Convolutions on compact groups and Fourier algebras of coset spaces
aa r X i v : . [ m a t h . F A ] M a y CONVOLUTIONS ON COMPACT GROUPS AND FOURIERALGEBRAS OF COSET SPACES
BRIAN E. FORREST, EBRAHIM SAMEI AND NICO SPRONK
Abstract.
In this note we study two related questions. (1) For a com-pact group G , what are the ranges of the convolution maps on A( G × G )given for u, v in A( G ) by u × v u ∗ ˇ v (ˇ v ( s ) = v ( s − )) and u × v u ∗ v ?(2) For a locally compact group G and a compact subgroup K , whatare the amenability properties of the Fourier algebra of the coset spaceA( G/K )? The algebra A(
G/K ) was defined and studied by the firstnamed author.In answering the first question, we obtain for compact groups whichdo not admit an abelian subgroup of finite index, some new subalgebrasof A( G ). Using those algebras we can find many instances in whichA( G/K ) fails the most rudimentary amenability property: operatorweak amenability. However, using different techniques, we show that ifthe connected component of the identity of G is abelian, then A( G/K )always satisfies the stronger property that it is hyper-Tauberian, whichis a concept developed by the second named author. We also establish acriterion which characterises operator amenability of A(
G/K ) for a classof groups which includes the maximally almost periodic groups. Under-lying our calculations are some refined techniques for studying spectralsynthesis properties of sets for Fourier algebras. We even find new setsof synthesis and nonsynthesis for Fourier algebras of some classes ofgroups.
In a recent article [28, § G , reduces to asking if the maps from A( G × G )to A( G ), given on elementary functions by u × v u ∗ ˇ v and u × v u ∗ v , aresurjective. We show in Sections 2 and 4 that this is not the case when G does not admit an abelian subgroup of finite index. Moreover, the rangesof both maps are quite different: the first gives us a new algebra A ∆ ( G ),and the second gives us an algebra A γ ( G ), which was originally discoveredby B. E. Johnson [16]. It is worth noting that Johnson used A γ ( G ) ina very clever way to show that for compact groups G , A( G ) is generallynot amenable, in fact not even weakly amenable. Johnson’s results were Date : August 26, 2018.2000
Mathematics Subject Classification.
Primary 43A30, 43A77, 43A85; Secondary47L25.
Key words and phrases. convolution, coset space, Fourier algebra.Research of the the first named author supported by NSERC Grant 90749-00. Researchof the second named author supported by an NSERC Post Doctoral Fellowship. Researchof the the third named author supported by NSERC Grant 312515-05. surprising when his article was published, since at the time the expectationwas that amenability of A( G ) would be equivalent to classical amenabilityfor the underlying group G . Recent applications of the theory of operatorspaces to the study of A( G ) have given us a much better understanding ofwhy A( G ) fails to be amenable as a Banach algebra for any group whichdoes not contain abelian subgroups of finite index. However, it is one of themain themes of this paper that subalgebras of A( G ) such as A γ ( G ) can shedfurther light on the amenability problem and allow us to deduce much moreabout the relationship between A( G ) and G .Motivated by Johnson’s beautiful theorem that the group algebra L ( G )of a locally compact group G is amenable as a Banach algebra if and onlyif G is amenable, together with the spectacular failure of the natural analogof this result for A( G ), Z.-J. Ruan demonstrated the tremendous value inrecognising the operator space structure on A( G ) by introducing the conceptof operator amenability and then using this to show that A( G ) is operatoramenable if and only if G is amenable [23]. This was followed by the thirdnamed author [27], and, independantly the second named author [25], eachestablishing that A( G ) is always operator weakly amenable. The generalquestion of when A( G ) is weakly amenable as a Banach algebra remainsopen. However, using, in part, techniques developed in the present article,the authors have recently shown A( G ) fails to be weakly amenable for any G which contains a connected nonabelian compact subgroup. This will appearin another article, soon.For a compact subgroup K of G , the Fourier algebra of the coset space,A( G/K ) was described by the first named author [6]. A(
G/K ) may be si-multaneously viewed as an algebra of continuous functions on the coset space
G/K and as a sublgebra of A( G ). The latter view allows us to define a nat-ural operator space structure on A( G/K ). It was shown in [6] that manyproperties of A(
G/K ) associated with amenability, such as existence of abounded approximate identity, and factorisation, are closely linked to suchproperties of A( G ). Thus we are naturally led to consider the amenabilityproperties of A( G/K ). Surprisingly, positive results are rather sparse, evenin the category of operator spaces. In Section 3 we establish that whenever G has a compact connected nonabelian subgroup K , then there exists a com-pact subgroup K ∗ of G × G such that A( G × G/K ∗ ) is not operator weaklyamenable. This contrasts sharply the positive result of [27, 25], mentionedabove. As a complement, we establish in Section 3.3 that A( G/K ) is hyper-Tauberian whenever the connected component of the identity G e in G isabelian. The hyper-Tauberian property, developed by the second namedauthor [26], implies weak amenability of a commutative semisimple Banachalgebra. Note that this result does not address the operator space structureof A( G ) at all. This generalises a result of the first named author and V.Runde [8] that A( G ) is weakly amenable when G e is abelian. In Section3.4 we obtain, for certain groups which we call [MAP] K -groups, a charac-terisation of when A( G/K ) is operator amenable. Using related techniques,
ONVOLUTIONS AND COSET SPACES 3 we also find new sets of local synthesis for A( G × G ) when G has an abelianconnected component of the identity, ones which are not known to be in theclosed coset ring of G × G .Many of the results of Section 3 rely heavily on constructions relating toconvolutions on compact groups. In Section 1 we develop a general frame-work in which to view the ‘twisted’ convolution f × g f ∗ ˇ g on functionson compact G . We realise the image of this map as a special example offunctions on a coset space. We find that our general framework naturallyaccommodates an easy, though far reaching generalisation of a result re-lating spectral information between various different algebras, obtained forabelian compact groups by N. Th. Varapolous [31], and generalised to ar-bitrary compact groups by L. Turowska and the third named author; seeTheorem 1.4. Since it takes us little extra effort, we prove our spectral re-sults for a class we call [MAP] K -groups, a class which includes maximallyalmost periodic groups. In Section 2 we apply our twisted convolution frame-work to A( G × G ). In doing so, we obtain not one, but an infinite sequenceof new subalgebras of A( G ), when G does not admit an abelian subgroup offinite index. In Section 4 we provide a framework for convolutions on com-pact G , and show how it differs from the twisted convolution when appliedto A( G × G ). In effect, we have an alternate method to obtain the algebraA γ ( G ) of Johnson.0.1. Background and notation.
