OPERATOR WEAK AMENABILITY of the FOURIER ALGEBRA Let G Be a Locally Compact Group. It Is Shown by Johnson In

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 12, Pages 3609–3617 S 0002-9939(02)06680-7 Article electronically published on June 11, 2002

OPERATOR WEAK AMENABILITY OF THE FOURIER ALGEBRA

NICO SPRONK

(Communicated by David R. Larson)

Abstract. We show that for any locally compact group G, the Fourier algebra A(G) is operator weakly amenable.

Let G be a locally compact group. It is shown by Johnson in [16] (and by Despi´c and Ghahramani in [6]) that the group algebra L1(G) is always weakly amenable. It is natural to ask whether the same holds for the Fourier algebra A(G). In [17] it is shown for G = SO(3) that A(G) is not weakly amenable. We note that A(G) is weakly amenable (and, in fact amenable) whenever G is Abelian, since 1 A(G) ∼= L (G)whereG is the dual group. Also, A(G) is weakly amenable whenever the connected component of the identity in G is Abelian [11]. It is conjectured that this characterizesb theb weak amenability of A(G). Since A(G) is the predual of the von Neumann algebra VN(G), it admits a natural structure as an operator space. Using this structure, Ruan [23] developed a completely bounded cohomology theory and proved that A(G) is operator amenable exactly when G is an amenable group. This is analogous to Johnson’s result [15] that L1(G) is amenable exactly when G is amenable. We note that the natural operator 1 space structure on L (G) as the predual of L∞(G) is such that all bounded maps from L1(G) into any operator space are automatically completely bounded. Thus the notions of amenability and operator amenability coincide on L1(G), making Ruan’s result truly a dual result of Johnson’s, in the sense that A(G) is the dual of L1(G) in a way which generalizes Pontryagin’s Duality Theorem (see [9]). The purpose of this note is to show that the natural operator space structure on A(G) allows us to obtain another analogous result to one for L1(G): we show that A(G) is always operator weakly amenable. We note that this result was obtained in [12], for the case that the connected component of the identity in G is amenable. The author would like to thank his doctoral advisor, Brian Forrest, for suggesting this problem.

1. Preliminaries

If is a Banach space we always let ∗ denote its dual and ( )denote the BanachX algebra of bounded operators onX .Thesymbol (possiblyB X with a X H Received by the editors July 6, 2001. 2000 Mathematics Subject Classiﬁcation. Primary 46L07; Secondary 43A07. Key words and phrases. Fourier algebra, operator space, weakly amenable Banach algebra. This work was supported by an Ontario Graduate Scholarship.

c 2002 American Mathematical Society

3609

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subscript) will always denote a Hilbert space and ( ) will denote the group of unitary operators on with the relativized weak operatorU H topology. Let G be a locallyH compact group. The Fourier and Fourier-Stieltjes algebras, A(G)andB(G), are defined in [10]. We recall that B(G) is the space of matrix coefficients of all continuous unitary representations of G; i.e. the space of functions of the form s π(s)ξ η where π : G ( π) is a continuous homomorphism 7→ h | i →UH and ξ,η π.B(G) is the dual of the enveloping group C*-algebra C∗(G)via ∈H a, π( )ξ η = π (a)ξ η ,whereπ :C∗(G) ( π) is the representation in- hducedh · by| πii. ItcanbeshownthatB(h ∗ | i ∗ G) is a commutative→BH Banach algebra (under pointwise operations) and A(G) is the closed ideal in B(G) generated by compactly supported matrix coefficients. If π is a continuous unitary representation of G,letAπ be the norm closure of span π( )ξ η : ξ,η π in B(G). Then, by [1, 2.2], Aπ∗ ∼= VNπ,whereVNπ is the von{h Neumann· | i algebra∈H } generated by π(G). If λ is the left regular representation 2 of G on L (G), then A(G)=Aλ and we write VN(G)=VNλ. Given two unitary representations π and σ of G,weletπ σ : G ( π σ) denote their direct ⊕ →U H ⊕H sum. If π and σ are disjoint (i.e. there are no subrepresentations π0 of π and σ0 of σ such that π0 is spatially equivalent to σ0), then

