) ≡ B (Γ) show thatthe usage of the notation S idconv ( G ) agrees with that from Subsection 4.1.Consider the operator space projective tensor product L ( G ) b ⊗ A ( G ) =( L ∞ ( G ) ⊗ vN( G )) ∗ . We note that, when equipped with the product, givenon elementary tensors by( f ⊗ u )( g ⊗ v ) = ( f ∗ g ) ⊗ ( uv ) , f, g ∈ L ( G ) , u, v ∈ A ( G ) , the operator space L ( G ) b ⊗ A ( G ) is a completely contractive Banach algebra.A map T ∈ B ( L ( G ) b ⊗ A ( G )) will be called a right multiplier of L ( G ) b ⊗ A ( G )if T ( ab ) = aT ( b ) , a, b ∈ L ( G ) b ⊗ A ( G ) . If, in addition, T is completely bounded, we write T ∈ M r cb ( L ( G ) b ⊗ A ( G )),and call T a right completely bounded multiplier of L ( G ) b ⊗ A ( G ). When G is abelian we have the identificationsM r cb ( L ( G ) b ⊗ A ( G )) = M cb ( A (Γ × G )) = B (Γ × G ) . Our goal is to generalise Theorem 4.1, identifying S idconv ( G ) with the spaceof right completely bounded multipliers M r cb ( L ( G ) b ⊗ A ( G )).If M is any of the von Neumann algebras L ∞ ( G ), vN( G ) or L ∞ ( G ) ⊗ vN( G ), T ∈ M and f ∈ M ∗ , we write f · T and T · f ∈ M for the operatorsgiven by h f · T, g i := h T, gf i , h T · f, g i := h T, f g i , g ∈ M ∗ , where h· , ·i is the pairing between M and M ∗ . We recall [11] that the supportof T ∈ vN( G ) is the closed set of all t ∈ G such that u · T = 0 whenever u ∈ A ( G ) and u ( t ) = 0. Lemma 4.3. If T ∈ M r cb ( L ( G ) b ⊗ A ( G )) then there exists a unique family { µ t } t ∈ G ⊆ M ( G ) such that T ∗ ( f ⊗ λ Gt ) = θ ( µ t )( f ) ⊗ λ Gt , f ∈ L ∞ ( G ) , t ∈ G. Proof.
Let f , f ∈ L ( G ), a , a ∈ A ( G ). The equality T (( f ⊗ a )( f ⊗ a )) = ( f ⊗ a ) T ( f ⊗ a )implies that, if g ∈ L ∞ ( G ) then(14) (cid:10) T ∗ ( g ⊗ λ Gt ) , ( f ⊗ a )( f ⊗ a ) (cid:11) = a ( t ) (cid:10) T ∗ ( g · f ⊗ λ Gt ) , f ⊗ a (cid:11) . Taking the limit along an approximate identity { f α } α ∈ A of L ( G ), we obtain(15) h T ∗ ( g ⊗ λ Gt ) , f ⊗ a a i = h a ( t ) T ∗ ( g ⊗ λ Gt ) , f ⊗ a i . For ω ∈ L ( G ), let R ω : L ∞ ( G ) ⊗ vN( G ) → vN( G ) be the slice map, definedby h R ω ( S ) , a i := h S, ω ⊗ a i , S ∈ L ∞ ( G ) ⊗ vN( G ) , a ∈ A ( G ) . After taking a limit along an approximate unit for L ( G ), equation (15)implies that a · R ω (cid:0) T ∗ ( g ⊗ λ Gt ) (cid:1) = a ( t ) R ω (cid:0) T ∗ ( g ⊗ λ Gt ) (cid:1) . It follows that R ω ( T ∗ ( g ⊗ λ Gt )) ∈ vN( G ) has support in { t } . By [11,Th´eor`eme 4.9], R ω ( T ∗ ( g ⊗ λ Gt )) = c ( ω, t ) λ Gt for some constant c ( ω, t ) and R ω ( T ∗ ( g ⊗ λ Gt )(1 ⊗ λ Gt − )) ∈ C I. By [22], T ∗ ( g ⊗ λ Gt )(1 ⊗ λ Gt − ) ∈ L ∞ ( G ) ⊗ C I and hence T ∗ ( g ⊗ λ Gt ) = g t ⊗ λ Gt for some g t ∈ L ∞ ( G ). ENTRAL AND CONVOLUTION MULTIPLIERS 31
The map Φ t : g g t is completely bounded, normal, and T ∗ ( g ⊗ λ Gt ) =Φ t ( g ) ⊗ λ Gt , t ∈ G . By (14),Φ t ( g · f ) = Φ t ( g ) · f , f ∈ L ( G ) , showing that (Φ t ) ∗ ( f ∗ f ) = f ∗ ((Φ t ) ∗ ( f )). Thus (Φ t ) ∗ is a right completelybounded multiplier of L ( G ). By [30, Theorem 3.2] (see also [29, Theorem4.5]), there exist { µ t } t ∈ G such that Φ t ( g ) = θ ( µ t )( g ). (cid:3) In what follows we will speak of a family Λ = { µ t } t ∈ G ⊆ M ( G ) beinga convolution multiplier or a (completely bounded) right multiplier. Fora right multiplier Λ of L ( G ) b ⊗ A ( G ) we denote by R Λ the mapping on L ∞ ( G ) ⊗ vN( G ), given by(16) R Λ ( f ⊗ λ Gr ) := θ ( µ r )( f ) ⊗ λ Gr . Theorem 4.4.