The
Fourier algebra A( G ) of a locallycompact group G was defined by Eymard [5]. For compact G , there is analternative description in [13, Chap. 34]. That the two descriptions coincidecan be seen by comparing [5, p. 218] with [13, (34.16)]. We note thatthe Fourier algebra is closed under both group actions of left and righttranslations ( s, u ) s ∗ u, s · u : G × A( G ) → A( G ), given by s ∗ u ( t ) = u ( s − t ) , s · u ( t ) = u ( ts ) . Moreover, these actions are continuous in G and isometric on A( G ). Wenote that A( G ) admits a von Neumann algebra VN( G ) as its dual space. Assuch it comes equipped with a natural operator space structure. See [4], forexample, for more on this. We use the same definitions as [4] for completelybounded map , complete isometry and complete quotient map . We note thatright and left translations on A( G ), being the preadjoints of multiplicationsby unitaries on VN( G ), are complete isometries.Our main references for amenability are [15] and [24]. A Banach algebra A is said to be amenable if for any Banach A -bimodule X , and any boundedderivation D : A → X ∗ , where X ∗ is the dual space with adjoint moduleactions, D is inner. For a commutative Banach algebra A , we say A is weaklyamenable if for any symmetric A -bimodule X , the only bounded derivation D : A → X is 0; this is equivalent to having the same happen for X = A ∗ .Weak amenability for commutative Banach algebras was introduced in [1]. BRIAN E. FORREST, EBRAHIM SAMEI AND NICO SPRONK
For both amenability and weak amenability there are some homologicalcharacterisations; see [17, 2, 11, 24].Operator space notions of amenability and weak amenability were intro-duced in [23] and in [9] respectively, specifically for use with A( G ). If A is acommutative Banach algebra which is also an operator space we say A is a completely contractive Banach algebra if the multiplication map A×A → A is completely contractive in the sense of [4, Chap. 7]. An operator space V is a completely contractive A -module if it is an A module for which themodule maps A×V , V×A → V are completely contractive. The class ofcompletely contractive A -modules is closed under taking dual spaces withadjoint actions. We say A is operator (weakly) amenable if every completelybounded derivation D : A → V ∗ (with V = A ) is inner (zero). Many ofthe homological characterisations alluded to above, carry over to this set-ting, though with Banach space projective tensor products ⊗ γ replaced byoperator space projective tensor products b ⊗ .Let A be a commutative semisimple (completely contractive) Banach al-gebra. Suppose A is regular on its spectrum X = Σ A ; we regard A as analgebra of functions on X . If ϕ ∈ A ∗ we definesupp ϕ = (cid:26) x ∈ X : for every neighbourhood U of x there is f in A such that supp f ⊂ U and ϕ ( f ) = 0 (cid:27) . Here supp f = { x ∈ X : f ( x ) = 0 } . An operator T : A → A ∗ is called a localmap if supp T f ⊂ supp f for every f in A . We say A is (operator) hyper-Tauberian if every (com-pletely bounded) bounded local map T : A → A ∗ is an A -module map.This concept was developed in [26] to study the reflexivity of the (com-pletely bounded) derivation space of A , and it generalises (operator) weakamenability.For a commutative semisimple Banach algebra which is regular on itsspectrum we have the following implications.amenable + weakly amenable hyper-Tauberian k s Moreover, if A is a completely contractive Banach algebra, each propertyimplies its operator analogue, and the operator analogues satisfy the sameimplications.1. Algebras on coset spaces and a twisted convolution oncompact groups
The basic construction.
Let G be a locally compact group. Let A ( G ) be a Banach algebra of continuous functions on G which is closedunder right translations and such that for any f in A ( G ) we have • k s · f k = k f k for any s in G, and • s s · f is continuous . ONVOLUTIONS AND COSET SPACES 5
If, moreover, A ( G ) is an operator space, we want that for each t in G that f t · f is a complete isometry.If K is a compact subgroup of G we let A ( G : K ) = { f ∈ A ( G ) : k · f = f for each k in K } which is a closed subalgebra of A ( G ) whose elements are constant on leftcosets of K . We let G/K denote the space of left cosets with the quotienttopology. We define two maps P : A ( G ) → A ( G ) , P f = Z K k · f dk and M : A ( G : K ) → C b ( G/K ) , M f ( sK ) = f ( s ) . The map P is to be regarded as a Bochner integral over the normalised Haarmeasure on K ; its range is A ( G : K ) and P is a (completely) contractiveprojection. The map M is well-defined by comments above, and its rangeconsists of continuous functions since A ( G : K ) ⊂ C b ( G : K ). We note that M is an injective homomorphism and denote its range by A ( G/K ). Weassign a norm (operator space structure) to A ( G/K ) in such a way that M is a (complete) isometry. We finally define two maps N = M − : A ( G/K ) → A ( G ) and Γ = M ◦ P : A ( G ) → A ( G/K )so N is a (completely) isometric homomorphism and Γ is a (complete) quo-tient map.Let us record some basic properties of A ( G/K ). For a commutative Ba-nach algebra A , we let Σ A denote its Gelfand spectrum. Proposition 1.1. (i)
Suppose A ( G ) is regular on G and G separates thepoints of A ( G ) . Then A ( G/K ) is regular on G/K and
G/K separates thepoints of A ( G/K ) .Moreover, if Σ A ( G ) ∼ = G via evaluation maps, and K is a set of spectralsynthesis for A ( G ) , then Σ A ( G/K ) ∼ = G/K via evaluation maps. (ii)
If the subalgebra A c ( G ) of compactly supported elements of A ( G ) isdense in A ( G ) , then the algebra A c ( G/K ) of compactly supported elementsin A ( G/K ) is dense in A ( G/K ) . We remark that (i) applies to the Fourier algebra A( G ) for any locallycompact group G and compact subgroup K by [12] or [30]. Proof. (i)
Same as [6, Thm. 4.1]. (ii)
It is obvious that Γ A c ( G ) = A c ( G/K ). If w in A ( G ) is the limit of asequence ( w n ) ⊂ A c ( G ), then k Γ w − Γ w n k ≤ k w − w n k −→ n → ∞ . (cid:3) For Fourier algebras, we have the important identificationA( G ) b ⊗ A( H ) ∼ = A( G × H )via f ⊗ g f × g , where b ⊗ denotes the operator projective tensor product.See [3]. This is known to fail when the usual projective tensor product ⊗ γ is used [18]. BRIAN E. FORREST, EBRAHIM SAMEI AND NICO SPRONK
Proposition 1.2. If G and H are locally compact groups with respectivecompact subgroups K and L , then there is a complete isometry identifying A( G/K ) b ⊗ A( H/L ) ∼ = A( G × H/K × L ) given on elementary tensors by f ⊗ g f × g . Note that there is a natural homeomorphism G × H/K × L ∼ = G/K × H/L ,which is in fact a G × H -space morphism. Proof.
We identify A(
G/K ) ∼ = A( G : K ), etc. We have the following com-muting digram A( G ) b ⊗ A( H ) f ⊗ g f × g / / P K ⊗ P L (cid:15) (cid:15) A( G × H ) P K × L (cid:15) (cid:15) A( G : K ) b ⊗ A( G : K ) ?(cid:31) O O f ⊗ g f × g / / ______ A( G × H : K × L )where the inclusion map A( G : K ) b ⊗ A( G : K ) ֒ → A( G ) b ⊗ A( H ) is a com-plete isometry, since each of P K and P L are complete quotient projections.Notice that this inclusion map is a right inverse to P K ⊗ P L . The desiredmap from A( G : K ) b ⊗ A( G : K ) to A( G × H : K × L ) completes this digram,and may be realised as the composition of the inclusion map (up arrow),with the identification map (top arrow), and then P K × L . It is clear that P K × L | spanA( G : K ) × A( H : L ) = id. Hence the desired map is injective. Sincethe map identifying A( G ) b ⊗ A( H ) ∼ = A( G × H ) is a complete isometry, and P K ⊗ P L and P K × L are complete quotient maps, our desired map is a com-plete quotient map. Hence we have an injective complete quotient map, i.e.a complete isometry. (cid:3) A twisted convolution.
Let G be a compact group for the remainderof this section. We use G × G in place of G above, and K = ∆ = { ( s, s ) : s ∈ G } . Since the map(1.1) ( s, e )∆ s : ( G × G ) / ∆ → G is a homeomorphism, we identify the coset space with G . We observe thatin this case that the map P : A ( G × G ) → A ( G × G ) satisfies P w ( s, t ) = Z G w ( sr, tr ) dr = Z G w ( st − r, r ) dr and the maps M : A ( G × G : ∆) → C ( G ) and N : C ( G ) → C ( G × G : ∆) satisfy(1.2) M w ( s ) = w ( s, e ) and N f ( s, t ) = f ( st − ) . We denote the range of M by A ∆ ( G ), and then N | A ∆ ( G ) = M − . The mapΓ = M ◦ P , above, can be regarded as a ‘twisted’ convolution, for if A ( G × G )contains an elementary function f × g , then for s in G Γ( f × g )( s ) = Z G f × g ( st, t ) dt = Z G f ( st ) g ( t ) dt = f ∗ ˇ g ( s ) . ONVOLUTIONS AND COSET SPACES 7
We list some examples of A ( G × G ) and A ∆ ( G ). (i) If A ( G × G ) = C ( G × G ), then C ∆ ( G ) = C ( G ), by easy computation. (ii) Let V( G × G ) = C ( G ) ⊗ h C ( G ) (Haagerup tensor product). ThenV ∆ ( G ) = A( G ), completely isometrically. See [29]. Note that by Grothedieck’sinequality V( G × G ) = C ( G ) ⊗ γ C ( G ) (projective tensor product) isomorphi-cally, though not isometrically. (iii) Let A ( G × G ) = A( G ) ⊗ γ A( G ). Then A ∆ ( G ) = A γ ( G ) is a sub-algebra of A( G ) considered by B. Johnson [16]. He used it to study theamenability of A( G ). (iv) Consider A( G × G ) ∼ = A( G ) b ⊗ A( G ) (operator projective tensor prod-uct). The algebra A ∆ ( G ), thus formed, will be an essential object of ourstudy.We summarise some basic properties of the algebras A ∆ ( G ) which clearlyapply to the examples above. We say a norm (operator space structure) α , on X ⊗ Y is (operator) homogeneous if for every pair of (completely)contractive linear maps S : X → X ′ , T : Y → Y ′ we have that S ⊗ T extends to a (complete) contraction from X ⊗ α Y to X ′ ⊗ α Y ′ . Proposition 1.3. (i)
If there is a (completely contractive) Banach algebra B ( G ) of continuous functions on G and a homogeneous (operator) norm(operator space structure) α on B ( G ) ⊗ B ( G ) so A ( G × G ) = B ( G ) ⊗ α B ( G ) ,then A ∆ ( G ) is a (completely contractive) subalgebra of B ( G ) . (ii) If A ( G × G ) is closed under left translations and the translation mapsare continuous in G × G and (completely) isometric on A ( G × G ) , then A ∆ ( G ) is closed under both left and right translations, and the translations are con-tinuous on G , and (completely) isometric on A ∆ ( G ) . Proof. (i)
The map M : A ( G × G : ∆) → A ∆ ( G ) is the restriction of the slicemap id ⊗ δ e : A ( G ) ⊗ α A ( G ) → A ( G ) where δ e is the evaluation functionalat e . (ii) For f ∈ A ∆ ( G ), r in G and any ( s, t ) ∈ G × G we have N ( r · f )( s, t ) = f ( st − r ) = f ( s ( r − t ) − ) = ( e, r ) ∗ N f.