(1.1) VNπ σ =VNπ VNσ and Aπ σ =Aπ 1 Aσ ⊕ ⊕∞ ⊕ ⊕ p by [22, 3.8.10] and [1, 3.13], respectively, where p denotes the ` -direct sum for p =1, . ⊕ Our∞ standard reference for operator spaces will be [8]. Given two operator spaces and ,wedenoteby ( , ) the space of completely bounded linear maps X Y CB X Y between and and denote the norm on it by cb.If = , we will denote the BanachX algebraY ( , )by ( ). Dual spacesk·k willX alwaysY be given the standard operator dualCB structureX X ([8,CB Sec.X 3.2], [3]). Given two operator spaces and , their direct product with the canonical product operator space structureX Y will be denoted . The direct sum 1 will be given the operator space X⊕∞ Y X⊕ Y structure it obtains from being imbedded into ( ∗ ∗)∗. The product and sum X ⊕∞ Y are denoted by CM and CL , respectively, in [7]. If is a von Neumann algebra, its predualX⊕ Ywill alwaysX⊕ beY given the operator spaceM structure it inherits M∗ from being imbedded in the dual ∗.Moreover, is then the standard dual of M M ([8, 4.2.2], [3]). In particular, each space Aπ ∼= (VNπ) will be endowed with thisM∗ predual operator space structure. ∗ The projective tensor product of Banach space theory admits an operator space analogue. Given two operator spaces and ,wedenotetheiroperator space projective tensor product by . WeX will notY need the explicit formula for the norm of this tensor product, butX ⊗Y note that it is a completion of the algebraic tensor product . We will useb two important properties of this operator tensor product.X⊗Y First,

(1.2) ∗ = ( , ∗)via x y,T = y,Tx . X ⊗Y ∼ CB X Y h ⊗ i h i See [8, 7.1.5] or [4].b This is analogous to the usual dual formula for the Banach space projective tensor product. Second, if and are von Neumann algebras, then M N

(1.3) = ( ¯ ) M∗⊗N∗ ∼ M⊗N ∗ b

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where ¯ is the von Neumann tensor product of and . See [8, 7.2.4]. In particular,M⊗N since VN(G G) = VN(G) ¯ VN(G) spatially,M viaN the unitary which 2 2 × 2∼ ⊗ implements L (G) L (G) ∼= L (G G), we thus have that A(G) A(G) ∼= A(G G) completely isometrically.⊗ This identity× holds isometrically for⊗ the Banach space× projective tensor product only when G is Abelian. See [20]. b If is an operator space which is also an algebra, it is called a completely A contractive Banach algebra if the multiplication map m0 : extends to a complete contraction m : . The Fourier algebraA⊗A→A A(G) is a completely contractive Banach algebraA since⊗A the → A multiplication map m :A(G) A(G) A(G) ⊗ → corresponds to restriction tob the diagonal subgroup, that is, the map R :A(G × G) A(G)givenbyRu(s)=u(s, s)(s G). Since the adjoint bR∗ :VN(G) → ∈ → VN(G G) is a *-homomorphism, it is a complete contraction and hence R ∼= m is a complete× contraction. If is a completely contractive Banach algebra and is an operator space which A X is also an -module for which the module multiplication maps ml,0 : A A⊗X →X and mr,0 : extend to complete contractions ml : and X⊗A→X A⊗X → X mr : ,then is called a completely contractive -module. A linear map X ⊗A → X X A D : is called a derivation if D(ab)=a D(b)+D(a) b for a,b in .If is A→X · · A X a completelyb contractive -module, then ∗ is, too. is called operator amenable A X A if every completely bounded derivation D : ∗,where is a completely A→X X contractive -module, is inner (i.e. D(a)=a f f a for some f in ∗). is called A · − · X A operator weakly amenable if every completely bounded derivation D : ∗ is inner. If is commutative, this is equivalent to saying that the onlyA→A completely bounded derivationA D : ,where is any symmetric completely contractive -module (i.e. a x = xA→Xa for a in , xXin ), is 0. See [2] for this result in the BanachA algebra case· and· [12] for theA adaptationX to the operator theoretic setting.