Let
Λ = { µ t } t ∈ G ⊆ M ( G ) . The following are equivalent:i. Λ ∈ S idconv ( G ) ;ii. Λ ∈ M r cb ( L ( G ) b ⊗ A ( G )) .The identification R Λ S F Λ is a completely isometric algebra isomorphism.Proof. (i) = ⇒ (ii) We identify C ( G ) ⋊ w ∗ β, id G with the von Neumann alge-bra crossed product L ∞ ( G ) ⋊ vN β G , and let Λ = { µ t } t ∈ G be a convolutionmultiplier. For f ∈ L ∞ ( G ), using (13) we have N ( F Λ )( s, t )( f ) = β t − (cid:0) F Λ ( ts − )( β t ( f )) (cid:1) = β t − (cid:0) θ ( µ ts − )( β t ( f )) (cid:1) = θ ( µ ts − )( f ) . Following similar arguments as in the proof of [27, Theorem 3.8], we obtainthat there exist a normal ∗ -representation ρ of L ∞ ( G ) on H ρ and V , W ∈ L ∞ ( G, B ( L ( G ) , H ρ )) such that θ ( µ ts − )( f ) = W ∗ ( t ) ρ ( f ) V ( s )and k Λ k S = esssup s ∈ G kV ( s ) k esssup t ∈ G kW ( t ) k .Define a map R Λ : L ∞ ( G ) ⊗ vN( G ) → B ( L ( G ) ⊗ L ( G )) by R Λ ( f ⊗ λ Gt ) := W ∗ ( ρ ( f ) ⊗ λ Gt ) V, where V, W ∈ B ( L ( G, H ρ ⊗ L ( G ))) are given by ( V ξ )( t ) = V ( t ) ξ ( t ),( W ξ )( t ) = W ( t ) ξ ( t ). Then R Λ ( f ⊗ λ Gt ) ξ ( s ) = W ∗ ( s ) ρ ( f ) V ( t − s ) ξ ( t − s ) = θ ( µ s ( s − t ) )( f ) ξ ( t − s )= ( θ ( µ t )( f ) ⊗ λ Gt ξ )( s ) . In particular, R Λ ( f ⊗ λ Gt ) ∈ L ∞ ( G ) ⊗ vN( G ), and hence R Λ is a normalcompletely bounded map on L ∞ ( G ) ⊗ vN( G ). Moreover, if f , f ∈ L ( G ), a , a ∈ A ( G ), g ∈ L ∞ ( G ), and ( R Λ ) ∗ is the predual of R Λ , we have (cid:10) g ⊗ λ Gt , ( R Λ ) ∗ (( f ⊗ a )( f ⊗ a )) (cid:11) = (cid:10) θ ( µ t )( g ) ⊗ λ Gt , f ∗ f ⊗ a a (cid:11) = h µ t · g, f ∗ f i (cid:10) λ Gt , a a (cid:11) = h g, ( f ∗ f ) ∗ µ t i (cid:10) a · λ Gt , a (cid:11) = h g · f , f ∗ µ t i (cid:10) a · λ Gt , a (cid:11) = h µ t · ( g · f ) , f i (cid:10) a ( t ) λ Gt , a (cid:11) = (cid:10) R Λ ( g · f ⊗ a ( t ) λ Gt ) , f ⊗ a (cid:11) = (cid:10) g · f ⊗ a ( t ) λ Gt , ( R Λ ) ∗ ( f ⊗ a ) (cid:11) = (cid:10) g ⊗ λ Gt , ( f ⊗ a )( R Λ ) ∗ ( f ⊗ a ) (cid:11) , i.e. ( R Λ ) ∗ (( f ⊗ a )( f ⊗ a )) = ( f ⊗ a )( R Λ ) ∗ ( f ⊗ a ) . Hence ( R Λ ) ∗ ( ab ) = a ( R Λ ) ∗ ( b ) for any a, b ∈ L ( G ) b ⊗ A ( G ) and therefore( R Λ ) ∗ is a right completely bounded multiplier of L ( G ) b ⊗ A ( G ). In