Our assumptions assure that the space A ( G × G : ∆) is closed under lefttranslations and each translation map is a (complete) isometry. For lefttranslations we note that for r in G and f ∈ A ∆ ( G ), we have N ( r ∗ f ) = ( r, e ) ∗ N f and we argue as above. (cid:3)
Relationships between ideals.
Let A be a Banach algebra containedin C ( X ) for some locally compact Hausdorff space X . We define for any BRIAN E. FORREST, EBRAHIM SAMEI AND NICO SPRONK closed subset E of X I A ( E ) = { f ∈ A : f ( x ) = 0 for x ∈ E } , I A ( E ) = { f ∈ A : supp f ∩ E = ∅ and supp f is compact } , andJ A ( E ) = { f ∈ I A ( E ) : supp f is compact } where supp f = { x ∈ X : f ( x ) = 0 } . If X = Σ A , and A is regular on X we say E is a set of spectral synthesis for A if I A ( E ) = I A ( E ), and of localsynthesis if I A ( E ) = J A ( E ).If G is a compact group we let b G denote the dual object of G , a setof representatives, one from each unitary equivalence class, of irreduciblecontinuous representation of G . If π ∈ b G , we fix an orthonormal basis { ξ π , . . . , ξ πd π } for H π and define(1.3) π ij : G → C , π ij ( s ) = (cid:10) π ( s ) ξ πj | ξ πi (cid:11) for i, j = 1 . . . d π . We recall the well-known fact that(1.4) T ( G ) = span { π ij : π ∈ b G, i, j = 1 , . . . , d π } is uniformly dense in C ( G ).If G is not compact, we let b G f denote the finite dimensional part of thedual object. We let T f ( G ) be defined as the span of matrix coefficients of b G f ,analogously as above. The almost periodic compactification is the compactgroup G ap = n ( π ( s )) π ∈ b G f : s ∈ G o ⊂ Y π ∈ b G f π ( G ) . There is a canonical identification T ( G ap ) ∼ = T f ( G ). We say G is maximallyalmost periodic [MAP] if b G f , or equivalently T f ( G ), separates the pointsof G . If K is a compact subgroup of G , we say G is [MAP] K if the map k ( π ( k )) π ∈ b G f : K → G ap is injective. Clearly, a [MAP] group is [MAP] K for any compact subgroup K .The following is an adaptation of [29, Thm. 3.1]. A change in perspectiveallows us to gain not only more general, but finer results than in [29]. Theorem 1.4.
Let G be a locally compact group, K a compact subgroup so G is [MAP] K , and A ( G ) be as in Section 1.1 and additionally satisfy that T f ( G ) A ( G ) ⊂ A ( G ) . If E is a closed subset of G/K let E ∗ = { s ∈ G : sK ∈ E } . Then (i) ΓI A ( G ) ( E ∗ ) = I A ( G/K ) ( E ) , and (ii) I A ( G ) ( E ∗ ) is the closed ideal generated by N I A ( G/K ) ( E ) . Note that if G is compact then in the case of Section 1.2 we have E ∗ = { ( s, t ) ∈ G × G : st − ∈ E } via the identification (1.1). Proof.
We will let I( E ∗ ) = I A ( G ) ( E ∗ ) and I( E ) = I A ( G/K ) ( E ), below. ONVOLUTIONS AND COSET SPACES 9 (i)
It is clear thatΓI( E ∗ ) ⊂ I( E ) and N I( E ) ⊂ I( E ∗ ) . Thus I( E ) = Γ ◦ N I( E ) ⊂ ΓI( E ∗ ) ⊂ I( E ) . (ii) Let w ∈ I( E ∗ ). For each π in b G we define ‘matrix-valued’ functions w π , ˜ w π : G → B ( H π ) by w π ( s ) = Z K w ( sk ) π ( k ) dk and ˜ w π ( s ) = π ( s ) w π ( s ) . Then for any i, j = 1 , . . . , d π we let w πij = D w π ( · ) ξ πj | ξ πi E and we have w πij = π ij | K · w where f · w = R G f ( k ) k · wdk for any f in L ( K ). We note that since w ∈ I( E ∗ ), k · w ∈ I( E ∗ ) for any k in K and hence f · w ∈ I( E ∗ ) for any f in L ( K ).Thus each w πij ∈ I( E ∗ ). Now for any s in G , i, j = 1 , . . . , d π we have˜ w πij ( s ) = d π X l =1 π il ( s ) w πlj ( s ) , i.e. ˜ w πij = d π X k =1 π ik w πlj . Hence since T f ( G ) A ( G ) ⊂ A ( G ) we have that each ˜ w πij ∈ I( E ∗ ). However,it is easily seen that ˜ w π ( sk ) = ˜ w π ( s ) for any s in G and k in K , so each˜ w πij ∈ A ( G : K ). Thus each˜ w πij ∈ I( E ∗ ) ∩ A ( G : K ) = N I( E ) . Now we use the relation w π ( s ) = π ( s − ) ˜ w π ( s ), for s in G , to see that foreach i, j we have w πij = d π X l =1 ˇ π il ˜ w πlj = d π X l =1 ¯ π li ˜ w πlj so w πij = π ij | K · w lies in the ideal generated by N I( E ).Since G is [MAP] K , we may regard K as a closed subgroup of G ap . Wehave that T f ( G ) | K = T ( G ap ) | K is a conjugation-closed point-separatingsubalgebra of C ( K ), thus is uniformly dense in C ( K ), and hence norm densein L ( K ). Thus there is a bounded approximate identity ( f β ) for L ( K ) forwhich each f β ∈ T f ( G ) | K . Then for each β we have f β · w ∈ span { π ij | K · w : π ∈ b G and i, j = 1 , . . . , d π } and is thus in the ideal generated by N I( E ). Hence, since A ( G ) is anessential L ( K )-module, we find that w = lim β f β · w and thus is in theclosed ideal generated by N I( E ). (cid:3) For any subalgebra B of a commutative normed algebra A , we let hBi = span { ab + b : a ∈ A , b , b ∈ B} and B = span { ab : a, b ∈ B} . Corollary 1.5.
With G , K , A ( G ) , E ⊂ G/K and E ∗ ⊂ G as in Theorem1.4, we have that (i) I A ( G ) ( E ∗ ) has a bounded approximate identity (b.a.i.) if and only if I A ( G/K ) ( E ) has a b.a.i.; (ii) I A ( G ) ( E ∗ ) = I A ( G ) ( E ∗ ) if and only if I A ( G/K ) ( E ) = I A ( G/K ) ( E ) ; and (ii’) I A ( G ) ( E ∗ ) = J A ( G ) ( E ∗ ) if and only if I A ( G/K ) ( E ) = J A ( G/K ) ( E ) . (iii) I A ( G ) ( E ∗ ) = I A ( G ) ( E ∗ ) if and only if I A ( G/K ) ( E ) = I A ( G/K ) ( E ) . Proof.