2. A theorem of Groenbaek In this section we adapt a theorem of Groenbaek [13] to the completely contractive Banach algebra setting. Let be a completely contractive Banach algebra and an operator space whichA is a completely contractive -module. Let X A 1 = 1 C be the unitization of .Then is a completely contractive 1- A A⊕ A X A module by setting 1 x = x and x 1=x for x in ,where1=0 1in 1. · · X ⊕ A Indeed, 1 = ( ) 1 and the left multiplication map m1 : 1 A ⊗X ∼ A⊗X ⊕ X A ⊗X → X corresponds to the complete contraction m id :( ) 1 given by X A⊗X ⊕ X→X m id ((ab x) yb)=a x + y. Similarly the right multiplication mapb can be X ⊗ ⊕ · extended. See [7]. b The projection π : 1 is a complete quotient map. Then if we let i : 1 be the natural injection,A →A we get that the identity map id factors as A→A A⊗X i id π id ⊗ 1 ⊗ . b A⊗X −→ A ⊗X −→ A ⊗X Hence we have for u in Mn( )(n n-matrices over )that Ab⊗X × b b A⊗X u π id cb (i id)nu u , k k≤kb ⊗ k k ⊗ k≤k bk from which it follows that (i id)nu = u ,so is completely isometrically k ⊗ k k k A⊗X imbedded in 1 . A ⊗X Consider the sequence b b i m 0 1 0 −→ K −→ A ⊗X −→ X −→ b

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where m denotes the left module multiplication map (denoted m1 above), =kerm K and i : 1 is the injection. Note that 1 is a completely contractive -moduleK→A via a⊗X(b x)=(ab) x and (b x) aA=⊗Xb (x a). Let A · ⊗ ⊗ ⊗ · ⊗ · (2.1) b [ ; ]=span a u u a : u band a K A { · − · ∈K ∈A} and note that [ ; ] . The following proposition is [13, Prop. 3.1]. We rework it here to ensureK thatA ⊂K it holds in our context.

Proposition 2.1. If an operator S in ( 1, ∗) = ( 1 )∗ is a derivation, CB A X ∼ A ⊗X then it annihilates [ ; ].Inparticular,0 is the only derivation in ( 1, ∗) if K A CB A X [ ; ]= . b K A K Proof. First note that

= u 1 m(u):u 1 = span b x 1 b x : b 1 and x . K { − ⊗ ∈A ⊗X } { ⊗ − ⊗ · ∈A ∈X} Then for any a and b in 1 and x in ,wehave A b X b x 1 b x, S a a S = x, S(ab) a S(b) b x, S(a) a S(1) h ⊗ − ⊗ · · − · i h − · i−h · − · i = x, S(ab) a S(b) x, S(a) b h − · i−h · i = x, S(ab) (a S(b)+S(a) b) h − · · i from which it follows that S a a S ⊥ for all a. However, S a a S ⊥ for all a, if and only if for all u in· −, · ∈K · − · ∈K K 0= u, S a a S = a u u a, S . h · − · i h · − · i Hence we obtain the ﬁrst statement of the proposition. Suppose that S in ( 1, ∗) is a derivation. For any u in 1 ,wecanwrite CB A X A ⊗X u =(u 1 m(u))+1 m(u), so 1 = (1 ). (Note that u u 1 m(u) − ⊗ ⊗ A ⊗X K⊕ ⊗X 7→ − ⊗ is a completely bounded projection from 1 onto , so the directb sum is one A ⊗X K of operator spaces.) Observe that b1 x, S = x, S1 =0foranyx in ,soif h ⊗ i h i X S = 0, then there must be u in such thatb u, S = 0. This is possible only if [ ;6 ] = . K h i6 K A 6 K

