COHOMOLOGY AND THE OPERATOR SPACE STRUCTURE OF THE FOURIER ALGEBRA AND ITS

SECOND DUAL

BY BRIAN FORREST AND PETER WOOD

Abstract. Let G be a locally compact group. We introduce the notion of operator weak amenability for a completely contractive Banach Algebra. We then study the potential operator weak amenability of the Fourier algebra A(G) of G and various sub-algebras of its second dual. In particular, we will show that A(G) is operator weakly amenable for a large class of groups which includes all [IN]-groups and all Hermitian groups.

1. Introduction

The classical theory of amenability for a locally compact group has spawned a number of very significant related notions such as Johnson’s amenability for Banach algebras. While the definition of amenability for a Banach algebra involves cohomology rather than the existence of an invariant mean, Johnson was able to establish a connection with the classical notion by showing that the group algebra L^{1}(G) is amenable as Banach algebra if and only if the locally compact group G is amenable [28]. Recently, Ruan [36] has introduced a weaker notion of amenability for completely contractive Banach algebras called operator amenability. In Ruan’s notion of amenability, the natural category is that of operator spaces and the derivations being considered are completely bounded maps. Ruan was able to show that for L^{1}(G) these two notions of amenability are equivalent. It follows that operator amenability of L^{1}(G) again characterizes amenability of the underlying group. For the Fourier algebra A(G) of a locally compact group, the situation is somewhat different. Johnson [27] showed that there exist compact groups for which the Fourier algebra fails to be amenable as a Banach algebra, though amenability of A(G) does indeed imply amenability for the group G. In fact, at this time the only groups for which the Fourier algebra is known to be amenable are those groups with an abelian subgroup of finite index. In contrast, Ruan proved that the completely contractive Banach algebra A(G) is operator amenable if and only if G is amenable as a locally compact group [36], thereby establishing the natural analog of Johnson’s fundamental theorem on L^{1}(G). Ruan’s theorem suggests that when considering problems involving cohomology of the Fourier algebra, the category of operator spaces is perhaps the most appropriate. In this paper, we shall consider questions concerning the completely bounded cohomology of the Fourier algebra and its second dual. In particular, we will introduce the notion of operator weak amenability for such algebras.

1991 Mathematics subject classification 43A85, 43A07, 43A15 Key words and phrases: Locally compact group, Fourier algebra, completely contractive Banach algebra, operator weak amenability Date: August 4, 1998.

2. Preliminaries

Let G be a locally compact group. We let A(G) denote the subspace of C_{0}(G) consisting of functions of the form

u(x) = ∑_{i=1}^{∞} { \atop (f_{i} ∗ g̃_{i})}(x)

where f_{i}, g_{i} ∈ L^{2}(G), ∑_{i=1}^{∞} ∥f_{i}∥_{2}∥g_{i}∥_{2} < ∞, f̂(x) = f(x^{−1}) and f̃(x) = \overline{f(x^{−1})} . A(G) is a commutative Banach

algebra with respect to pointwise operations and the norm

∥u∥_{A(G)} = inf \{∑_{i=1}^{∞} ∥f_{i}∥_{2}∥g_{i}∥_{2} | u = ∑_{i=1}^{∞}({ \atop f_{i} ∗ g̃_{i}})\} ,

called the Fourier algebra of G [14]. If G is abelian, then A(G) is simply the Fourier transform of the group algebra of G. A(G) was introduced for non-commutative groups by Eymard in [14]. If F is a proper closed subset of G, then the set

I(F) = {u ∈ A(G) |u(x) = 0 for all x ∈ F }

is a closed ideal in A(G). For any closed ideal I of A(G), we let

Z(I) = {x ∈ G |u(x) = 0 for every u ∈ I }.

Since each u ∈ A(G) is continuous, it follows that Z(I) is a closed subset of G.

Let VN(G) denote the closure of L^{1}(G) , considered as an algebra of convolutions operators on L^{2}(G), with respect to the weak operator topology on B(L^{2}(G)). The von Neumann algebra

VN(G)

can be identified with the Banach space dual of A(G) [14]. Let ϕ ∈ VN(G). Let u, v ∈ A(G). We define u ⊙ ϕ ∈ VN(G) by ⟨u ⊙ ϕ, v⟩ = ⟨ϕ, uv⟩. If Γ ∈ A(G)^{∗∗}. Γ ⊙ ϕ will denote the functional

on A(G) defined by ⟨Γ ⊙ ϕ, u⟩ = ⟨Γ, u ⊙ ϕ⟩. A subset X of VN(G) is said to be topologically invariant if u ⊙ ϕ ∈ X for each u ∈ A(G) and ϕ ∈ X. If X is a topologically

invariant subspace of VN(G), then X is said to be topologically introverted if for each Γ ∈ X^{∗} and each ϕ ∈ X, we have Γ ⊙ ϕ ∈ X. If X is topologically introverted, then X^{∗} is a Banach algebra with respect to the product Φ ⊙ Γ where ⟨Φ ⊙ Γ, ϕ⟩ = ⟨Φ, Γ ⊙ ϕ⟩ for ϕ ∈ X. In case X = VN(G), the product on VN(G)^{∗} = A(G)^{∗∗} is simply one of the Arens products introduced

in [2]. Moreover, if X is topologically introverted and Γ̃ and Φ̃ are extensions to VN(G) of Γ and Φ respectively, then Γ̃ ⊙ Φ̃ extends

Γ ⊙ Φ. For

this reason, we may view X^{∗} as a quotient algebra of A(G)^{∗∗}. We refer the reader to [29] for further details.

Let λ : G ↦ B(L^{2}(G)) denote the left regular representation of G. Then λ(g)h(x) = h(g^{−1}x) for each g ∈ G, h ∈ L^{2}(G). It is well known that λ lifts to L^{1}(G) with λ(f)(h) = f ∗ h for f ∈ L^{1}(G) and h ∈ L^{2}(G). Let C_{λ}^{∗}(G) denote the C^{∗}-subalgebra of VN(G) generated by {λ(f) : f ∈ L^{1}(G)} and C_{δ}^{∗}(G) be the C^{∗}-algebra generated by {λ(g) : g ∈ G}. As usual C^{∗}(G) will denote the enveloping C^{∗}-algebra of L^{1}(G). C^{∗}(G) is called the group C^{∗}-algebra of G, while C_{λ}^{∗}(G) is called the reduced C^{∗}-algebra of G. C^{∗}(G) and C_{λ}^{∗}(G) are known to be *-isomorphic precisely when G is amenable. In general C_{λ}^{∗}(G) is a quotient of C^{∗}(G). The Banach space dual of C^{∗}(G) is the space B(G) of all linear combinations of continuous positive definite functions on G [14]. Eymard showed that with respect to the dual norm and pointwise operations B(G) is a commutative Banach algebra called the

Fourier-Stieltjes algebra of G. The dual of C_{λ}^{∗}(G) can be identified

with a closed ideal of B(G) which we denote B_{λ}(G). B_{λ}(G) is the weak-* closure of A(G), and moreover A(G) is

also a closed ideal of both B_{λ}(G) and B(G) [14]. The dual of C_{δ}^{∗}(G) is denoted by B_{δ}(G_{d}). It is a weak-* closed subalgebra of B(G_{d}) where G_{d} denotes the group G with the discrete topology. In general, B_{δ}(G_{d}) contains A(G_{d}) and hence B_{λ}(G_{d}). Moreover B_{δ}(G_{d}) = B(G_{d}) if and only of G is amenable.[4] Let UCB(Ĝ) denote the closed linear span of A(G)⊙VN(G). Then