addition,(17) k R Λ k cb ≤ esssup s ∈ G kV ( s ) k esssup t ∈ G kW ( t ) k = k Λ k S . (ii) = ⇒ (i) Assume now that Λ = { µ t } t ∈ G ∈ M r cb ( L ( G ) b ⊗ A ( G )), i.e. themap f ⊗ λ Gt θ ( µ t )( f ) ⊗ λ Gt extends to a normal right L ( G ) b ⊗ A ( G )-modularcompletely bounded map R Λ on L ∞ ( G ) ⊗ vN( G ). By [19, Proposition 4.3],there exists a unique vN( G ) ⊗ L ∞ ( G )-bimodule map f R Λ ∈ CB σ ( B ( L ( G × G ))) such that f R Λ | L ∞ ( G ) ⊗ vN( G ) = R Λ and k f R Λ k cb = k R Λ k cb . We have, inparticular,(18) f R Λ ( g ⊗ f λ Gt ) = θ ( µ t )( g ) ⊗ f λ Gt , f, g ∈ L ∞ ( G ) . Note that L ( G × G ) ≡ L ( G, L ( G )) and let π : L ∞ ( G ) → B ( L ( G × G ))be the *-representation, given by π ( f ) ξ ( t ) = β t − ( f )( ξ ( t )) , ξ ∈ L ( G × G ) , f ∈ L ∞ ( G ) . Let f ∈ L ∞ ( G ) and note that π ( f ) ∈ L ∞ ( G × G ). Thus, there exists a net { ω α } α ∈ A ⊆ span { g ⊗ h : g, h ∈ L ∞ ( G ) } , with ω α → α ∈ A π ( f ) in the weak*topology. Write ω α = P n α i =1 g i,α ⊗ h i,α . Using (13) and (18), we have (cid:0) f R Λ ( π ( f )(1 ⊗ λ Gr ) (cid:1) = lim α ∈ A n α X i =1 θ ( µ r )( g i,α ) ⊗ h i,α λ Gr = π (cid:0) θ ( µ r )( f ) (cid:1) (1 ⊗ λ Gr ) . Since ( g S F Λ ( π ( f )(1 ⊗ λ Gt )) = ( π ( θ ( µ t )( f ))(1 ⊗ λ Gt ) , the restriction of f R Λ to the crossed product C ( G ) ⋊ β,r G coincides with S F Λ , implying the converse statement. Note, in addition, that(19) k S F Λ k cb ≤ k f R Λ k cb = k R Λ k cb . By (17), k R Λ k cb ≤ k S F Λ k cb , and together with (19) this shows that k R Λ k cb = k S F Λ k cb . Moreover, by Lemma 2.7 the map F Λ
7→ N ( F Λ ) isa complete isometry, and by [19, Proposition 4.3] the map R Λ f R Λ is a ENTRAL AND CONVOLUTION MULTIPLIERS 33 complete isometry, therefore the norm inequalities hold on all matrix levels,implying that the identification S F Λ R Λ is a complete isometry.The homomorphism claim follows from Lemma 2.7 and the fact that theidentification in [19, Proposition 4.3] is a homomorphism. (cid:3) We observe that the product of the convolution multipliers Λ = { µ t } t ∈ G and Ξ = { ν t } t ∈ G is given by ΛΞ = { µ t ∗ ν t } t ∈ G . We write S cent ( A, G, α ) forthe central Herz–Schur (
A, G, α )-multipliers.
Proposition 4.5.