Now we will let = ,andm : 1 1 1 be the multiplication map. Put X A A ⊗A →A (2.2) 1 =kerm, = 1 ( 1 )and 0 = 1 ( ). K K K ∩ A b⊗A K K ∩ A⊗A Since imbeds into 1 , it is easily seen that isthesameasinthe A⊗A A ⊗A b K b notation above. If is commutative, then 1 and hence and 0 are closed A K K K ideals inb 1 1. b With onlyA ⊗A trivial modiﬁcations to his proof, we get the following theorem of Groenbaek [13].b Theorem 2.2. If is a commutative completely contractive Banach algebra, then the following are equivalent:A (i) is operator weakly amenable. (ii) A[ ; ]= . K2A K (iii) = 1. K1 K (iv) 2 = . K2 K 2 (v) = and = ( ) 1. A A K0 A⊗A ·K b

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Furthermore, if has a bounded approximate identity, the above conditions are equivalent to A 2 (iv) = 0. K0 K 3. The Fourier algebra We would now like to apply the above result to the Fourier algebra A(G)ofa locally compact group G. Unless it is specified otherwise, let G be a non-compact locally compact group. If π : G ( π)andσ : G ( σ) are continuous unitary representations of →U H →U H G,letπ σ : G G ( π σ)bethe Kronecker product of π and σ,givenby × × →U H ⊗H π σ(s, t)=π(s) σ(t). We note, for future reference, that VNπ ¯ VNσ =VNπ σ. × 2⊗ ⊗ × Let λ : G (L (G)) be the left regular representation of G and 1 : G T ∼= (C) be the trivial→U representation. → U Proposition 3.1. The representations 1 1, 1 λ, λ 1 and λ λ of G G are all disjoint (i.e. no two of these representations× × have× subrepresentations× × which are unitary equivalent). Proof. Since G is non-compact, λ hasnofixedpoints,forafixedpointwouldgivea constant function in A(G).Hencenoneof1 λ, λ 1orλ λ have any fixed points for all of G G, so they are all disjoint from× 1 1.× 1 λ ×fixes the entire subgroup G e while× neither λ 1norλ λ have any fixed× points× for that subgroup, so 1 λ is×{ disjoint} from λ 1and× λ λ.× Similarly, λ 1andλ λ are disjoint. × × × × × Since λ and 1 are disjoint representations of G, we find that VNλ 1 = VNλ ⊕ ∼ ⊕∞ VN1 = VN(G) C, by (1.1), and hence obtain the completely isometric iden- ⊕∞ tification A(G)1 =A(G) 1 C ∼= Aλ 1. The implied map is clearly an algebra isomorphism. ⊕ ⊕ If u and v are complex functions on G,letu v denote the complex valued function on G G given by u v(s, t)=u(s)v(t). Also,× let 1 denote the constant function on G (as× well as the trivial× representation). Proposition 3.2. We have the following completely isometric identifications: (i) A(G)1 A(G) ∼= A(λ λ) (1 λ) =spanu, 1 v : u A(G G) and v A(G) . ⊗ × ⊕ × { × ∈ × ∈ } (ii) A(G)1b A(G)1 ∼= A(λ λ) (1 λ) (λ 1) (1 1) =span u, 1 v, v 1, 1 1:u A(G G)⊗and v A(G)×. ⊕ × ⊕ × ⊕ × { × × × ∈ × ∈ } These are implementedb by algebra isomorphisms.