UCB(Ĝ) is a topologically introverted C^{∗}-subalgebra of VN(G) (See [30]). Given ϕ ∈ VN(G), let Ob(ϕ) = {u ⊙ ϕ | u ∈ A(G), ∥u∥ ≤ 1}. ϕ is

(weakly) almost periodic if Ob(ϕ) is relatively (weakly)

compact. Let W AP (Ĝ) and AP (Ĝ) denote the spaces of weakly almost periodic

and almost periodic functionals on A(G) respectively. AP (Ĝ) and W AP (Ĝ)

are also topologically introverted subspaces of VN(G). Moreover, each

of the spaces UCB(Ĝ), W AP (Ĝ), and AP (Ĝ) contain C_{δ}^{∗}(G) as a closed

subspace, while UCB(Ĝ) and W AP (Ĝ) contain C_{λ}^{∗}(G). When G is abelian, UCB(Ĝ) is the Fourier transform of the C^{∗}-

algebra of uniformly continuous functions on Ĝ. W AP (Ĝ) and AP (Ĝ) are the Fourier transforms of the C^{∗}-algebra of weakly almost periodic functions and almost periodic functions of G respectively. In general it is not known if AP (Ĝ) or W AP (Ĝ) are C^{∗}-algebras. Fi-

nally if G is compact, then UCB(Ĝ) = VN(G) and if G is discrete, C_{δ}^{∗}(G) = C_{λ}^{∗}(G) = UCB(Ĝ). We refer the reader to [23] and [30] for these

and other properties of the above spaces. An operator space is a vector space V together with a family ∥ ∥_{n} of Banach space norms on 𝕄_{n}(V), the space of n x n matrices with entries in V such that

(i) ∥[ \begin{matrix}A & 0 \\ 0 & B \end{matrix} ]∥_{n+m} = max \{∥A∥_{n} , ∥B∥_{m}\}

for each A ∈ 𝕄_{n}(V), B ∈ 𝕄_{m}(V), and

(ii)∥([a_{ij}])A([b_{ij}])∥_{n} ≤ ∥[a_{ij}]∥∥A∥_{n}∥[b_{ij}]∥

for each [a_{ij}], [b_{ij}] ∈ 𝕄_{n}(ℂ) and A ∈ 𝕄_{n}(V) Let X and Y be operator spaces and let Γ : X ↦ Y. For each n ∈ ℕ define Γ^{(n)} : 𝕄_{n}(X) ↦ 𝕄_{n}(Y) by Γ^{(n)}[x_{ij}] = [Γx_{ij}] The map

Γ is

said to be completely bounded if sup{∥Γ^{(n)}∥} < ∞. In this case we

let ∥Γ∥_{cb} = sup{∥Γ^{(n)}∥} . We say that Γ is a complete isometry if each Γ^{(n)} is an isometry and Γ is a complete contraction if each

Γ^{(n)} is a contraction. It is a fundamental result in the theory that every operator space is completely isometrically isomorphic to a norm closed subspace S of B(H), the algebra of bounded operators

on the Hilbert space H, where the operator space structure on S is the structure inherited from B(H) ([12]). Given a general Banach space X, there there exists maximal and minimal operator space structures, denoted MAX(X) and MIN(X) respectively, such

that for any operator space Y, any bounded linear maps S, T

S : MAX(X) ↦ Y and T : Y ↦ MIN(X)

are automatically completely bounded with ∥T ∥_{cb} = ∥T ∥ and ∥S∥_{cb} = ∥S∥ (see [8], and [5]). If we let CB(X, Y) denote the space of all completely bounded maps from X to Y, then CB(X, Y) has a natural operator structure which can be obtained by identifying 𝕄_{n}(CB(X, Y)) with CB(X, 𝕄_{n}(Y)). It is important to note that continuous linear functionals are completely bounded. In fact, since we can identify X^{∗} with CB(X, ℂ), X^{∗} is also an operator space called the standard dual

of X (see [6]). Given two operator spaces, there is an operator space analogue of

the projective tensor product which we denote X ˆ⊗Y. In particular, there is a complete

isometry

between (X ˆ⊗Y)^{∗} and CB(X, Y^{∗}). An associative algebra A which is also an operator space and is such that the multiplication

m : A ˆ⊗A ↦ A

is completely contractive is called a completely contractive Banach algebra. If A is a completely contractive Banach algebra

and J is a closed ideal of A, it is easy to see that both J and A/J are completely

contractive Banach algebras with respect to the operator space structure each inherits from A. We note that if X is the predual of a von Neumann algebra, A it inherits a natural operator space structure as follows: for [x_{ij}] ∈ 𝕄_{n}(X) we set

∥[x_{ij}]∥_{n} = sup{∥[f_{kl}(x_{ij})]∥_{nm} : [f_{kl}] ∈ 𝕄_{m}(A), ∥[f_{kl}]∥_{m} ≤ 1}

Thus both the Fourier and Fourier-Stieltjes algebras can be given natural operator structures by virtue of their being preduals of von Neumann algebras. In each case, this operator space structure results in a completely contractive Banach algebra (see [36] and [6]). Let A be a Banach algebra. A Banach A-module is a two sided A-module X that is itself a Banach space and for which

∥ax∥_{X} ≤ ∥a∥_{A}∥x∥_{X} and ∥xa∥ ≤ ∥a∥_{A}∥x∥_{X}

for each a ∈ A and each x ∈ X. If in addition, A is a completely contractive Banach algebra and X is a operator space such that the A-module operations (a, x) ↦ ax and (a, x) ↦ xa are complete bounded from A ˆ⊗X onto X, then X is called an operator A-

module.

If X is a Banach A-module, then X^{∗} becomes a Banach A-module with respect to the actions

(aϕ)(x) = ϕ(xa) and (ϕa)(x) = ϕ(ax).

We call X^{∗} a dual Banach A-module. Moreover, if A is a completely

contractive Banach algebra and X is an operator A-module, then with respect to the dual operator space structure on X^{∗} and with respect to the actions indicated above, X^{∗} is also an operator A-module (see [36]).

In general, we will call an A-module X symmetric if ax = xa for every a ∈ A and x ∈ X. In particular, A^{∗} is symmetric if and only if A is commutative. A linear map D : A ↦ X is called a derivation if

D(ab) = aD(b) + D(a)b ∀a, b ∈ A.

D is said to be inner if there exists an x ∈ X such that D(a) = xa − ax. Let ϕ be a continuous multiplicative functional on A. A point derivation of A at ϕ is a linear functional d : A ↦ ℂ such that d(ab) = ϕ(a)d(b) + ϕ(b)d(a). We call a Banach algebra A amenable if every bounded derivation from A into a dual A-module X^{∗} is inner. A is weakly

amenable if every bounded derivation from A into A^{∗} is inner. The theory of amenable and weakly amenable Banach algebras is extremely rich. The reader is referred to [28] and [10] for the basic properties of amenable algebras and to [9] and [25] for properties of weakly amenable algebras. Following [36], we will call a completely contractive Banach algebra A operator amenable if every completely bounded deriva-

tion

from A into a dual operator A-module is inner. We recall that for any locally

compact group A(G) is operator amenable if and only if G is amenable.

3. Operator Weak Amenability

One of the principal goals of this paper is to show that when considering problems of cohomology for A(G), the operator space setting is the most appropriate. Ruan’s result in [36] is certainly strong evidence to support this point of view. In this section, we will introduce the notion of operator weak amenability. We will then extend many of the fundamental results

from the Banach algebra setting to the operator space setting. These will be used in section 5 to study operator weak amenability for A(G). The following is adapted from the definition of weak amenability for

Banach algebras:

Definition:

We say that a completely contractive Banach Algebra A is operator weakly amenable if every completely bounded deriva-

tion D from A into A^{∗} is inner.