We let I( E ∗ ) = I A ( G ) ( E ∗ ), I( E ) = I A ( G/K ) ( E ), etc. (i) If ( f α ) is a b.a.i. for I( E ), then ( N f α ) is a b.a.i. for the subalgebra N I( E ). It is readily checked that ( N f α ) is a b.a.i. for h N I( E ) i .If ( w α ) is a b.a.i. for I( E ∗ ), then (Γ w α ) is a b.a.i. for I( E ). Indeed wehave that Γ ◦ N = id and, since P is an idempotent, we have the following‘expectation property’:(1.5) Γ( w N f ) = Γ( w ) f for w ∈ A ( G × G ) and f ∈ A ( G/K ). Thus if f in I( E ) thenΓ( w α ) f − f = Γ( w α N f − N f ) −→ . (ii) & (ii’) It is clear that f ∈ I ( E ) ⇔ N f ∈ I ( E ∗ ) and f ∈ J ( E ) ⇔ N f ∈ J ( E ∗ ) . It can then be proved exactly as in Theorem 1.4 that (cid:10) N I ( E ) (cid:11) = I ( E ∗ ) and (cid:10) N J ( E ) (cid:11) = J ( E ∗ ) . Indeed, it is sufficient to note that if supp f is compact then so too issupp( π ij | K · f ), for each π ∈ b G f and i, j = 1 , . . . , d π . (iii) If I( E ) = I( E ), then N (cid:0) I( E ) (cid:1) = (cid:0) N I( E ) (cid:1) is dense in N I( E ).Hence we haveI( E ∗ ) = h N I( E ) i = D ( N I( E ) (cid:1) E = h N I( E ) i = I( E ∗ ) . Conversely, if I( E ∗ ) = I( E ∗ ) we have from the theorem above thatI( E ∗ ) = I( E ∗ ) = I( E ∗ ) N I( E ) . But it follows from (1.5) thatI( E ) = ΓI( E ∗ ) = Γ (cid:0) I( E ∗ ) N I( E ) (cid:1) = ΓI( E ∗ ) I( E ) = I( E ) ⊂ I( E )whence I( E ) = I( E ) . (cid:3) ONVOLUTIONS AND COSET SPACES 11 Some subalgebras of Fourier algebras
The algebra A ∆ ( G ) . In this section, we will always let G denote acompact group. We have the following characterisation of the Fourier alge-bra in [13, (34.4)]: for f ∈ C ( G )(2.1) f ∈ A( G ) ⇔ k f k A = X π ∈ b G d π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) < ∞ where ˆ f ( π ) = R G f ( s )¯ π ( s ) ds and k·k denotes the trace class norm. We alsorecall the following orthogonality relations [13, (27.19)]: if π, σ ∈ b G then inthe notation of (1.3) we have(2.2) Z G π ij ( s ) σ kl ( s ) ds = 1 d π δ πσ δ jl δ ik where i, j = 1 , . . . , d π , k, l = 1 , . . . , d σ and each δ αβ is the Kronecker δ -symbol.We first wish to characterise A ∆ ( G ), as defined in the previous section.We will make use of the following lemma. We let for π, σ ∈ b G , π × σ ∈ \ G × G be the Kronecker product representation. Also, we let N : C ( G ) → C ( G × G )be given by N f ( s, t ) = f ( st − ) for ( s, t ) in G × G , as suggested by (1.2). Lemma 2.1.
For any f ∈ C ( G ) and π ∈ b G , we have (cid:13)(cid:13)(cid:13)d N f (¯ π × π ) (cid:13)(cid:13)(cid:13) p = 1 √ d π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) where k·k p is the Schatten p -norm for ≤ p ≤ ∞ . Proof.
We first note that d N f (¯ π × π ) = Z G Z G f ( st − ) π ( s ) ⊗ ¯ π ( t ) dsdt = Z G Z G f ( s ) π ( st ) ⊗ ¯ π ( t ) dsdt = (cid:20)Z G f ( s ) π ( s ) ⊗ ¯ π ( e ) ds (cid:21) ◦ (cid:20)Z G π ( t ) ⊗ ¯ π ( t ) dt (cid:21) (2.3) = [ ˆ f (¯ π ) ⊗ I H ¯ π ] ◦ P where P is a rank 1 projection, as we shall see below. Indeed, the Schurorthogonality relations [13, (27.30)] tell us that π ⊗ ¯ π contains the trivial rep-resentation 1 with multiplicity 1. Thus when we decompose into irreducibles π ⊗ ¯ π ∼ = L σ ∈ b G m σ · σ , we obtain Z G π ( t ) ⊗ ¯ π ( t ) dt ∼ = M σ ∈ b G m σ · Z G σ ( t ) dt where R G σ ( t ) dt = 0 unless σ = 1, and hence the formula above reduces toa rank 1 projection, P . Let { ξ , . . . , ξ d π } be an orthonormal basis for H π ; we claim that P = h·| η i η where η = 1 √ d π d π X k =1 ξ k ⊗ ¯ ξ k . Indeed, for any d π × d π unitary matrix U we have that P d π k =1 U ξ k ⊗ U ξ k = P d π k =1 ξ k ⊗ ¯ ξ k . Hence η is a norm 1 vector, invariant for π ⊗ ¯ π , and theformula for P follows.Thus d N f (¯ π ⊗ π ) = [ ˆ f (¯ π ) ⊗ I H ¯ π ] ◦ P = h·| η i [ ˆ f (¯ π ) ⊗ I H ¯ π ] η and, using the standard formula for rank 1 operators, kh·| ζ i ϑ k p = k ζ k k ϑ k ,we obtain(2.4) (cid:13)(cid:13)(cid:13)d N f (¯ π ⊗ π ) (cid:13)(cid:13)(cid:13) p = (cid:13)(cid:13)(cid:13) [ ˆ f (¯ π ) ⊗ I H ¯ π ] η (cid:13)(cid:13)(cid:13) H π ⊗ H ¯ π . Letting ˆ f (¯ π ) kl = D ˆ f (¯ π ) ξ l | ξ k E we have[ ˆ f (¯ π ) ⊗ I H ¯ π ] η = 1 √ d π d π X k =1 h ˆ f (¯ π ) ξ k i ⊗ ¯ ξ k = 1 √ d π d π X k =1 d π X l =1 ˆ f (¯ π ) kl ξ k ⊗ ¯ ξ l . Since { ξ l ⊗ ¯ ξ k : k, l = 1 . . . , d π } is an orthonormal basis for H π ⊗ H ¯ π weobtain that(2.5) (cid:13)(cid:13)(cid:13) [ ˆ f (¯ π ) ⊗ I H ¯ π ] η (cid:13)(cid:13)(cid:13) H π ⊗ H ¯ π = 1 d π d π X k =1 d π X l =1 | ˆ f (¯ π ) kl | = 1 d π (cid:13)(cid:13)(cid:13) ˆ f (¯ π ) (cid:13)(cid:13)(cid:13) . The result is obtained by combining (2.4) with (2.5) and the fact that (cid:13)(cid:13)(cid:13) ˆ f (¯ π ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) . (cid:3) We now obtain a characterisation of A ∆ ( G ) in the spirit of (2.1). Theorem 2.2. If f ∈ C ( G ) , then f ∈ A ∆ ( G ) ⇔ X π ∈ b G d / π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) < ∞ . Moreover, the latter quantity is k f k A ∆ = inf {k w k A : w ∈ A( G × G ) with Γ w = f } . Proof.