Proof. (i) We have A(G)1∗ ∼= VNλ 1 and A(G)∗ ∼= VN(G)=VNλ.Thus,using (1.3), (1.1) and the lemma above, we⊕ obtain

(A(G)1 A(G))∗ = VNλ 1 ¯ VNλ =(VNλ VN1) ¯ VNλ ⊗ ∼ ⊕ ⊗ ⊕∞ ⊗ =(VNλ ¯ VNλ) (VN1 ¯ VNλ) b ⊗ ⊕∞ ⊗ =VNλ λ VN1 λ =VN(λ λ) (1 λ), × ⊕∞ × × ⊕ × where the last space is the dual of A(λ λ) (1 λ). The latter equality in the state- × ⊕ × ment (i) is a straightforward identiﬁcation of A(λ λ) (1 λ) in B(G G). That the identiﬁcations are implemented by algebra isomorphisms× ⊕ × is clear. × The proof of (ii) is similar.

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Theorem 3.3. If G is a locally compact group, then A(G) is operator weakly amenable. Proof. If G is compact, then A(G) is operator amenable by [8, 16.2.3] (or Theorem 4.2, infra) and hence operator weakly amenable. Hence we are left to consider non-compact G.Letπ =(1 1) (1 λ) (λ 1) (λ λ)sothatAπ is the sub- algebra of B(G G) indicated× in⊕ Proposition× ⊕ × 3.2(ii)⊕ above.× In the identiﬁcation × A(G)1 A(G)1 = Aπ, the multiplication map m :A(G)1 A(G)1 A(G)1 cor- ⊗ ∼ ⊗ → responds to the map R :Aπ Aλ 1, which restricts functions to the diagonal → ⊕ subgroupb GD = (s, s):s G = G, i.e. Ru(s)=u(s, s)forb s in G. Letting 1 { ∈ } ∼ K and 0 be as in (2.2), we obtain identiﬁcations K 1 = ker R =span u 1 R(u),u R(u) 1:u Aπ K ∼ { − × − × ∈ } and

0 = ker R A(G G)=I(GD). K ∼ ∩ × Here, I(GD) denotes the ideal in A(G G)withhullGD:sinceGD is a subgroup, × it is a set of spectral synthesis by [14] (or see [17])†. From spectral synthesis we 2 obtain that I(GD) =I(GD)so 2 (3.1) = 0. K0 K Since A(G G)isanidealinAπ,weget × (3.2) (A(G) A(G)) 1 = A(G G) ker R A(G G) ker R =I(GD) = 0. ⊗ ·K ∼ × · ⊂ × ∩ ∼ K On the other hand, again using that GD is a set of spectral synthesis for A(G G), b × (3.3) 0 = I(GD)=A(G G) I(GD ) A(G G) ker R = (A(G) A(G)) 1. K ∼ × · ⊂ × · ∼ ⊗ ·K We thus have, assembling (3.1) and the inclusions (3.2) and (3.3), b 2 (A(G) A(G)) 1 = . ⊗ ·K K0 Hence condition (v) of Theorem 2.2 is satisﬁed, since A(G)2 =A(G) by the Taube- rian Theorem for A(G). b If is a completely contractive Banach algebra and ϕ is a character of ,then A A C can be made into an -module via a z = ϕ(a)z = z a for a in and z in C.Ifϕ A · · A is continuous, it is automatically completely bounded, and hence C is a completely contractive -module. A point derivation is a derivation D : C.If admits continuous non-zeroA point derivations, then it is not (operator)A→ weakly amenable.A Corollary 3.4. A(G) has no continuous point derivations.

In contrast to the case for A(G), if G is Abelian and non-compact, then B(G) ∼= M(G) (the measure algebra of the non-discrete Abelian group G) admits continuous point derivations by [5], and hence is not (operator) weakly amenable. If G is a locallyb compact group containing a closed non-compact Abelianb subgroup H such that the restriction map RH :B(G) B(H) is surjective, then B(G)isnot → (operator) weakly amenable. Note that if RH is surjective, then R∗ :W∗(H) H → W∗(G) (enveloping von Neumann algebras) is a *-homomorphism, and hence a complete contraction. See [11] for further results on the weak amenability of B(G).

†Note added in proof : The spectral synthesis result, in full generality, can be found in [24].