We begin with a few simple observations which are well known for weak amenability (see [9] and [25] for analogous results). While it is not surprising that the proofs in the category of operator spaces

are often essentially the same as in the category of Banach spaces, we caution that some care must

be used to

verify that the mappings involved are completely bounded when required.

Indeed, not all of the standard results on amenability and weak amenability hold for the category of operator spaces. We will therefore include the proofs for completeness.

Lemma 3.1:

Let A be a completely contractive Banach algebra such that A^{2} is not dense in A. Then A is not operator weakly amenable.

Proof:

Let ϕ ∈ A^{∗} be nonzero with ϕ(A^{2}) = 0. Then D(a) = ϕ(a)ϕ is a derivation from A into A^{∗}. Moreover, D is completely bounded since ϕ ∈ A^{∗} . Assume that D(a) = ζa − aζ for some ζ ∈ A^{∗}. Let b ∈ A be such that ϕ(b) ≠ 0. Then D(b)(b) = (ζb − bζ)(b) = 0, where as D(b)(b) = ϕ(b)ϕ(b) ≠ 0. It follows that D cannot be inner and hence that A is not weakly amenable.

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Proposition 3.2:

Let A be a commutative completely contractive Banach algebra. Then A is operator weakly amenable if and only if every completely bounded derivation from A into a symmetric operator A-module X

is trivial.

Proof:

If A is commutative, A^{∗} is symmetric. Therefore the ”if” direction is trivial.

Assume that A is operator weakly amenable, and that X is a symmetric operator A-module. Let D : A ↦ X is a non-zero completely bounded derivation. We may

by

Lemma 3.1 assume that A^{2} is dense in A. Hence there exists a ∈ A with D(a^{2}) ≠ 0. Choose ϕ ∈ X^{∗} such that ϕ(D(a^{2})) ≠ 0. For each x ∈ X, define R_{x} ∈ A^{∗} by R_{x}(a) = ϕ(ax). The map R : X ↦ A^{∗} with R(x) = R_{x} is

completely bounded. To see this observe that

∥R^{(n)}∥ = sup {∥R^{(n)}([x_{ij}])∥ : ∥[x_{ij}]∥ ≤ 1, [x_{ij}] ∈ 𝕄_{n}(X)}
= sup{∥[R_{x_{ij}}]∥ : ∥[x_{ij}]∥ ≤ 1, [x_{ij}] ∈ 𝕄_{n}(X)}
= sup{∥[R_{x_{ij}}(a)]∥ : ∥[x_{ij}]∥ ≤ 1, [x_{ij}] ∈ 𝕄_{n}(X) ∥a∥ ≤ 1}
= sup{∥ϕ^{(n)}([ax_{ij}])∥ : ∥[x_{ij}]∥ ≤ 1, [x_{ij}] ∈ 𝕄_{n}(X) ∥a∥ ≤ 1}
≤ ∥ϕ∥_{cb} = ∥ϕ∥.

It is now straight forward to see that the map D̃ : A ↦ A^{∗} given by D̃(a) = R(D(a)) is a completely

bounded derivation. Finally,

D̃(a)(a) = R_{D(a)}(a) = ϕ(aD(a)) = 1/2 ϕ(D(a^{2})) ≠ 0.

Since A is operator weakly

amenable, this is impossible.

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Proposition 3.3:

Let A and B be commutative completely contractive Banach algebras. Let ϕ : A ↦ B be a completely bounded homomorphism with dense range. If A is operator weakly amenable, then so is B.

Proof:

Suppose that B is not operator weakly amenable. We may assume that ∥ϕ∥_{cb} = 1. Observe that B^{∗} becomes a symmetric operator A-module via the action:

(a, Γ) ↦ ϕ(a) · Γ = Γ · ϕ(a)

for each a ∈ A, Γ ∈ B^{∗}. Let D : B ↦ B^{∗} be a nonzero completely bounded derivation. Then D̃(a) = D(ϕ(a)) is a completely bounded derivation from A into B^{∗}. Moreover D̃ is non-zero since ϕ has dense range. Now apply Proposition 3.2 to conclude that A is not weakly amenable.

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The analogue of Proposition 3.3 may fail for Banach algebras without the assumption of commutativity (see [26]). We can show

that this is also true in our setting. Assume that we have a Banach algebra A which is weakly amenable and a bounded homomorphism ϕ onto a dense sub-algebra of B where B is not weakly amenable. Let D : B ↦ B^{∗} be bounded but not inner. If we provide both A and B with the MAX operator space structure, then ϕ is completely bounded. Moreover, the bounded derivation D : B ↦ B^{∗} is completely bounded but not inner. This shows that B is not operator weakly amenable, while A, being weakly amenable, will be operator weakly amenable. We conclude that Proposition 3.3 would fail without the assumption of commutativity.

Proposition 3.4:

Let A be a completely contractive Banach algebra. If A has a nonzero point derivation, then A is not operator weakly amenable.

Proof:

Let d : A ↦ ℂ be a nonzero point derivation at at ϕ. We may assume that A^{2} is dense by Lemma 3.1. In particular, ϕ is nonzero. Let D(a) = d(a)ϕ. Then it is routine to verify that D is a nonzero derivation of A into A^{∗}. Again, since d is completely bounded, so is D. Assume that D is inner with D(a) = ζa − aζ. Then d(a)ϕ(a) = D(a)(a) = (ζa − aζ)(a) = 0 for every a ∈ A. It follows that d(a^{2}) = 2 d(a)ϕ(a) = 0 for all a. Finally, since d is continuous and A^{2} is dense, we have d(b) = 0 for all b ∈ A which is impossible.

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Let A be a commutative completely contractive Banach algebra. Let X be a symmetric operator A-module. Let Hom_{cb}(A, X) denote

the space of completely bounded module maps from A into X. With respect to its natural operator space structure inherited from CB(A, X), and with respect to the action (bΓ)(a) = (Γb)(a) = Γ(ab) for each Γ ∈ Hom_{cb}(A, X), a, b ∈ A, Hom_{cb}(A, X) becomes a symmetric operator A-module. Moreover, just as is indicated in [25], if I is an ideal in the commutative completely contractive algebra A, then the above action with a ∈ A makes Hom_{cb}(I, X) into a symmetric operator A-module in such a way that the restriction to I is the natural action of I on Hom_{cb}(I, X). Again following [25], we define the map j : X ↦ Hom_{cb}(A, X) by j(x)(a) = ax. A routine calculation

similar to that in the proof of Proposition 3.2 shows that j is completely bounded A-module homomorphism. We are now able to establish the following useful analog of [25, Corollary 1.3] with essentially the same proof:

Proposition 3.5: Let A be an operator weakly amenable commutative completely contractive Banach algebra. Let I be a closed ideal in A. Then I is operator weakly amenable if and only if \overline{I^{2}} = I.

Proof:

The only if direction is clear from Lemma 3.1. Hence we shall assume that \overline{I^{2}} = I. Let D : I ↦ X be a completely bounded derivation into a symmetric operator I-module X. Let j : X ↦ Hom_{cb}(I, X) be as above. Then since j is a completely

bounded module map j^{∘}D is a completely bounded derivation. Define the bilinear map D̃ : I×A ↦ Hom_{cb}(I, X) by

D̃(m, a) = D(ma) − aD(m).

Let m ∈ I^{2} . By [25, Theorem 1.1], the map a ↦ D̃(m, a) is a derivation of A into Hom_{cb}(I, X). Moreover, it is clear from its definition that this map is completely bounded and thus since A is operator weakly amenable it is identically 0. Again from [25, Theorem 1.1 (i)], we get that I^{2}D(I) ⊂ ker j and hence that I^{3}D(I) = 0. However, \overline{I^{2}} = I implies that D = 0. Hence it follows from Proposition 3.2 that I is weakly amenable.