We have S conv ( G ) ∩ S cent ( C ( G ) , G, β ) = M cb A ( G ) .Proof. Suppose that F : G → CB( C ( G )) is a central multiplier which isalso a convolution multiplier. Then for each r ∈ G there is a r ∈ C b ( G ) suchthat F ( r )( a ) = a r a . Also, since F is a convolution multiplier, by (13) F ( r )satisfies β t (cid:0) F ( r )( a ) (cid:1) = F ( r ) (cid:0) β t ( a ) (cid:1) , r, t ∈ G, a ∈ C ( G ) . Combining these two identities, and allowing a to vary, gives a r ( st ) = a r ( t )for all s, t ∈ G , so a r is a scalar multiple of the identity. The conclusionfollows from [27, Proposition 4.1]. (cid:3) Idempotent multipliers
Given standard measure spaces (
X, µ ) and (
Y, ν ), a well-known open prob-lem asks for the identification of the idempotent Schur multipliers on X × Y .A characterisation of the contractive idempotent Schur multipliers, based ona combinatorial argument, combined with an observation of Livshitz [24],was given by Katavolos–Paulsen in [21].In a similar vein, for a general locally compact group G , there is no knowncharacterisation of the idempotent Herz–Schur multipliers. Some partialresults are known: the idempotent measures in M ( G ) of norm one werecharacterised by Greenleaf [12] — a measure µ has the properties µ ∗ µ = µ and k µ k = 1 if and only if µ = γm H , where m H is the Haar measureon a compact subgroup H and γ is a character of H . Such µ is positiveif and only if γ above is equal to 1. Dually, the idempotent elements of B ( G ) were characterised by Host [17]; using Host’s method, Ilie and Spronk[18] characterised contractive idempotents — a function u ∈ B ( G ) has theproperties u = u and k u k = 1 if and only if u = χ C , where C is anopen coset of G . Such u is positive if and only if C is a subgroup of G .Stan [40] extended this characterisation to norm one idempotent elementsof M cb A ( G ).In this section we use the aforementioned results of Katavolos–Paulsenand Stan to study the idempotent central and the idempotent convolutionmultipliers.5.1. Central idempotent multipliers.
We fix standard measure spaces(
X, µ ) and (
Y, ν ) and a separable, non-degenerate C ∗ -algebra A ⊆ B ( H ).Suppose ϕ ∈ L ∞ ( X × Y ) is an idempotent Schur multiplier, so the map k ϕ · k on L ( Y × X ) gives rise to a bounded idempotent map S ϕ on thespace of compact operators; we have that ϕ ( x, y ) k ( y, x ) = ϕ ( x, y ) k ( y, x )almost everywhere for all k ∈ L ( Y × X ), which implies that ϕ = ϕ .By [21, Proposition 11], ϕ = χ E almost everywhere for some ω -open and ω -closed E ⊆ X × Y .Recall from [21] that a subset E ⊆ X × Y is said to have the provided that given any distinct pair of points x = x in X andany pair of distinct pairs y = y in Y , whenever 3 of the 4 ordered pairs( x i , y j ) belong to E then the fourth one also belongs to E .For a subset W ⊆ C × Z , where C is a set (which will below be equal toeither X or Y ), and an element z ∈ Z , we write W z = { t ∈ C : ( t, z ) ∈ W } .The following result generalises [21, Theorem 10]. Proposition 5.1.
Let ( X, µ ) and ( Y, ν ) be standard measure spaces and Z a locally compact Hausdorff space. Let ϕ : X × Y × Z → C be a measurablefunction, continuous in the Z -variable. The following are equivalent:i. ϕ is a contractive idempotent central Schur C ( Z ) -multiplier;ii. for each z ∈ Z , there exist families ( A zi ) i ∈ N and ( B zi ) i ∈ N of pairwise dis-joint measurable subsets of X and Y , respectively, such that ϕ ( x, y, z ) = P ∞ i =1 χ A zi ( x ) χ B zi ( y ) almost everywhere.Proof. (i) = ⇒ (ii) By Theorem 3.6, ϕ z is a contractive idempotent Schurmultiplier for every z ∈ Z . By [21, Theorem 10], there exist families ( A zi ) ∞ i =1 and ( B zi ) ∞ i =1 of pairwise disjoint measurable subsets of X and Y , respectively,such that ϕ z ( x, y ) = P ∞ i =1 χ A zi ( x ) χ B zi ( y ) almost everywhere.(ii) = ⇒ (i) By [21, Theorem 10], ϕ z is a contractive idempotent Schurmultiplier for every z ∈ Z ; thus, by Theorem 3.6, ϕ is a central C ( Z )-multiplier, which is easily seen to be idempotent. Since each ϕ z is contractivewe have ϕ is contractive by Theorem 3.6. (cid:3) Remark 5.2.
The statement holds when the standard measure spaces arereplaced by discrete spaces X and Y with counting measures, but in this casethe families ( A zi ) i , ( B zi ) i might be uncountable if X or Y is uncountable. Inthis case (i) is also equivalent to ϕ = χ W , where W z has the 3-of-4 propertyfor each z ∈ Z , see [21, Lemma 2].Let Z be a locally compact Hausdorff space equipped with an action α ofa locally compact group G . In the subsequent results, we view the set Z × G as a groupoid as in Section 3.4. We provide a combinatorial characterisationof the contractive central Herz–Schur ( C ( Z ) , G, α )-multipliers. It is easy tosee that in this case ψ ( x, t ) = χ V ( x, t ) for some subset V ⊆ Z × G . Theorem5.3 generalises the result of Stan [40, Theorem 3.3]. Theorem 5.3.