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It has been recently announced by H. G. Dales, F. Ghahramani and A. Ya. Helemskii that the measure algebra M(G) has continuous point derivations whenever G is not discrete. Hence we deduce that M(G) is weakly amenable if and only if G is discrete. The reasonable dual conjecture to this is: B(G) is operator weakly amenable if and only if G is compact.

4. Operator amenability of the Fourier algebra In this section we give a quick proof that A(G) is operator amenable when G is an amenable [SIN]-group. This proof uses elements of both [8, Sec. 16] (i.e. of [23, 3.5]) and [17, 5.3]. We say that G has the small invariant neighbourhood property, or that G is a [SIN]-group, if there is a neighbourhood base of the identity in G such that 1 V sV s− = V for V in and s in G, i.e. each V in is invariant under inner automorphisms. Any compact,V Abelian or discrete groupV is a [SIN]-group. [SIN]- groups are all unimodular. See [21] for more information. We let I(GD) be as in the proof of Theorem 3.3. If we let I0(GD)denotetheset of functions in A(G G) which are compactly supported with support disjoint from × GD,thenI0(GD)=I(GD), since GD is a set of spectral synthesis for A(G G). × Lemma 4.1. If G is a [SIN]-group, then there is a bounded net uV V in { } ∈V B(G G) such that uV (s, s)=1for s in G and uV v 0 for v in I(GD). × → Proof. Let be a neighbourhood base of the identity in G consisting of relatively compact neighbourhoods,V each of which is invariant under inner automorphisms. Let π : G G (L2(G)) be given by π(s, t)f = λ(s)ρ(t)f for f in L2(G), where λ and ρ are× the→U left and right regular representations of G, respectively. Then π is a 1 continuous unitary representation of G G.Forv in ,letuV = π( )χV χV , × V µ(V ) h · | i where χV is the indicator function of V and µ is the Haar measure. Then for (s, t)inG G, × 1 1 1 µ(sV t− V ) u (s, t)= χ (s− rt)χ (r)dµ(r)= ∩ . V µ(V ) V V µ(V ) ZG Clearly uV (s, s)=1fors in G. Hence, since each uV is positive deﬁnite, we have uV = uV (e, e)=1.Ifv I0(GD), then there is V in such that uV v = 0. Hence k k ∈ V uV v 0forv in I(GD), where V runs through decreasing elements of . → V

We remark that if G is discrete, we can use the positive deﬁnite function χGD in place of the net above. Theorem 4.2. If G is an amenable [SIN]-group, then A(G) is operator amenable. By [8, 16.1.4] (i.e. [23]), it suﬃces to show that A(G) has an operator bounded approximate diagonal, i.e. a bounded net wβ β B in A(G G) ∼= A(G) A(G) such that { } ∈ × ⊗

(i) Rwβ β B is a bounded approximate identity for A(G), and b { } ∈ (ii) u 1 wβ wβ 1 u 0 for all u in A(G). k × − × k→ Recall that R :A(G G) A(G)isthemapRu(s)=u(s, s)fors in G.Let × → uα α A be a bounded approximate identity for A(G) (which we can obtain since { } ∈ A G is amenable; see [19]), and uV V be as in the lemma above. Let B = A { } ∈V ×V be the product directed set. For each β =(α, (Vα )α A)inB let 0 0∈ wβ = uV uα uα. α ×

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2 Then Rwβ = u , so (i) is satisﬁed. To see (ii), let v A(G), so uα uα(v 1 1 v) α ∈ × × − × ∈ I(GD)foreachα, and hence

lim u 1 wβ wβ 1 u = lim uVα uα uα(v 1 1 v) β k × − × k β=(α,(Vα )α A) k × × − × k 0 0∈ ( ) = lim lim uV uα uα(v 1 1 v) † α V k × × − × k = lim 0=0. α

The equality of limits at ( ) is from [18, p. 69]. †

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Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 E-mail address: [email protected]

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