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4. Automatic Complete Boundedness for Linear Maps on A(G) and B(G)

In [36], Ruan showed that amenability for the Banach algebra L^{1}(G) is

equivalent to operator amenability when L^{1}(G) is given the operator space structure that it inherits as the predual of the von Neumann algebra L^{∞}(G). This follows from the fact that L^{1}(G) has the MAX operator space structure and as such every bounded map from L^{1}(G) into an operator space X in completely bounded. Moreover, any Banach L^{1}(G)- module can be given the MAX operator space structure which will make it into

an operator L^{1}(G)-module. As such any bounded derivation into a dual L^{1}(G)-module can be viewed as a completely bounded derivation into a dual operator L^{1}(G)-module. When considering the Fourier algebra of G the situation can be quite different. In fact, we have the following:

Proposition 4.1 Let G be a locally compact group. Then A(G) = A(G)_{MAX} if and only if G is abelian.

Proof:

The standard dual of A(G) is the von Neumann algebra VN(G) and A(G) = A(G)_{MAX} if and only if VN(G) = VN(G)_{MIN} (see

[6]). Since VN(G)

is a unital operator algebra, VN(G) = VN(G)_{MIN} if and only if VN(G) is abelian [7]. However VN(G) is abelian if and only if G is abelian.

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Clearly if G is abelian, then since A(G) = A(G)_{MAX} and A(G) is completely isometrically isomorphic with L^{1}(Ĝ), there is little to be gained by looking at A(G) as a completely contractive Banach algebra rather than simply as a Banach algebra. We will

now show however, that if G does not contain

an abelian subgroup of finite index, then the nature of A(G) when viewed

as a completely contractive Banach algebra is fundamentally different from that of A(G) viewed as a Banach algebra. This we believe explains why it is necessary to retain the operator space structure when studying the cohomology of the Fourier algebra. Before providing the main result of this section, we require a few useful observations.

Proposition 4.2:

Let H be a closed subgroup of G. Then A(G)/I(H) is completely isometrically isomorphic with A(H).

Proof:

Let ũ ∈ A(G)/I(H). Define Γ : A(G)/I(H) ↦ A(H) by Γ(ũ) = v |_{H} where v is chosen so that Q(v) = ũ and Q : A(G) ↦

A(G)/I(H) is the quotient map. It is known that Γ is an isometric isomorphism of A(G)/I(H) onto A(H) [20]. To see that Γ is a complete isometry observe that I(H)^{⊥} = V N_{H}(G) = {T ∈ VN(G) : supp(T) ⊂ H}. Also V N_{H}(G) is a von Neumann subalgebra of VN(G) which is *-isomorphic with VN(H) [14]. It follows from [6] that (A(G)/I(H))^{∗} is completely isometrically isomorphic to V N_{H}(G) and hence to VN(H). Thus (A(G)/I(H)) is completely isometrically isomorphic

to

A(H).

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Proposition 4.3:

Let H be an open subgroup. Then 1_{H}A(G) is completely isometrically isomorphic with A(H).

Proof:

Let u ∈ A(H). Let u^{∘}∈ A(G) be such that u^{∘} (x) = u(x) if x ∈ H and u^{∘} (x) = 0 otherwise. It is well known that Γ : A(H) ↦

A(G) defined by Γ(u) =u^{∘} is an isometric isomorphism of A(H) into 1_{H}A(G) [14]. Let [u_{ij}] ∈ 𝕄_{n}(A(H)) with ∥[u_{ij}]∥_{n} = 1. It follows from

Proposition 4.2 that ∥[u_{ij}^{∘}˜]∥_{n} = 1. Let ϵ > 0. We can find [v_{ij}] ∈ 𝕄_{n}(A(G)) such that [v_{ij}˜] = [u_{ij}^{∘} ˜] and ∥v_{ij}]∥ ≤ 1 + ϵ. Now

∥Γ^{(n)}([u_{ij}])∥ = ∥[u_{ij}^{∘}]∥_{n} = ∥P ([v_{ij}])∥_{n}

where P (v) = 1_{H}v. However 1_{H} ∈ B(G) and ∥1_{H}∥ = 1. It follows that

∥P ∥_{cb} = 1. Hence

∥Γ^{(n)}([u_{ij}])∥_{n} ≤ 1 + ϵ

Therefore we can conclude that ∥Γ∥_{cb} ≤ 1. To complete the proof, observe that Γ^{−1} : 1_{H}A(G) ↦ A(H) is simply the restriction of the quotient map Q : A(G) ↦ A(G)/I(H) composed with the complete isometry of Proposition 4.2. It follows

that

∥Γ^{−1}∥_{cb} =1 and hence that Γ is a complete isometry.

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Lemma 4.4:

Let t ∈ G. The map L_{t} : A(G) ↦ A(G) defined by L_{t}u(x) = u(tx) is a complete isometry of A(G). Proof:

Let L_{t}^{∗} : VN(G) ↦ VN(G). Then L_{t}^{∗}(T) = δ_{t} ∗ T where

∗ is the product in VN(G). It follows that L_{t}^{∗} and hence L_{t} is completely bounded with ∥L_{t}∥_{cb} = 1. Clearly we also have that ∥L_{t}^{−1}∥_{cb} = ∥L_{t^{−1}}∥_{cb} = 1, and hence L_{t} is a complete isometry.

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Theorem 4.5 The following are equivalent

(i) Every bounded map from A(G) into any operator space is completely bounded. (ii) Every bounded map from A(G) into VN(G) is completely bounded. (iii) G has an abelian subgroup of finite index.

Proof:

(i) ⇒ (ii) trivial (ii) ⇒ (iii): Recall that

CB(A(G), VN(G)) = (A(G) ˆ⊗A(G))^{∗}

and

B(A(G), VN(G)) = (A(G) ⊗_{γ} A(G))^{∗}

where ˆ⊗ and ⊗_{γ} denote the operator space tensor product and Banach space tensor product respectively. Since from (ii) we have that B(A(G), VN(G)) = CB(A(G), VN(G)), we conclude

from the open mapping theorem that there exists some number N ∈ ℕ such that ∥Φ∥_{cb} ≤ N∥Φ∥ for each Φ ∈ B(A(G), VN(G)) = CB(A(G), VN(G)). Let u ∈ A(G) ⊗ A(G). Then

∥u∥_{A(G) ˆ⊗A(G)} = sup{Φ(u) : Φ ∈ CB(A(G), VN(G)), ∥Φ∥_{cb} ≤ 1}
≥ sup{Φ(u) : Φ ∈ B(A(G), VN(G)), ∥Φ∥ ≤ 1/N}
= 1/N∥u∥_{A(G)⊗_{γ}A(G)}

This shows that the natural injection i of (A(G) ⊗ A(G), ∥ · ∥_{γ}) into (A(G) ⊗ A(G), ∥ · ∥_{∧}) extends to an isomorphism of A(G) ⊗_{γ} A(G) onto A(G) ˆ⊗A(G). To complete the proof, observe that the canonical injection of A(G) ˆ⊗A(G) into A(G × G) is known to be surjective ([13]). It follows that the same will be true of the canonical injection of A(G) ⊗_{γ} A(G) into A(G × G). Using a result of Losert [34], we conclude that G has an abelian subgroup of finite index.