Assume that V ⊆ Z × G is a subset that is both closed andopen. The following are equivalent:i. F χ V is a contractive central Herz–Schur ( C ( Z ) , G, α ) -multiplier; ENTRAL AND CONVOLUTION MULTIPLIERS 35 ii. if ( x, t ) , ( x, s ) , ( xr, r − s ) ∈ V then ( xr, r − t ) ∈ V ; equivalently, if ( x, t ) , ( y, s ) , ( z, p ) ∈ V and the product ( z, p )( y, s ) − ( x, t ) is well defined then ( z, p )( y, s ) − ( x, t ) ∈ V .In particular, if V = Z × A for some A ⊆ G then A is an open coset of G .Proof. Let W = { ( x, s, t ) ∈ Z × G × G : ( xt − , ts − ) ∈ V } . By Corollary 3.18, F χ V is a Herz–Schur ( C ( Z ) , G, α )-multiplier if and onlyif the map N ( F χ V ), given by N ( F χ V )( s, t )( a )( x ) = χ V ( xt − , ts − ) a ( x ) = χ W ( x, s, t ) a ( x ) , is a Schur C ( Z )-multiplier.We first show that condition (ii) is equivalent to W z := { ( s, t ) ∈ G × G : ( z, s, t ) ∈ W } having the 3-of-4 property for all z ∈ Z . Suppose that( z, t , s ), ( z, t , s ) and ( z, t , s ) ∈ W , which is equivalent to ( zt − , t s − ),( zt − , t s − ), ( zt − , t s − ) ∈ V . Writing zt − = x , t s − = t , t s − = s and t t − = r , we get zt − = xr , t s − = r − t and t s − = r − s and hence( x, t ), ( x, s ), ( xr, r − s ) ∈ V . The condition ( z, t , s ) ∈ W is equivalentto ( xr, r − t ) ∈ V , giving the statement. We note that ( z, p )( y, s ) − ( x, t ) =( z, p )( ys, s − )( x, t ) is well defined if and only if y = x and z = xsp − ; letting r = sp − , we have ( z, p ) = ( xr, r − s ). We have shown that condition (ii) isequivalent to the 3-of-4 property for each W z .Assume first that G is a locally compact second countable group andhence ( G, m G ) is a standard measure space.(i) = ⇒ (ii) If (i) holds then N ( F χ V ) is a contractive idempotent Schur C ( Z )-multiplier. By Theorem 3.6, ϕ z = χ W z is a contractive idempotentSchur multiplier for each z ∈ Z . By [21, Theorem 10], there exist countablecollections { I m } and { J m } of mutually disjoint Borel subsets of G , such that,if E = ∪ m I m × J m , then χ W z = χ E almost everywhere.As χ W z is continuous and hence ω -continuous and χ E is ω -continuous,by [38, Lemma 2.2], χ W z = χ E marginally almost everywhere. Hence thereexists a null set N z such that χ W z = χ E on N cz × N cz . In particular, W z ∩ ( N cz × N cz ) has the 3-of-4 property. To see that the whole W z has the property,take t , t , s , s such that ( t , s ), ( t , s ), ( t , s ) ∈ W z , but some of t , s , t , s belong to N z . Using the fact that W z is open and m ( N z ) = 0 we canfind sequences ( t n ) n , ( s n ) n , ( t n ) n , ( s n ) n of elements in N cz such that ( t n , s n ),( t n , s n ), ( t n , s n ) ∈ W z and t ni → t i , s ni → s i , i = 1 ,
2. Hence ( t n , s n ) ∈ W z ,and as 1 = χ W z ( t n , s n ) → χ W z ( t , s ), we obtain that ( t , s ) ∈ W z . Hence(ii) holds.(ii) = ⇒ (i) As W z is open and hence ω -open, W z is marginally equivalentto a countable union of Borel rectangles. Hence W z ∩ ( N cz × N cz ) = ∪ ∞ m =1 A zm × B zm , where m G ( N z ) = 0 and each A zm × B zm is Borel. By [21, Lemma 2]and the second paragraph in the proof, W z and hence W z ∩ ( N cz × N cz )has the 3-of-4 property for each z ∈ Z and there exist families { X zi } i ∈ I and { Y zi } i ∈ I of pairwise disjoint sets of G , such that W z ∩ ( N cz × N cz ) = ∪ i ∈ I X zi × Y zi . Arguing as in the proof of [21, Theorem 10] one shows that theindex set I can be chosen countable and each X zi × Y zi is a Borel rectangle.Hence χ W z is a contractive Schur multiplier. By Proposition 5.1 χ W isa contractive idempotent central Schur multiplier, so χ V is a contractiveidempotent central Herz–Schur ( C ( Z ) , G, α )-multiplier.If G is discrete, the statement follows from Remark 5.2. Finally, if V = Z × A then χ V ( x, t ) = χ A ( t ) which is a Herz–Schur ( C ( Z ) , G, α )-multiplierif and only if χ A is a Herz–Schur multiplier. It is of norm at most 1 if andonly if A is an open coset of G . (cid:3) Remark 5.4.