Since N : A ∆ ( G ) → A( G × G : ∆) ⊂ A( G × G ) is an isometry we have f ∈ A ∆ ( G ) ⇔ N f ∈ A( G × G )in which case k f k A ∆ = k N f k A . Recall that \ G × G = { π × σ : π, σ ∈ b G } ;see [13, (27.43)], for example. Note that analogous computations to (2.3),combined with (2.2), show that(2.6) d N f ( π × σ ) = 0 if σ = ¯ π. ONVOLUTIONS AND COSET SPACES 13
We have that
N f ∈ A( G × G ) exactly when k N f k A = X ( σ,π ) ∈ b G × b G d σ d π (cid:13)(cid:13)(cid:13)d N f ( σ × π ) (cid:13)(cid:13)(cid:13) by (2.1)= X π ∈ b G d π (cid:13)(cid:13)(cid:13)d N f (¯ π × π ) (cid:13)(cid:13)(cid:13) by (2.6)= X π ∈ b G d / π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) by Lemma 2.1and the latter quantity is finite. (cid:3) We prove some consequences of the result above. Let us note for any d × d matrix the well-known inequalities(2.7) 1 √ d k A k ≤ k A k ≤ k A k . These inequalities are sharp with scalar matrices αI serving as the extremecase for the left inequality, and rank 1 matrices serving for the extreme caseon the right.We let ZL ( G ) be the centre of the convolution algebra L ( G ). Corollary 2.3.
We have that A( G ) ∩ ZL ( G ) = A ∆ ( G ) ∩ ZL ( G ) with k f k A = k f k A ∆ for f in this space. Proof.
It is well-known that f ∈ ZL ( G ) only if ˆ f ( π ) = α f,π I H π for somescalar α f,π , for each π in b G . Thus by the left extreme case of (2.7) we have X π ∈ b G d / π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) = X π ∈ b G d π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) . Hence it follows that f ∈ A( G ) ∩ ZL ( G ) if and only if f ∈ A ∆ ( G ) ∩ ZL ( G ),with k f k A = k f k A ∆ . (cid:3) Corollary 2.4.
We have that A ∆ ( G ) = A( G ) if and only if G admits anabelian subgroup of finite index. Proof.
We invoke the well-known result of [20] that G admits an abelian subgroupof finite index ⇔ d G = sup π ∈ b G d π < ∞ . Now if d G < ∞ , then for any u in A( G ) we have, using (2.7), that X π ∈ b G d / π k ˆ u ( π ) k ≤ d / G X π ∈ b G d π k ˆ u ( π ) k < ∞ so u ∈ A ∆ ( G ). Conversely, if A( G ) = A ∆ ( G ), then since k·k A ≤ k·k A ∆ , theopen mapping theorem provides us with a constant K such that k·k A ∆ ≤ K k·k A . For any π in b G we let π be as in (1.3). We haveˆ π ( σ ) = Z G h π ( s ) ξ π | ξ π i ¯ σ ( s ) ds = ( d π (cid:10) ·| ¯ ξ π (cid:11) ¯ ξ π if σ = π d / π k ˆ π ( π ) k = d / π k ˆ π ( π ) k = k ˆ π k A ∆ ≤ K k ˆ π k A = Kd π k ˆ π ( π ) k so d π ≤ K . Thus d G ≤ K < ∞ . (cid:3) We remark that despite the identification ( G × G ) / ∆ ∼ = G , the result abovetells us that for the Fourier algebra over this coset space, A( G × G/ ∆) ∼ =A ∆ ( G ) is not naturally ismorphic to A( G ). We will see in Section 3 thatA ∆ ( G ) can fail to be operator weakly amenable, while A( G ) is always oper-ator weakly amenable.2.2. Some subalgebras of A ∆ ( G ) . Let us begin with a variant of Propo-sition 1.2.
Proposition 2.5. (i)
There is a completely isometric identification A ∆ ( G ) b ⊗ A ∆ ( H ) ∼ = A ∆ ( G × H ) given on elementary tensors by f ⊗ g f × g . (ii) If K is a closed subgroup of G , and L is a closed subgroup of H , thenwe obtain a completely isometric identification A ∆ ( G/K ) b ⊗ A ∆ ( H/L ) ∼ = A ∆ ( G × H/K × L ) . Proof. (i)
We have, using Proposition 1.2, completely isometric identifica-tions A ∆ ( G ) b ⊗ A ∆ ( H ) ∼ = A( G × G/ ∆ G ) b ⊗ A( H × H/ ∆ H ) ∼ = A( G × G × H × H/ ∆ G × ∆ H ) . And we have a completely isometric identificationA ∆ ( G × H ) ∼ = A( G × H × G × H/ ∆ G × H ) . Thus we must show that(2.8) A( G × G × H × H/ ∆ G × ∆ H ) ∼ = A( G × H × G × H/ ∆ G × H ) . Let ς be the topological group isomorphism ( s , t , s , t ) ( s , s , t , t ) : G × H × G × H → G × G × H × H . This map induces a completely isometricisomorphism u u ◦ ς from A( G × G × H × H ) to A( G × H × G × H ). Moreover,the following digram commutes.A( G × G × H × H ) u u ◦ ς / / P ∆ G × ∆ H (cid:15) (cid:15) A( G × H × G × H ) P ∆ G × H (cid:15) (cid:15) u u ◦ ς − o o A( G × G × H × H : ∆ G × ∆ H ) u u ◦ ς / / A( G × H × G × H : ∆ G × H ) u u ◦ ς − o o ONVOLUTIONS AND COSET SPACES 15
Since each of the top row maps are complete isometries, and each of the maps P ∆ G × ∆ H and P ∆ G × H are complete quotient maps, we have that the bottomrow maps must each be complete quotient maps which are mutual inverses,hence complete isometries. By standard identifications, (2.8) follows. (ii) This can be proved exactly as Proposition 1.2 using (i), above, inplace of the identification A( G ) b ⊗ A( H ) ∼ = A( G × H ). (cid:3) We now define a sequence of subalgebras of A( G ). Theorem 2.6.
Let A ∆ ( G ) = A ∆ ( G ) and for each n ≥ let A ∆ n +1 ( G ) = Γ(A ∆ n ( G × G )) . (i) Each A ∆ n ( G ) is a subalgebra of A( G ) , which is closed under both leftand right translations and for which all translations are complete isometries.Also, for each n , the map Γ : A ∆ n ( G × G ) → A ∆ n +1 ( G ) is a complete quotientmap. (ii) If f ∈ C ( G ) , then f ∈ A ∆ n ( G ) ⇔ X π ∈ b G d (2 n +1) / π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) < ∞ . Moreover, the latter quantity is the norm, k f k A ∆ n . Proof. (i)
We use induction. If the ascribed properties hold for A ∆ n ( G ),and we have that(2.9) A ∆ n ( G ) b ⊗ A ∆ n ( G ) ∼ = A ∆ n ( G × G ) . then it follows from Proposition 1.3 (i) that A ∆ n +1 ( G ) is a subalgebra ofA( G ), which is completely isometrically isomorphic to a completely contrac-tively complemented subspace of A ∆ n ( G × G ). It follows from Proposition1.3 (ii) that A ∆ n +1 ( G ) is closed under left and right translations. The for-mula (2.9) holds for n = 1 by Proposition 2.5 (i). Moreover, if (2.9) holdsfor n , then we can use the proof of Proposition 2.5 (ii), then (i), to see that(2.9) holds for n + 1. (ii) This follows exactly as the proof of Theorem 2.2, using Lemma 2.1in the case p = 2, and induction. (cid:3) We note that a simple modification of Corollary 2.4 shows that if G ad-mits no abelian subgroup of finite index then { A ∆ n ( G ) } ∞ n =1 is a properlynested sequence of subalgebras of A( G ). We also note that it follows fromProposition 1.1 (i), that Σ A ∆ ( G ) ∼ = G . We suspect the same holds for eachA ∆ n ( G ) ( n ≥ A ∆2 ( G ) ∼ = G . Indeed, we have that ∆ is not a set of spectralsynthesis for A ∆ ( G × G ) when G is a nonabelian connected Lie group, by(3.1), below. Amenability properties
Failure of operator weak amenability of A ∆ ( G ) . We adapt argu-ments from [16, Thm. 7.2 & Cor. 7.3]. We also use the Fourier series of any f in T ( G ) ( T ( G ) is defined in (1.4)): f ( s ) = X π ∈ b G d π d π X i,j =1 ˆ f ( π ) ij π ij ( s ) = X π ∈ b G d π trace h ˆ f ( π ) π ( s ) t i where ˆ f ( π ) ij = D ˆ f ( π ) ξ πj | ξ πi E and A t is the transpose of a matrix A . This isa variant of the formula given in [13, (34.1)]. Theorem 3.1. If G is a nonabelian connected compact Lie group, then A ∆ ( G ) ∼ = A( G × G/ ∆) is not operator weakly amenable. Proof.