(iii) ⇒ (i): Assume that G has an abelian subgroup H of finite index in G. Let {x_{i}H} be a complete set of cosets of G. Then 1_{H} is a completely bounded projection of A(G) onto a sub-algebra of A(G) which is completely isometrically isomorphic to A(H). Since H is abelian, A(H) has the MAX operator structure. It follows that every bounded linear map from A(H) into any operator space is completely bounded. Thus if T is a bounded linear map from A(G) to any operator space, the above argument and Lemma 4.4 shows that T^{∘}1_{x_{i}H} is completely bounded. However, since T = ∑_{i=1}^{n} T^{∘}1_{x_{i}H} it follows that T is also completely bounded.

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Corollary 4.6 Let G be a locally compact group. Then every bounded map from B(G) into an arbitrary operator space X is completely bounded if and only if G has an abelian subgroup of finite index.

Proof:

If H is abelian, we note that B(H) is again the predual of

an abelian von Neumann algebra and is hence a MAX operator space.

We can proceed as before to show that 1_{H}B(G) is c.b. isomorphic to B(H) when H is an open subgroup of G. Thus when G has an abelian subgroup of finite index, that every bounded map from B(G) into an arbitrary operator space is completely bounded follows in the same manner as for A(G) above. To prove the converse, first note that A(G) is a complemented ideal in B(G) and the pro-

jection P is induced by the central projection in W^{∗}(G) corresponding to the left regular representation. As such P is completely bounded. If G does not have an abelian subgroup of finite index, then by Theorem 4.5 there is an operator space X and a linear map Γ : A(G) ↦ X which is bounded but not completely bounded. Then Γ^{∘}P is the desired map.

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It is worthwhile to shed further light on the previous two results. Let π be a continuous unitary representation of G. Let A_{π} denote

the closed subspace of B(G) generated by the coefficient functions of π. Then A_{π} inherits an operator space structure from B(G). Moreover, the central projection P_{π} in the von Neumann algebra B(G)^{∗} = W^{∗}(G) associated to π is such that A_{π} = P_{π}B(G) and A_{π}^{∗} = V N_{π} = P_{π}W^{∗}(G) (see [1]). If π is irreducible, then V N_{π} = B(H_{π}) and hence A_{π} = T C(H_{π}) the trace class operators on the Hilbert space H_{π}. Moreover, since the operator space structure on B(G) is the standard predual operator space structure, the induced structure on A_{π} agrees with the standard operator space structure T C(H_{π}) inherits from B(H_{π}).

Let G be a locally compact group with an infinite dimensional irreducible representation π. ¿From the remark above, we see that A_{π} = T C(H_{π}). Let Γ : B(H_{π}) ↦ B(H_{π}) be the transpose map. Since Γ is weak-* to weak-* continuous and has norm 1 , the preadjoint map Γ_{∗} : T C(H_{π}) ↦ T C(H_{π}) also has norm 1. However, since it is well known that Γ is not completely bounded, neither is Γ_{∗} which we view as a bounded map from A_{π} onto itself. It follows that the map Γ^{∘}P_{π} is a bounded map from B(G) into B(G) which is not completely bounded. Now let G be a locally compact group which has irreducible finite dimensional representations of arbitrarily large degree. Then we can construct in a similar manner as above a bounded map from B(G) into B(G) which is not completely bounded. Let {π_{n}} be a sequence of irreducible finite dimensional representations with dim π_{n} > dim π_{m} whenever n > m. Let π = ∑ ⊕π_{n} be the direct sum of the π_{n}’s. Then by [1], the space A_{π} = ∑ ⊕_{L^{1}}A_{π_{n}} = ∑ ⊕_{L^{1}}T C(H_{π_{n}}). It follows from

[6] that as an operator space A_{π} is the operator predual of V N_{π} = ∑ ⊕_{L^{∞}}V N_{π_{n}} = ∑ ⊕_{L^{∞}}𝕄_{dim π_{n}}. If we define Γ : A_{π} ↦ A_{π}

by

Γ = ∑ ⊕(Γ_{n})_{∗} where Γ_{n} : 𝕄_{dim π_{n}} ↦ 𝕄_{dim π_{n}} is the transpose map and (Γ_{n})_{∗} is its preadjoint,

then once again Γ is a map of norm one. However, since ∥Γ_{n}∥_{cb} ≥ dim π_{n}, Γ is not completely bounded.

If we again let P_{π} denote the central projection in W (G) associated to π, then the Γ^{∘}P_{π} is once more a bounded map from B(G) onto A_{π} which is not completely bounded. Finally, since G has finite dimensional irreducible representations of bounded degree if and only if G has an abelian subgroup of finite

index,

we have established an alternate proof of Corollary 4.6. We have also

provided a proof for the following proposition:

Proposition 4.7 Let π be a continuous unitary representation of G. If either π contains an infinite dimensional irreducible subrepresentation or if π contains finite dimensional subrepresentations of arbitrarily large degree, then there exists a contractive linear map Γ : A_{π} ↦ A_{π}

which is not completely bounded.

A locally compact group is said to belong to the class [AR] if the left regular representation λ is completely reducible. It is well known that for G ∈ [AR], A(G) has the Radon-Nikodym Property (in fact this characterizes the groups in [AR] [37]) and hence that the unit ball of A(G) is the closed convex hull of its extreme points. [AR] contains not only all compact groups but also a variety of noncompact groups (see [3]). The ax + b group is in [AR] as its left regular representation is the direct sum of two infinite dimensional irreducible representations.

Proposition 4.8 Let G be either a noncompact [AR] group or a compact group which does not contain an open abelian subgroup. Then there exists an isometry from A(G) onto A(G) which is not completely bounded.

Proof:

In either case, the map Γ constructed as in the remark preceding Proposition 4.7 is an isometry of A(G) onto itself which is not completely bounded.

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We can make a case for the significance of the results of this section for our study if we recall what is currently known concerning the bounded cohomology of A(G). At present, the only

known

groups for which A(G) is amenable are those which contain an abelian

subgroup of finite index. That the Fourier algebra of such a group is amenable was originally proved for compact groups by Johnson

[27].

The same result was later established for general locally compact groups by Lau, Loy and Willis in [32] and independently by the first author in [15]. However, Johnson showed that for a finite group, the minimal norm for a virtual diagonal for A(G) is given by

\frac{Σ_{π∈Ĝ}(dim π)^{3}}{Σ_{π∈Ĝ}(dim π)^{2}} .

This provided the first concrete evidence to suggest that in the presence

of either an infinite dimensional irreducible representation or of finite dimensional representations of arbitrarily large degree, (precisely the setting of Proposition 4.7) A(G) would fail to be amenable. In [32], Johnson’s work was refined considerably and the case for the link between amenability of A(G)

and ”approximate commutativity” of G was strengthened.

Finally, note that the fact that A(G) ˆ⊗A(G), the operator space tensor product of A(G) with itself, is completely isometrically isomorphic to A(G × G) is at the heart of Ruan’s theorem that A(G) is operator amenable precisely when G is amenable. Moreover, the failure of the corresponding result for the Banach al-

gebra projective tensor product in the absence of an abelian subgroup of finite index (as demonstrated by Losert [34]) is the essence of our proof of Theorem 4.5. Indeed, one might view Theorem 4.5 as a

restatement within our current context of Losert’s theorem. Once again, we are led to conclude that Banach algebra amenability for A(G) is a com-

mutative phenomenon, whereas the category of operator spaces with its richer structure provides the appropriate setting for studying the cohomol-

ogy of the Fourier algebra of a typical locally compact group.

5. Operator Weak Amenability of the Fourier Algebra

In [15], the first author established a link between weak amenability of A(G) and commutativity of the connected component of G. For example, it was shown that if G is a [SIN] group, then A(G/K) is weakly amenable for each compact normal subgroup K of G if and only if G has an abelian connected component. Moreover, Johnson [27] has shown the existence of compact groups

for

which A(G) is not weakly amenable. In contrast, in the this section, we will show that operator weak

amenability of A(G) holds for a large class of locally compact groups which

included all [IN] groups.