It follows from Proposition 5.3 that if F χ V is a contractiveHerz–Schur ( C ( Z ) , G, α )-multiplier and the points ( x, t ), r (( x, t )) = ( x, e )and d (( x, t )) = ( xt, e ) all belong to V then ( x, t ) − = ( xt, t − ) ∈ V . More-over, if ( x, t ), d (( x, t )) = ( xt, e ) and ( xt, s ) ∈ V then ( x, t )( xt, s ) = ( xt, ts ) ∈ V .The following corollary is an immediate consequence of Remark 5.4. Corollary 5.5.
With the notation of Theorem 5.3, assume that G ⊆ V .We have that F χ V is a contractive Herz–Schur ( C ( Z ) , G, α ) -multiplier ifand only if V is a subgroupoid of G . Positive central idempotent multipliers.
The following descrip-tion of positive contractive Schur multipliers can be obtained in a similarmanner to [21, Theorem 10], and we omit its proof.
Proposition 5.6.
Let ( X, µ ) be a standard measure space and E ⊆ X × X .The following are equivalent:i. χ E is a positive contractive Schur multiplier;ii. E is equivalent, with respect to product measure, to a subset of the form ∪ ∞ m =1 I m × I m , where { I m } ∞ m =1 is a collection of disjoint Borel subset of X . Remark 5.7.
The standard measure space (
X, µ ) can be replaced by dis-crete space X with counting measure. In this case the collection of disjointsubsets of X might be uncountable.The following positive version of Proposition 5.1 and its discrete versioncan be proved using similar ideas, and we omit the detailed argument. Proposition 5.8.
Let ( X, µ ) and ( Y, ν ) be standard measure spaces and Z a locally compact Hausdorff space. Let ϕ : X × Y × Z → C be a measurablefunction which is continuous in the Z -variable. The following are equivalent:i. ϕ is a positive contractive idempotent central Schur C ( Z ) -multiplier;ii. for each z ∈ Z , there exists a family ( A zi ) i of pairwise disjoint mea-surable subsets of X , such that ϕ ( x, y, z ) = P ∞ i =1 χ A zi ( x ) χ A zi ( y ) almosteverywhere. ENTRAL AND CONVOLUTION MULTIPLIERS 37
Proposition 5.1 and the transference theorem of [27] give an implicit char-acterisation of the positive central idempotent Herz-Schur ( C ( Z ) , G, α )-multipliers. In Theorem 5.9 below, we give a more direct description of thepositive central idempotent Herz-Schur multipliers of norm not exceeding 1. Theorem 5.9.