We will establish that A ∆ ( G ) admits a point derivation at e . Thisimplies that I A ∆2 ( G ) ( { e } ) = I A ∆2 ( G ) ( { e } ). Indeed, the point derivation willvanish on I A ∆2 ( G ) ( { e } ) , but not on I A ∆2 ( G ) ( { e } ). Hence it follows from theconstruction of A ∆ ( G ) (Theorem 2.6), and Corollary 1.5 (iii), that(3.1) I A ∆ ( G × G ) (∆) = I A ∆ ( G × G ) (∆) . This condition implies that A ∆ ( G ) is not operator weakly amenable by [11,Thm. 3.2], which was shown to be a valid characterisation of operator weakamenability in [27].It has been shown in [22] that under the assumptions given, there is asubgroup T ∼ = T in G such that for any π in b Gπ | T ∼ = d π M k =1 χ n k with each | n k | < d π where χ n ( z ) = z n for n in Z ∼ = b T . We let θ t θ : ( − π, π ] → T be theparameterisation of T which corresponds to θ e iθ : ( − π, π ] → T . For each π in b G we can choose an orthonormal basis for H π with respect to which π ( t θ ) = diag( e in θ , . . . , e in dπ θ ); and it follows by elementary estimates that (cid:13)(cid:13) π ( t θ ) t − I (cid:13)(cid:13) θ = max k =1 ,...,d π | e in k θ − || θ | < d π for θ in a neighbourhood of 0.We have for f in T ( G ) that f ( t θ ) − f ( e ) θ = X π ∈ b G d π trace (cid:20) ˆ f ( π ) ( π ( t θ ) t − I ) θ (cid:21) . We note that for small θ we have, using (2.7) (cid:12)(cid:12)(cid:12)(cid:12) f ( t θ ) − f ( e ) θ (cid:12)(cid:12)(cid:12)(cid:12) ≤ X π ∈ b G d π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) π ( t θ ) t − I (cid:13)(cid:13) | θ | ≤ X π ∈ b G d / π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) where the last quantity is k f k A ∆2 , by Theorem 2.6 (ii). Hence, since eachlim n →∞ ( π ( t θ ) − I ) /θ exists, we have that d ( f ) = ddθ (cid:12)(cid:12)(cid:12) θ =0 f ( t θ ) = lim θ → f ( t θ ) − f ( e ) θ . exists, and | d ( f ) | ≤ k f k A ∆2 for f in T ( G ). Hence d extends to a contractivepoint derivation on A ∆ ( G ) at e . (cid:3) We remark that Johnson [16, Cor. 7.3] showed that the point derivation d : T ( G ) → C extends to a bounded map on A γ ( G ) (see Section 4), andhence established that A( G ) is not weakly amenable. Since ∆ is a set ofspectral synthesis for A( G × G ), we can proceed as in the first paragraph ofthe proof above to see that d cannot be extended to A ∆ ( G ).As an application of the above result, we obtain a new set of non-synthesis. Corollary 3.2.
Let G be a compact connected non-abelian Lie group. Then (∆ G × ∆ G )∆ G × G fails spectral synthesis for A( G × G × G × G ) . Proof.
It follows from (3.1) that ∆ G is not a set of spectral synthesis forA ∆ ( G × G ). We then appeal to Proposition 2.5 (ii) to see thatA ∆ ( G ) ˆ ⊗ A ∆ ( G ) ∼ = A ∆ ( G × G ) ∼ = A( G × G × G × G/ ∆ G × G ) . In the identification (1.1) we have ∆ G ∼ = { ( s, s, e, e )∆ G × G : s ∈ G } . It thenfollows from Corollary 1.5 (iii) that∆ ∗ G ∼ = (∆ G ×{ } )∆ G × G = (∆ G × ∆ G )∆ G × G is not spectral for A( G × G × G × G ). (cid:3) Failure of operator weak amenability of A( G/K ) . Let us nowturn our attention to general locally compact groups. We will let G denotea locally compact group. Let us collect some useful facts. Lemma 3.3.
Let H be a closed subgroup of G and K be a compact subgroupof H . Then the restriction map u u | H maps A( G : K ) onto A( H : K ) . Proof.
This is [10, Lem. 3.6 (ii)]. (cid:3)
Lemma 3.4.
Let N be a compact normal subgroup of G , and q : G → G/N be the canonical quotient map. If ˜ K is a compact subgroup of G/N and K = q − ( ˜ K ) , then the algebras A( G : K ) and A( G/N : ˜ K ) are completelyisometrically isomorphic. Proof.
The map u u ◦ q : A( G/N ) → A( G ) is a complete isometry withrange A( G : N ). If u ∈ A( G/N : ˜ K ), then for s in G and k in K we have u ◦ q ( sk ) = u (cid:0) q ( s ) q ( k ) (cid:1) = u (cid:0) q ( s ) (cid:1) = u ◦ q ( s ) so u ◦ q ∈ A( G : K ). Conversely, let v ∈ A( G : K ) ⊂ A( G : N ). Let u in A( G/N ) be such that v = u ◦ q . For any ˜ s in G/N and ˜ k ∈ ˜ K , find s ∈ G and k ∈ K so q ( s ) = ˜ s and q ( k ) = ˜ k . We have u (˜ s ˜ k ) = u ◦ q ( sk ) = v ( sk ) = v ( s ) = u (˜ s )so u ∈ A( G/N : ˜ K ). (cid:3) We now obtain a generalisation of Theorem 3.1.
Theorem 3.5.
Suppose G contains a connected nonabelian compact sub-group K . Then there is a compact subgroup K ∗ of G × G such that A( G × G/K ∗ ) is not operator weakly amenable. Proof.
There exists a closed normal subgroup N of K such that K/N isa compact Lie group [13, (28.61)(c)]. Moreover, we may have that
K/N is nonabelian. Indeed, if st = ts in K , find a neighbourhood U of e in K such that ts stU ; find in U a compact subgroup N so K/N is a Liegroup. Then by Theorem 3.1, A(
K/N × K/N : ∆
K/N ) is not operator weaklyamenable. Let q : K × K → K/N × K/N be the quotient map and K ∗ = q − (∆ K/N ). Then by Lemma 3.4, A( K × K : K ∗ ) ∼ = A( K/N × K/N : ∆
K/N )is not operator weakly amenable. Moreover, by Lemma 3.3, A( G × G/K ∗ ) ∼ =A( G × G : K ∗ ) admits A( K × K : K ∗ ) as a quotient, and hence is not operatorweakly amenable either. (cid:3) Examples of hyper-Tauberian A( G/K ) . In this section we will al-ways let G denote a locally compact group. In this section we generalise thefact that A( G ) is hyper-Tauberian when the connected component of theidentity is abelian [26, Theo. 21]. Our approach is inspired by that of [8,Thm. 3.3]. However, in dealing with coset spaces some extra technicalitiesarise. The following lemma deals with some of these technicalities. Lemma 3.6.
Suppose G contains an open subgroup G and a compact sub-group K for which A( G /G ∩ K ) is hyper-Tauberian. Then A( G/K ) ishyper-Tauberian. Proof.