The following lemma is related to Proposition 4.2. Lemma 5.1:

Let H be a closed subgroup of G. Then the restriction map R : A(G) ↦ A(H) is a completely contractive homomorphism of

A(G) onto A(H).

Proof:

It is well known that R is a continuous homomorphism of A(G) onto A(H). Let V N_{H}(G) be the weak closure in VN(G) of span {λ_{G}(h) :

h ∈ H}. Then V N_{H}(G) is a von Neumann subalgebra of VN(G). Moreover, R^{∗} : VN(H) ↦ VN(G) is a *-isomorphism of VN(H) onto V N_{H}(G) [14]. It follows that R^{∗} is completely contrac-

tive

and hence that R is also completely contractive.

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Proposition 5.2:

Let H be a closed subgroup of G. Assume that A(G) is operator weakly amenable. Then A(H) is operator weakly amenable.

Proof:

This follows immediately from Lemma 5.1 and Proposition 3.3. ■

Since A({e}) is always operator weakly amenable, the converse to Proposition 5.2 can only hold if A(G) is operator weakly amenable for each G. We can however establish the converse if H is assumed to be open.

Lemma 5.3:

Let H be an open subgroup of G. Then A(G) is operator weakly amenable if and only if A(H) is operator weakly amenable.

Proof:

If A(G) is operator weakly amenable, then by Proposition 5.2, so is A(H). Assume that A(H) is operator weakly amenable and that

D : A(G) ↦ VN(G) is a completely bounded derivation. Let u ∈ A(G)

be such that supp u is compact. We can find x_{1}, , ..., x_{n} such that suppu ⊂ ⋃ x_{i}H . Let u_{i} = 1_{x_{i}H}u . Then u = ∑_{i=1}^{n} u_{i}. If we let D_{i} be the restriction of D to the algebra 1_{x_{i}H}A(G), then D_{i} is a completely bounded derivation from the algebra 1_{x_{i}H}A(G) into the

symmetric operator module VN(G). By Lemma 4.3 and Lemma 4.4, 1_{x_{i}H}A(G) is

completely isometrically isomorphic to A(H). Since A(H) is weakly operator amenable, Proposition 3.2 shows that D_{i}(u_{i}) = 0. Thus

D(u) = 0 since elements with compact support are dense in A(G). Hence A(G) is operator weakly amenable.

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Theorem 5.4: Let G a locally compact group with an amenable connected component G_{0}. Then A(G) is operator weakly amenable.

Proof:

Let π : G ↦ G/G_{0} be the canonical projection. Since G/G_{0} is totally disconnected it has an open compact subgroup \overline{C}. Then H = π^{−1}(\overline{C}) is an open almost connected subgroup of G which is amenable if G_{0} is amenable. It follows from [36] that A(H) is operator amenable and hence is clearly operator weakly amenable. We now simply apply Lemma 5.3 to conclude that A(G) is operator

weakly amenable.

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This theorem provides us with a significant class of groups for which A(G) is operator weakly amenable. Indeed, recall the following classes of locally compact groups:

[MAP] G is Maximally Almost Periodic if the finite

dimensional representations of G separate points.

[IN] G is an Invariant Neighborhood group if the identity has a compact neighborhood which is invariant under all inner automorphisms.

[Her] G is Hermitian if L^{1}(G), the group algebra of G, is a hermitian Banach *-algebra.

[NF] G is in the class [NF] if G has no uniformly discrete free semigroup on two generators.

Corollary 5.5: Let G be a locally compact group such that G belongs to [MAP], [IN], [Her] or [NF]. Then A(G) is operator weakly amenable.

Proof:

In each of the classes above the connected component must be amenable (see [35]). It follows from Theorem 5.4 that A(G) is oper-

ator

weakly amenable.

■

Remark:

We do not know if for every locally compact group A(G) is operator weakly amenable. However, if there is an example of a

group for which A(G) is not operator weakly amenable, our next result shows that there

must be a connected Lie group with this property.

Theorem 5.6: If A(H) is operator weakly amenable for each connected Lie group H, then A(G) is operator weakly amenable for each locally compact group G. Proof: Assume that A(H) is operator weakly amenable for each connected Lie group H. Let G be a locally compact group for which A(G) is not operator weakly amenable. Then G must contain an open almost connected subgroup G_{1}. Lemma 5.3 shows that A(G_{1}) must also fail to be operator weakly amenable. Hence there exists a non-zero completely bounded derivation D : A(G_{1}) ↦ VN(G_{1}).

If K is a normal subgroup of G_{1}, then A(G_{1}/K) is completely isometrically isomorphic to the subalgebra A(G_{1} : K) of A(G_{1}) consisting of functions which are constant on left cosets of G. (This follows from the fact that μ_{K} ∗ VN(G_{1}) is *-isomorphic with VN(G_{1}/K) where μ_{K} is the Haar measure on K). It follows that if A(G_{1}/K) is operator weakly amenable, then so is A(G_{1} : K). But VN(G_{1}) is a symmetric A(G_{1} : K) operator module. Therefore the restriction of D to A(G_{1} : K) must be identically 0 whenever A(G_{1} : K) is operator weakly amenable.

The almost connected group G_{1} is a projective limit of Lie groups [35]. Therefore for any u ∈ A(G) we can find a compact normal subgroup K_{n} such that G_{1}/K_{n} is a Lie group and such that there exists a function u_{n} ∈ A(G_{1} : K_{n}) with ∥u_{n} − u∥_{A(G)} ≤ 1/n ([16]). Let H_{n} be the connected component of G_{1}/K_{n}. Then H_{n} is a connected Lie group, and hence by assumption A(H_{n}) is operator weakly amenable. But H_{n} is open in G_{1}/K_{n}, so by Lemma 5.3, A(G_{1}/K_{n}) ≅ A(G_{1} : K_{n}) is also operator weakly amenable. We can conclude as above that D(u_{n}) = 0. However, since D is continuous, we get that D(u) = 0 for each u ∈ A(G) contradicting the assumption that D was non-zero. Hence A(G_{1}) must also have been operator weakly amenable and in turn so was A(G). ■

6. Operator Amenability and Weak Operator Amenability of the Second Dual of A(G)

Let A be a completely contractive Banach algebra. Then as always A^{∗∗} can be made into a Banach algebra with either of the two Arens multiplications. Observe also that A^{∗∗}, the standard second dual of A, is an operator space under the dual structure inherited from A^{∗} and hence from A. We claim that A^{∗∗} is a completely contractive Banach algebra with respect to either Arens product. To see this note that 𝕄_{n}(A^{∗∗}) = (𝕄_{n}(A))^{∗∗} and the closed unit ball of 𝕄_{n}(A) is weak-* dense in 𝕄_{n}(A^{∗∗}). As such, every contractive element in 𝕄_{n}(A^{∗∗}) is the weak-* limit of a net of contractive elements in 𝕄_{n}(A). From this we deduce that with respect to either Arens product, A^{∗∗} is completely contractive. Finally, if X is a quotient of A^{∗∗}, X also becomes a completely contractive Banach algebra. Therefore, A(G)^{∗∗} and any of its quotients for which we are concerned below are completely contractive Banach algebras.

Theorem 6.1: Let X be a topologically introverted subspace of VN(G) which contains C_{δ}^{∗}(G). If X^{∗} is operator weakly amenable, then every abelian subgroup H of G is finite. Moreover, G is discrete.