Let ( C ( Z ) , G, α ) be a C ∗ -dynamical system and V ⊆ Z × G be a closed and open subset. The following are equivalent:i. F χ V is a positive, contractive Herz–Schur ( C ( Z ) , G, α ) -multiplier;ii. V is a subgroupoid of Z × G .Proof. We will prove the theorem for G a locally compact second countablegroup. The case when G is discrete can be treated in a similar but simplerway.(i) = ⇒ (ii) Let W = { ( z, s, t ) ∈ Z × G × G : ( zt − , ts − ) ∈ V } . If F χ V is a positive contractive Herz–Schur ( C ( Z ) , G, α )-multiplier then thefunction N ( F χ V ), given by N ( F χ V )( s, t )( a )( z ) = χ W ( z, s, t ) a ( z ), is a posi-tive Schur C ( Z )-multiplier. By Theorem 3.13, χ W z is a positive Schur mul-tiplier for each z ∈ Z . Note also that, as it is continuous, it is ω -continuous.Using [38, Lemma 2.2], we see that there exist a weakly measurable function v z : G → ℓ and a null set N z ⊆ G such that χ W z ( s, t ) = h v z ( s ) , v z ( t ) i , s, t / ∈ N z . Let ( x, t ) ∈ V ; as in Remark 5.4, it suffices to show that ( x, e ) and ( xt, e ) ∈ V . Assume that ( x, e ) / ∈ V , and note that χ V ( x, e ) = χ V (( xt ) t − , tt − ) = χ W ( xt, t, t ) and χ V ( x, t ) = χ W ( xt, e, t ) . If t / ∈ N xt and e / ∈ N xt then χ W ( xt, t, t ) = k v xt ( t ) k = 0 and χ W ( xt, e, t ) = h v xt ( e ) , v xt ( t ) i = 0 , giving a contradiction. If one or both of e or t are in N xt , say t ∈ N xt but e / ∈ N xt , then, as m ( N xt ) = 0 there exists a sequence s n / ∈ N xt such that s n → t . As χ W is continuous, we obtain k v xt ( s n ) k = χ W ( xt, s n , s n ) → χ W ( xt, t, t ) = 0 , while h v xt ( e ) , v xt ( s n ) i = χ W ( xt, e, s n ) → χ W ( xt, e, t ) , forcing χ W ( xt, e, t ) = 0, a contradiction. The other cases are treated simi-larly. To see that ( xt, e ) ∈ V observe that χ V ( xt, e ) = χ W ( x, t − , t − ) and χ V ( x, t ) = χ W ( x, t − , e )and apply similar analysis.(ii) = ⇒ (i) Let now V be an open subgroupoid. Arguing as in the proofof Proposition 5.3 we see that W z has the 3-of-4 property for each z ∈ Z . Moreover, if ( x, s, t ) ∈ W we have that ( xt − , ts − ) ∈ V and hence r ( xt − , ts − ) = ( xt − , e ) ∈ V and d ( xt − , ts − ) = ( xs − , e ) ∈ V , implying ( x, t, t ) ∈ W and ( x, s, s ) ∈ W . Therefore the projections W z and W z of W z on the first and the second coordinates are equal and { ( s, s ) : s ∈ W z } ⊆ W z .It follows easily now that for each z ∈ Z there exists disjoint sets { X zt } t ∈ T such that W z = ∪ t ∈ T X zt × X zt . Arguing as in [21, Theorem 10], there is aBorel subset N z , m G ( N z ) = 0 such that ( X zt ∩ N cz ) × ( X zt ∩ N cz ) is a Borelrectangle and W z ∩ ( N cz × N cz ) is a countable union of ( X zt ∩ N cz ) × ( X zt ∩ N cz ).By Proposition 5.6 χ W z is a positive contractive Schur multiplier. Therefore χ W is a positive contractive Schur C ( Z )-multiplier by Theorem 3.13. (cid:3) Idempotent convolution multipliers.
We next provide some exam-ples of idempotent convolution multipliers. The following is immediate fromTheorem 4.1 and [18, Theorem 2.1].
Corollary 5.10.
Suppose G is an abelian locally compact group and W ⊆ G × Γ is a measurable set, such that χ W ∈ S idconv . Then k χ W k S ≤ if andonly if W is an open coset of G × Γ . It is clear that if G is abelian, and C and D are open cosets of G andΓ respectively, then C × D is an open coset of G × Γ and therefore χ C × D is an idempotent convolution multiplier of norm 1 by Corollary 5.10. Thefollowing example shows that not all idempotent convolution multipliers ofnorm 1 are of this product form. Example 5.11.
Consider the abelian group G = R × Z , and note that G is isomorphic to its dual group Γ. Define H := { ( a, , b, , ( c, , d,
1) : a, b, c, d ∈ R } . It is clear that H is an open subgroup of G × Γ, but H cannot be writtenas a product of subgroups of G and Γ. Remark 5.12.