We will identify A(
G/K ) ∼ = A( G : K ), etc., so we may work withinthe algebra A( G ).Let H = G ∩ K . We will first establish that A( G : H ) is boundedlyisomorphic to a certain subalgebra of A( G : K ).Since H is open in K , it is of finite index. Thus there is a finite set { k , k , . . . , k n } ⊂ K for which(3.2) K = n [ i =1 Hk i and Hk i ∩ Hk j = ∅ if i = j. It then follows easily that G K = n [ i =1 G k i and G k i ∩ G k j = ∅ if i = j ONVOLUTIONS AND COSET SPACES 19 and thus GK is a union of open cosets. Then the indicator function 1 GK ∈ B( G ), and it is clear that k · G K = 1 G K for each k ∈ K . Hence 1 G K A( G : K ) is a closed subalgebra of A( G : K ).Now if u ∈ A( G : K ), then u | G ∈ A( G : H ) by Lemma 3.3. Therestriction map(3.3) u u | G : 1 G K A( G : K ) → A( G : H )is injective, for if u in A( G : K ) satisfies u ( s ) = 0, for some s = s k in G K ,where s ∈ G and k ∈ K , then u ( s ) = u ( s ) = 0. Let us see the map in(3.3) is surjective. Indeed, if v ∈ A( G : H ) we define ˜ v, w in A( G ) by˜ v ( s ) = ( v ( s ) if s ∈ G w = n X i =1 k − i · ˜ v. Then w ∈ G K A( G ) with w | G = v . Now, if s ∈ G , h ∈ H and k i is asabove, then, since s h ∈ G and v ∈ A( G : H ), we have w ( s hk i ) = v ( s h ) = v ( s ) = w ( s )and hence by (3.2), w ( s k ) = w ( s ) for s in G and k in K . Thus if s ∈ G ,and k ∈ K , then either s ∈ G K and we can find s in G and k i as abovesuch that s = s k i , so we have w ( sk ) = w ( s k i k ) = w ( s ) = w ( s k i ) = w ( s );or s G K , so for any k in K , sk G K and thus w ( sk ) = 0 = w ( s ). Weconclude that w ∈ A( G : K ). Thus the map in (3.3) is a contractive bijectionwhich is also a homomorphism. It follows from the open mapping theoremthat this map is an isomorphism. Thus, since A( G : H ) is hyper-Tauberian,1 G K A( G : K ) is hyper-Tauberian.Now let T : A( G : K ) → A( G : K ) ∗ be a bounded local operator, i.e.so N ∗◦ T ◦ N : A( G/K ) → A( G/K ) ∗ is a local operator. We will show for u , u , u in A( G : K ), each having compact support, that(3.4) h T ( u u ) , u i = h u T ( u ) , u i . Then it follows from Proposition 1.1 (ii), that (3.4) holds for any u , u , u in A( G : K ).Since G is open, there are t , . . . , t n in G such that [ j =1 supp u j ⊂ n [ i =1 t i G ⊂ n [ i =1 t i G K. The map u t i ∗ u is an isometric isomorphism from 1 G K A( G : K ) to1 t i G K A( G : K ) for each i , so each 1 t i G K A( G : K ) is hyper-Tauberian. Nowlet w = 1 t G K , and w i = 1 t i G K − i − X k =1 w k ! for i = 2 , . . . , n. Then each w i is an idempotent in B( G ) with k · w i = w i for each k in K .Moreover, w i w j = 0 if i = j . For each i the map u w i u from 1 t i G K A( G : K ) → w i A( G : K ) is a surjective homomorphism, so w i A( G : K ) is hyper-Tauberian by [26, Thm. 12]. Then the algebra A = n X i =1 w i A( G : K ) ∼ = n M i =1 w i A( G : K )is hyper-Tauberian, by [26, Cor. 13], and contains each u j . The inclusionmap ι : A → A( G : K ) is an A -module map, so ι ∗◦ T ◦ ι : A → A ∗ is an A -localoperator. Hence ι ∗◦ T ◦ ι is an A -module map and (3.4) holds. (cid:3) Theorem 3.7. If G has abelian connected component of the identity, G e ,then for any compact subgroup K , A( G/K ) is hyper-Tauberian. Proof.
As in the proof of the lemma above, we will identify A(
G/K ) ∼ =A( G : K ), etc.We will first assume that G is almost connected.Let U be a neighbourhood identity in G . Then, by [19], there is a compactnormal subgroup N U ⊂ U such that ˜ G = G/N U is a Lie group. Let q : G → ˜ G be the canonical quotient map and ˜ K = q ( K ), so ˜ K is a compact subgroupof ˜ G . We have that the connected component of the identity of ˜ G satisfies˜ G e = q ( G e ) by [14, (7.12)]. Then ˜ G e is abelian, and open in ˜ G since thelatter is a Lie group. It then follows that ˜ G e ∩ ˜ K is normal in ˜ G e andA( ˜ G e : ˜ G e ∩ ˜ K ) ∼ = A( ˜ G e / ˜ G e ∩ ˜ K ) is hyper-Tauberian by [26, Prop. 18]. Thenit follows from Lemma 3.6 that A( ˜ G : ˜ K ) is hyper-Tauberian.We let K U = q − ( ˜ K ) = KN U . Then Lemma 3.4 tells us that A( G : K U ) isisometrically isomorphic to A( ˜ G : ˜ K ), and hence is hyper-Tauberian. Since K U ⊃ K , it follows that A( G : K U ) ⊂ A( G : K ).Let u ∈ A( G : K ), and ε >
0. Fix a compact neighbourhood V of e . Finda neighbourhood U of e such that U ⊂ V and k s · u − u k A < εm ( KV ) for s ∈ U. Then find normal a subgroup N U ⊂ U as above and let P U u = Z K U k · u dk ∈ A( G : K U ) . We note that for any k = k ′ n ∈ KN U with k ′ in K and n in N U we have k k · u − u k A = (cid:13)(cid:13) k ′ · ( n · u ) − u (cid:13)(cid:13) A = (cid:13)(cid:13) n · u − k ′− · u (cid:13)(cid:13) A = k n · u − u k A < εm ( KV )since u ∈ A( G : K ) and right translation on A( G ) is an isometry. Hence wefind k P U u − u k A ≤ Z K U k k · u − u k A dk < εm ( KV ) m ( K U ) ≤ ε. Thus we have lim U ց{ e } P U u = u . ONVOLUTIONS AND COSET SPACES 21
Now let T : A( G : K ) → A( G : K ) ∗ be a bounded local operator. Sincethe inclusion ι : A( G : K U ) → A( G : K ) is an A( G : K U )-module map,and A( G : K U ) is hyper-Tauberian, ι ∗◦ T ◦ ι : A( G : K U ) → A( G : K U ) ∗ isan A( G : K U )-local map and hence an A( G : K U )-module map. Hence if u , u , u ∈ A( G : K ), then h T ( P U u P U u ) − P U u T ( P U u ) , P U u i = 0Taking U ց { e } , as above, we obtain h T ( u u ) − u T ( u ) , u i = 0and hence T is an A( G : K )-module map too.Finally, if G is not almost connected, we can find an almost connectedopen subgroup G . Then, from above, A( G : G ∩ K ) is hyper-Tauberian.We then appeal to Lemma 3.6. (cid:3) We say G is a [SIN]-group if there is a neighbourhood basis at e consistingof sets invariant under inner automorphisms. We obtain for such groups, apartial converse of Theorem 3.7, which is similar to [8, Theo. 3.7]. Corollary 3.8. If G is a [SIN]-group, then A( G × G/K ∗ ) is hyper-Tauberianfor every compact subgroup K ∗ of G × G if and only if G e is abelian. Proof.
Sufficiency is an obvious consequence of Theorem 3.7. To see ne-cessity, we first note that G e is a [SIN]-group and the Freudenthal-WeilTheorem (see [21, 12.4.28]) tells us that G e ∼ = V × K where V is a vectorgroup and K a connected compact group. If K is nonabelian, we appealto Theorem 3.5 to see that there exists a subgroup K ∗ of G × G such thatA( G × G/K ∗ ) is not operator weakly amenable, hence not weakly amenableand not hyper-Tauberian. (cid:3) Operator amenability of A( G/K ) . We recall that [MAP] K -groupswere defined in Section 1.3. We note that T f ( G )A( G ) ⊂ A( G ), since T f ( G )is a subalgebra of the Fourier-Stieltjes algebra B( G ). Theorem 3.9.
Let G be an amenable locally compact group and K a com-pact subgroup so that G is [MAP] K . Then the following are equivalent: (i) A( G/K ) is operator amenable; and (ii) ( K × K )∆ is in the closed coset ring of G × G .Moreover, If G is compact, each of the above is equivalent to (iii) I A ∆ ( G ) ( K ) has a bounded approximate identity. Proof.