Proof:

It follows from the proof of [17, Theorem 3.2] that if G has an infinite abelian subgroup, then X^{∗} has a nonzero point derivation. Proposition 3.4 would then imply that X^{∗} could not be operator weakly amenable. We can therefore assume that G has no such

subgroups.

Let G_{o} denote the connected component of e. Then since G is periodic, so is G_{0}. Moreover, the same is true for any homomorphic image of G_{0}. Let U be a neighborhood of e in G_{0}.

Then

there exists a compact normal subgroup N ⊆ U such that G_{0}/N is a

periodic, connected Lie group. Therefore G_{0}/N is trivial. Since this is true for all such U, G_{0} must be trivial. In particular, G is totally disconnected. If G is nondiscrete and totally disconnected, then G contains an infinite compact subgroup K. However, by [38, Theorem 2] the infinite compact group K contains an infinite abelian subgroup which is impossible since G contains no such subgroup. It follows that G is discrete.

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Corollary 6.2 Let X be be any of the spaces AP (Ĝ), W AP (Ĝ), or UCB(Ĝ). If X^{∗} is operator amenable, then G is an amenable discrete group. If X = VN(G), then X^{∗} is operator amenable if and only if G is finite.

Proof:

Assume X^{∗} is operator amenable where X is any of the spaces above. Then by Theorem 6.1 G is discrete. Moreover, since X^{∗} is

operator amenable it has a bounded approximate identity [36]. If X = AP (Ĝ), W AP (Ĝ) or UCB(Ĝ), then B_{δ}(G_{d}) is a quotient of X^{∗} [17]. However, since G is discrete, B_{δ}(G_{d}) = B_{λ}(G), the reduced Fourier-Stieltjes algebra of G. In particular, B_{λ}(G) also has a bounded approximate identity. It is easy to see that 1_{G} is a weak-* cluster point of this approximate identity in B(G). Since B_{λ}(G) is weak-* closed we get 1_{G} ∈ B_{λ}(G). Moreover, since

B_{λ}(G) is an ideal in B(G), B_{λ}(G) = B(G) and hence G is amenable.

If X = VN(G), then it follows from [29, Proposition 3.2 (b)] that G is compact and thus finite. Conversely if G is finite, then VN(G)^{∗} = A(G), which is clearly operator amenable. ■

For any of the spaces X = AP (Ĝ), W AP (Ĝ), UCB(Ĝ) or VN(G), we get from Theorem 6.1 that if X^{∗} is operator weakly amenable, then G is periodic with no infinite abelian subgroups. This is a severe restriction of the nature of G. The following two corollaries are obtained in the same manner as [17, Corollary 3.1 and 3.5]

Corollary 6.3:

Let G be a locally compact group which satisfies one of the following conditions:

i) G is locally finite; ii) G is an elementary group; iii) G is locally solvable; or iv) G is isomorphic to a subgroup of GL(n, 𝔽) for some n and any field 𝔽. If X is a topologically introverted subspace of VN(G) which contains C_{δ}^{∗}(G_{d}), then X is operator weakly amenable if and only if

G is finite.

Recall that a discrete group G has polynomial growth if for every finite set F ⊂ G there exists a p ∈ ℕ such that |F | = O(n^{p}). It follows form the proof of [17, Corollary 3.5] that if

G is infinite, it must contain an infinite abelian subgroup H. Theorem 6.1 implies:

Corollary 6.4: Let G be a discrete group of polynomial growth. Let X be a topologically introverted subspace of VN(G) which contains C_{δ}^{∗}(G_{d}), and such that X is operator weakly amenable, then G is finite. The results of this section are natural analogs of earlier results obtained in [17],[24] and [31] for the second dual of A(G) viewed as

a

Banach algebra. For example, in Corollary 6.2 we showed that A(G)^{∗∗}

is operator amenable if and only if G is finite. This was proved by Granirer in [24] for amenability in the category of Banach alge-

bras.

We note however that it is known that for the second dual A^{∗∗} to be amenable, A must itself be amenable [22]. Since we have already stated that A(G) is in general not amenable even for compact groups, Granirer’s result is quite natural. However, since A(G) is

operator

amenable for any amenable group, one might expect that it would be

more likely that A(G)^{∗∗} would be operator amenable. Corollary 6.2 shows that this is not the case. In fact the significance of this section is that all the known results for amenability and weak amenability for quotients of A(G)^{∗∗} hold in the new category. This shows that unlike the case of A(G) itself, barriers for amenability and weak amenability of quotients of A(G)^{∗∗} do not disappear with the addition of the operator space structure. Curiously, the reason for this could this could be, as suggested by the proof of Theorem 6.1, that the obstacles arise from the presence of abelian subgroups where our two notions of amenability agree once again.

7. Automatic Complete Boundedness and Derivations

In section 4 we showed that unless G contained an abelian subgroup of finite index, there are bounded linear maps from A(G) into

VN(G) which are not completely bounded. Conversely, if G does contain such a subgroup, then it is clear that each bounded

derivation from A(G) into VN(G) is completely bounded. Moreover, since any such group is amenable, every derivation from A(G) into any

Banach A(G)- module is automatically continuous [18] and hence is automatically completely bounded.

Since one of our main interests in this paper is to study the potential operator weak amenability of A(G) as compared with the usual no-

tion of

weak amenability of A(G), a natural question arises: For which locally

compact groups are all (bounded) derivations from A(G) into VN(G) automatically completely bounded? More generally, when is any

derivation

of A(G) into any arbitrary A(G)-module automatically completely bounded?

Recall that in [15] the first author showed that any locally compact group with an abelian connected component is such that

A(G)

is weakly amenable. It follows that in this case the only bounded derivation from

A(G) to VN(G) is zero, which is trivially completely bounded. We can speculate about the possibility that these are the only such groups without bounded derivations into VN(G) that are not com-

pletely bounded. We begin with a proposition which is somewhat analogous to [18,

Theorem 1].

Proposition 7.1:

Let G be a locally compact group such that every derivation from A(G) into a finite dimensional, symmetric operator A(G)-

module is completely bounded, then G is amenable.

Proof:

Assume that G is not amenable. Then A(G)^{2} is not closed in A(G) [33]. We follow the construction in [11]. In particular, we can find a closed cofinite ideal I in A(G) and another cofinite ideal K which is not closed and satisfies I^{2} ⊂ K ⊂ I. Let X be the radical of the finite dimensional algebra A(G)/K. Then X =

I/K. By Wedderburn’s Theorem there exists a subalgebra B of A(G)/K

such that

A(G)/K = B ⊕ X

with multiplication in B ⊕ X given by

(b_{1}, x_{1})(b_{2}, x_{2}) = (b_{1}b_{2}, b_{1}x_{2} + x_{1}b_{2}).

Let Γ : A(G) ↦ A(G)/K be the quotient map and define a symmetric

A(G)-module structure on X by

a · x = Γ(a)x = x · a

Since the kernel of the map a ↦ a · x is closed for each x ∈ X, the finite dimensional module X is a Banach A(G)-module with respect to any norm on X. Now fix any norm on X. Then X

can be given the MIN operator structure to make it into a symmetric operator A(G)-module. Finally, if Π is the projection of A(G)/K onto its second coordinate, then D = Π^{∘}Γ is a derivation from A(G) into X. Moreover, since (ker D) ∩ J = K is not closed, D is not continuous and hence is not completely bounded.

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We can obtain a converse to the previous proposition for the class of locally compact groups with an abelian subgroup of finite index. Proposition 7.2

Let G be a locally compact group with an abelian subgroup of finite index. Then every derivation from A(G) into an operator

A(G)

module is completely bounded. Proof:

Since G is amenable, every derivation from A(G) into any Banach A(G)

module is automatically bounded [18]. It now follows from Theorem 4.2 that every derivation into an operator A(G) module is automatically completely bounded.