Let G be an abelian locally compact group; by Theorem 4.1,a contractive idempotent Herz–Schur convolution multiplier, say F , corre-sponds to a characteristic function χ W , for an open coset W ⊆ G × Γ. In thefollowing, we show more precisely how the family ( F ( r )) r ∈ G ⊆ CB( C ∗ r (Γ))arises. Suppose that W = xH for an open subgroup H of G × Γ and x ∈ G × Γ. Let ν be the representation of G × Γ on ℓ (( G × Γ) /H ), given by ν ( z ) δ yH := δ zyH ( z, y ∈ G × Γ), { δ yH } y be the standard orthonormal basisin ℓ (( G × Γ) /H )), and write ν for the unitary representation γ ν ( e, γ )of Γ. For r ∈ G , let u r ∈ B (Γ) be the function given by u r : Γ → C ; u r ( γ ) := (cid:10) ν ( γ ) δ ( r,e ) H , δ W (cid:11) . Then S χ W ( λ Γ γ ⊗ λ Gr ) = χ W ( r, γ )( λ Γ γ ⊗ λ Gr ) = (cid:10) δ ( r,γ ) H , δ W (cid:11) ( λ Γ γ ⊗ λ Gr )= h ν ( r, γ ) δ H , δ W i ( λ Γ γ ⊗ λ Gr )= (cid:10) ν ( γ ) δ ( r,e ) H , δ W (cid:11) ( λ Γ γ ⊗ λ Gr )= u r ( γ ) λ Γ γ ⊗ λ Gr , ENTRAL AND CONVOLUTION MULTIPLIERS 39 so the idempotent element χ W ∈ B ( G × Γ) corresponds to the Herz–Schurconvolution multiplier F ( r ) := u r .It is immediate from Host’s theorem that if G is a connected locally com-pact group then B ( G ) does not have non-trivial idempotent elements. Weobserve that this extends to idempotent convolution multipliers on abeliangroups. Indeed, let ψ be an idempotent convolution multiplier of the dy-namical system ( C ∗ r (Γ) , G, β ) and write ψ = χ W for some W ⊆ G × Γ. For x ∈ Γ and s ∈ G , let W x := { t ∈ G : ( t, x ) ∈ W } and W s := { y ∈ Γ : ( s, y ) ∈ W } . Proposition 5.13.
Let ψ = χ W ∈ S idconv ( G ) and k ψ k S ≤ . Then W x (resp. W s ) is an open coset of G (resp. Γ ) for all x ∈ Γ (resp. s ∈ G ).Proof. Since for any x ∈ Γ, s ∈ G , we have ψ x = χ W x and ψ s = χ W s , thestatement follows from [18, Theorem 2.1], as ψ x ∈ B ( G ) and ψ s ∈ B (Γ). (cid:3) If ψ = χ W ∈ S idconv ( G ) is contractive, as ψ is separately continuous, weobtain that W s = W s ′ if s and s ′ are in the same connected component of G . Similarly, we have W x = W x ′ for x, x ′ in the same connected componentof Γ. This implies the following corollary. Corollary 5.14.
If the group G (resp. Γ ) is connected then any contractiveidempotent multiplier ψ ∈ S idconv ( G ) is given by ψ = 1 ⊗ χ A (resp. ψ = χ A ⊗ ), where A is an open coset of Γ (resp. G ). In particular, we have that C ∗ r ( R ) ⋊ β,r R has no non-trivial idempotentHerz–Schur convolution multipliers, and any idempotent Herz–Schur convo-lution multiplier of C ( T ) ⋊ β,r Z is given by χ A ⊗
1, where A is a coset of Z . Example 5.15.
Let G be a locally compact group. Since M cb L ( G ) = M ( G ), we have that γm H ⊗ χ C ∈ M cb ( L ( G ) b ⊗ A ( G )), where C is an opencoset of G , H is a compact subgroup and γ is a character of H . The cor-responding convolution multiplier Λ = ( µ t ) t ∈ G is given by µ t = χ C ( t ) γm H .In fact, if R is the completely bounded map R ( f ⊗ g ) = (( γm H ) ∗ f ) ⊗ χ C g, f ∈ L ( G ) , g ∈ A ( G ) , then R ∗ ( h ⊗ λ t ) = θ ( γm H )( h ) ⊗ χ C λ t = θ ( γm H ) h ⊗ χ C ( t ) λ t . Remark 5.16.
For a (not necessarily abelian) locally compact group G the algebra C ( G × ˆ G ) := C ( G ) ⊗ C ∗ r ( G ) can be considered as a quantumgroup with the comultiplication induced from comultiplications of the factors C ( G ) and C ∗ r ( G ). In [31] the authors give a characterisation of contractiveidempotent functionals on C ∗ -quantum groups in terms of compact quantumsubgroups and group-like unitaries of the subgroup. It would be interestingto use this characterisation to describe contractive convolution multipliersin the non-abelian case. At present, however, a lack of examples of compact quantum subgroups of C ( G × ˆ G ) impedes the application of the resultsof [31] to convolution multipliers. Acknowledgements.
We would like to thank Przemyslaw Ohrysko forhelpful conversations during the preparation of this paper. The second au-thor was partially supported by the Ministry of Sciences of Iran and School ofMathematics of Institute for research in fundamental sciences (IPM). Mostof this work was completed when the second author was visiting ChalmersUniversity of Technology. She would like to thank the Department of Math-ematical Sciences of Chalmers University of Technology and the Universityof Gothenburg for warm hospitality. She would also like to thank AlirezaMedghalchi and Massoud Amini for their support and encouragement duringthis work.
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