Since the map u ˇ u is an isomorphism on A( G × G ), I A( G × G ) (( K × K )∆) has a b.a.i. if and only if I A( G × G ) (∆( K × K )) has a b.a.i. It followsfrom [7] that I A( G × G ) (∆( K × K )) has a b.a.i. if and only if ∆( K × K ) is inthe closed coset ring of G × G . An application of Corollary 1.5 (i), tells usthat I A( G × G ) (∆( K × K )) has a b.a.i. if and only if I A( G × G/K × K ) (∆ G/K ) hasa b.a.i., where ∆
G/K = { ( s, s ) K × K : s ∈ G } . Under the identificationA( G/K ) b ⊗ A( G/K ) ∼ = A( G × G/K × K ) provided by Proposition 1.2, I A( G × G/K × K ) (∆ G/K ) corresponds to the ker-nel of the multiplication map m : A( G/K ) b ⊗ A( G/K ) → A( G/K ). SinceA(
G/K ) has a bounded approximate identity, this kernel has a boundedapproximate identity if and only if A(
G/K ) is operator amenable, by thecompletely bounded version of a splitting result from [17] (also see [2, 3.10]).Hence the equivalence of (i) and (ii), above, is established.If G is compact then by Corollary 1.5 (i), I A ∆ ( G ) ( K ) has a b.a.i. if andonly if I A( G × G ) ( K ∗ ) has a b.a.i., where(3.5) K ∗ = ( K ×{ e } )∆ = ( K × K )∆ . Hence it again follows [7] that properties (ii) and (iii), above, are equivalent. (cid:3)
The situation that ( K × K )∆ is in the coset ring of G × G seems rare. Itdoes occur, for example, when K contains a subgroup N , of finite index,which is normal in G . Thus it is only in such cases that we know A( G/K )is operator amenable.However, we gain some situations in which ( K × K )∆ is a set of localsynthesis for A( G × G ). Theorem 3.10. If G has abelian connected component of the identity and K is a compact subgroup of G so that G is [MAP] K , then (i) ( K × K )∆ is a set of local synthesis for A( G × G ) .Moreover, if G is compact then (ii) K is a set of spectral synthesis for A ∆ ( G ) . Proof. (i)
By Theorem 3.7, A(
G/K ) is hyper-Tauberian. Hence it is opera-tor hyper-Tauberian. Thus, by [26, Thm. 6], ∆
G/K = { ( s, s ) K × K : s ∈ G } ,is a set of local synthesis for A( G × G/K × K ), since the latter is isomorphicto A( G/K ) b ⊗ A( G/K ) by Proposition 1.2. (We note that [26, Thm. 6] isproved for the projective tensor product of a hyper-Tauberian algebra withitself. However, an inspection of the proof, coupled with the formula repre-senting an arbitrary element of the operator projective tensor product in [4,10.2.1], shows that it holds for an operator hyper-Tauberian algebra with theoperator projective tensor product.) Then it follows from Corollary 1.5 (ii’)that ∆ ∗ G/K = ∆( K × K ) is a set of local synthesis for A( G × G ). Since u ˇ u is an isomorphism on A( G × G ), ( K × K )∆ is also a set of local synthesis. (ii) By Corollary 1.5 (ii), K is spectral for A ∆ ( G ) ∼ = A( G × G/ ∆) if andonly if K ∗ = ( K × K )∆ (see (3.5)) is spectral for A( G × G ). We appeal to(i), above. (cid:3) Convolution
Convolution on compact groups.
We close this article by address-ing, in part, the case of what happens if we replace the map Γ, in Section1.2, with convolution.
ONVOLUTIONS AND COSET SPACES 23
We let A ( G × G ) be as in Section 1.2, and insist further that the groupaction of left translation is isometric on A ( G × G ) and continuous on G × G .We then define a group action ( r, f ) r ⋄ f : G ×A ( G × G ) → A ( G × G ) by r ⋄ w ( s, t ) = w ( sr, r − t ) = ( r, e ) · [( e, r ) ∗ w ]( s, t ) = ( e, r ) ∗ [( r, e ) · w ]( s, t ) . We let ˇ∆ = { ( t, t − ) : t ∈ G } and define A ( G × G : ˇ∆) = { f ∈ A ( G × G ) : r ⋄ f = f for every r in G } . We note that for w ∈ A ( G × G : ˇ∆), w ( s, t ) = w ( s , t ), provided ( ss − , tt − ) ∈ ˇ∆, even though ˇ∆ is not a subgroup unless G is abelian. We then defineˇ P : A ( G × G ) → A ( G × G ) , ˇ P w = Z G r ⋄ wdr and ˇ M : A ( G × G : ˇ∆) → C ( G ) , ˇ M f ( s ) = f ( s, e ) . Then ˇ P is a contractive idempotent whose range is A ( G × G : ˇ∆), in particularˇ P is a quotient map. The map ˇ M is injective; we denote its range A ˇ∆ ( G )and assign it the norm which makes ˇ M an isometry. Then ˇ M has inverseˇ N : A ˇ∆ ( G ) → A ( G × G : ˇ∆) , ˇ N f ( s, t ) = f ( st ) . Finally, we define ˇΓ : A ( G × G ) → A ˇ∆ ( G ) , ˇΓ = ˇ M ◦ ˇ P . If A ( G × G ) contains an elementary product f × g , then ˇΓ f × g = f ∗ g . Thus,ˇΓ may be regarded simply as the convolution map.4.2. Convolution on the Fourier algebras.
We will consider, now, onlythe case where A ( G × G ) = A( G × G ). As in Section 1.2, it is easy to verifythat ˇΓ : A( G × G ) → A ˇ∆ ( G ) is a complete quotient map and A ˇ∆ ( G ) ⊂ A( G ).We recall that A γ ( G ) is the subalgebra of A( G ) defined in Section 1.2,Example (iii). For this algebra we have(4.1) f ∈ A γ ( G ) ⇔ X π ∈ b G d π (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) < ∞ and the latter quantity is the norm k f k A γ , by [16, Prop. 2.5]. Theorem 4.1.
We have A ˇ∆ ( G ) = A γ ( G ) , isometrically. Morevoer, A ˇ∆ ( G ) =A( G ) if and only if G admits an abelian subgroup of finite index. Proof.
We have computation similar to that in Lemma 2.1. For f in C ( G )and π in b G we have d ˇ N f (¯ π × ¯ π ) = Z G Z G f ( st ) π ( s ) ⊗ π ( t ) dt = Z G Z G f ( s ) π ( st − ) ⊗ π ( t ) dt = (cid:20)Z G f ( s ) π ( s ) ⊗ π ( e ) ds (cid:21) ◦ (cid:20)Z G π ( t − ) ⊗ π ( t ) dt (cid:21) = [ ˆ f ( π ) ⊗ I H π ] ◦ d π U where U is a unitary, in fact a permutation matrix, as we shall see below.Indeed, identify the linear operators on H π with the matrix space M d π viaan orthonormal basis, and then identify M d π ⊗ M d π ∼ = M d π . We obtain,using (2.2), ij, kl th entry (cid:18)Z G π ( t − ) ⊗ π ( t ) dt (cid:19) ij,kl = Z G π ij ( t − ) π kl ( t ) dt = Z G π ji ( t ) π kl ( t ) dt = 1 d π δ il δ jk where δ il and δ jk are the Kronecker δ -symbols.Thus it follows that (cid:13)(cid:13)(cid:13)d ˇ N f (¯ π × ¯ π ) (cid:13)(cid:13)(cid:13) = 1 d π (cid:13)(cid:13)(cid:13) ˆ f ( π ) ⊗ I H π (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ˆ f ( π ) (cid:13)(cid:13)(cid:13) . If π = σ in b G then it can be shown, just as above, that d ˇ N f ( π × σ ) = 0.We then obtain that f ∈ ranˇΓ ⇔ X ( π,σ ) ∈ \ G × G d π d σ (cid:13)(cid:13)(cid:13)d ˇ N f ( π × σ ) (cid:13)(cid:13)(cid:13) = X π ∈ b G d π (cid:13)(cid:13)(cid:13) ˆ f (cid:13)(cid:13)(cid:13) < ∞ . This is precisely the characterisation obtained for A γ ( G ) in (4.1).It can be shown, similarly as in Corollary 2.4, that A γ ( G ) = A( G ) if andonly if G has an abelian subgroup of finite index. (cid:3) Note that it follows from (2.7) and Theorems 2.2 and 2.6 that there arecontractive inclusions A ∆ ( G ) ⊂ A γ ( G ) ⊂ A ∆ ( G ) . Also, note that since u ˇ u is an isometric isomorphism on A( G ), thedefinition of A γ ( G ) given in Section 1.2, Example (iii), yields the equalityˇΓ (cid:0) A( G ) b ⊗ A( G ) (cid:1) = ˇΓ (cid:0) A( G ) ⊗ γ A( G ) (cid:1) . ONVOLUTIONS AND COSET SPACES 25
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