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While for amenable groups derivations from A(G) into VN(G) are automatically bounded even for groups such as SO(3) it is not the case that they are always completely bounded. In fact, it turns out that for compact Lie groups, Proposition 7.2 was the best we

could do.

Corollary 7.3:

Let G be a compact Lie group. Then every derivation from A(G) into VN(G) is completely bounded if and only if G has an abelian subgroup of finite index.

Proof:

The ”if” direction follows immediately form the previous proposition. Conversely, since G is amenable, A(G) is operator amenable, and hence clearly operator weakly amenable. It follows that the only completely bounded derivation from A(G) into VN(G) is the zero map. However, A(G) is not weakly amenable [27]. This means there must exist a nonzero bounded derivation D

from

A(G) into VN(G) which cannot be completely bounded.

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For noncompact Lie groups we have the following:

Proposition 7.4:

i) Let G be a Lie group. Assume that every bounded derivation from A(G) into a symmetric operator A(G)-module is completely bounded. Then every compact subgroup K of G has an abelian subgroup of finite index. ii) Let G be a semisimple Lie group with a compact connected component K. Then every bounded derivation from A(G)

into VN(G) is completely bounded if and only if G is discrete. iii) Let G be a semisimple Lie group with a compact connected component. Then every derivation from A(G) into a symmetric operator A(G)-module is completely bounded if and only

if G is amenable and discrete.

Proof:

By Lemma 5.1 the restriction map R establishes a completely contractive homomorphism from A(G) onto A(K). This allows us to view VN(K) as a symmetric operator A(G)-module with respect to the action defined by u · T = R(u) ⊙ T = T ⊙ R(u) = T · u for each u ∈ A(G), T ∈ VN(K) (here ⊙ denotes the usual action of A(K) on VN(K)). If K does not have an abelian subgroup of finite index, then Corollary 7.3 shows that there exists a derivation D : A(K) ↦ VN(K) which is bounded but not completely bounded. Let D̃(u) = D(R(u)) for each u ∈ A(G). Then it is easy to see that D̃ defines a bounded derivation from A(G) into VN(K). We claim that D̃ is not completely bounded.

Observe that R = Γ^{∘}Q where Q : A(G) ↦ A(G)/I(K) is the quotient map and Γ : A(G)/I(K) ↦ A(K) is the map defined in Proposition 4.2. Let M > 0. Since D is not completely bounded, there exists [u_{ij}] ∈ 𝕄_{n}(A(K)) with ∥[u_{ij}]∥_{n} ≤ 1 but ∥D([u_{ij}]∥_{n} > M. However since Q is a complete quotient map and Γ is a complete isometry there exists [v_{ij}] ∈ 𝕄_{n}(A(G)) with ∥[v_{ij}]∥_{n} ≤ 2 and R^{(n)}([v_{ij}]) = (Γ^{∘}Q)^{(n)}([v_{ij}]) = [u_{ij}]. Thus ∥D̃^{(n)}([v_{ij}])∥_{n} = ∥D^{(n)}[u_{ij}]∥ > M. Hence D̃ is not completely bounded.

ii) Assume that K is not abelian. Then as before A(K) is not weakly amenable and hence by [15, Lemma 2.1] A(G) cannot be weakly amenable. However K is open since G is a Lie group. Furthermore A(K) is operator weakly amenable since K is compact. Thus by Lemma 5.3, A(G) is operator weakly amenable. This means there exists a nonzero bounded derivation from A(G) into VN(G) which is not

completely bounded. Since this is impossible, K must be abelian. However, since G is semisimple, this means that K is trivial and hence that G is discrete. For the converse, we observe that if G is discrete, then A(G) is weakly amenable [18] and the only bounded derivation of

A(G)

into VN(G) is the zero map which is obviously completely bounded. iii) Assume that every derivation from A(G) into a symmetric operator A(G)-module is completely bounded, then by (ii) G is discrete. Moreover by Proposition 7.1, G must be amenable. The converse follows trivially since, if G is discrete A(G) is operator weakly amenable and if G is amenable then every derivation

is bounded [18]. As such the only derivation from A(G) into a symmetric operator A(G)-module is once again the zero map.

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We can apply the previous Corollary to the family ℝ^{n} ⋉_{ρ} SO(n) of Euclidean motion groups to conclude that:

Corollary 7.5:

Let G be the semidirect product ℝ^{n} ⋉_{ρ} SO(n) where the action of SO(n) on ℝ^{n} is the natural action. If n ≥ 3, there exists a bounded derivation D : A(G) ↦ VN(G) which is not completely

bounded.

The various types of amenability which we have mentioned have one major structural flaw in common – they are not inherited by subalgebras. In particular, we do not know if given a compact normal subgroup K, A(G) being (operator) weakly amenable implies that A(G : K) ≅ A(G/K) is also (operator) weakly amenable. Were this the case, we would be able to make significant progress is classifying those groups for which A(G) is (operator) weakly amenable. For example, if we knew that A(G/K) always inherited weak amenability from A(G), we could conclude that for G ∈ [SIN], weak amenability of A(G) would be equivalent to G having an abelian connected component [15, Theorem 2.7]. In this case, it would be true that for any [SIN]-group with nonabelian connected component there would be a bounded derivation from A(G) into VN(G) which was not completely bounded. At present the best we can do is the following related result:

Proposition 7.6:

Let G be a [SIN] group for which G_{0} is not abelian. Then

for every neighborhood U of the identity, there exists a compact normal K_{U} ⊂ U and a bounded map Γ_{U} : A(G) ↦ VN(G/K_{U}) which is such that the restriction of Γ_{U} to A(G : K_{U}) is a derivation that is not completely bounded.

Proof:

Let G_{0} be the connected component of G. Then G_{0} = V ×K where V is a vector group and K is a nonabelian compact connected group

( see [35]). Let U be a neighborhood of e ∈ G. Then, since G is a projective limit of Lie groups, there exists a compact normal sub-

group

K_{U} ⊂ U such that G/K_{U} is a Lie group and if ϕ : G ↦ G/K_{U} is the quotient map, then ϕ(K) is a nonabelian compact

connected subgroup of G. It follows from [27] and [15] that A(G/K_{U}) is not

weakly amenable. Hence there exists a nonzero bounded derivation D_{U} from

A(G/K_{U}) into VN(G/K_{U}). Since A(G/K_{U}) is completely isometrically isomorphic to A(G : K_{U}), we may assume that D_{U} maps A(G : K_{U}) into VN(G/K_{U}). Finally, let P_{K_{U}} be the projection of A(G) onto A(G : K_{U}). Then Γ_{U} = D_{U}^{∘}P_{K_{U}} is the desired map.

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8. Amenability and Weak Amenability of Ideals in A(G)

In this section, we will take a brief look at operator amenability and operator weak amenability

for ideals in A(G). We know that if A(G) is operator amenable, then G must be amenable.

We can ask if there are any structural implications of the existence of a closed ideal in A(G) which is operator amenable. Since the Fourier algebra of a discrete group always contains such an ideal, namely the one-dimensional ideal of functions supported on the identity, perhaps the best one could hope for would be to show that our group must contain an open amenable subgroup. In fact, we shall show

that this

is indeed the case. The key to our proof will be the fact that each operator

amenable completely contractive Banach algebra contains a bounded approximate

identity.

Lemma 8.1:

Let G be a locally compact group. Assume that A(G) has a nonzero ideal I with a bounded approximate identity. Then G has an open amenable subgroup.

Proof:

Let G_{1} be an open almost connected subgroup of G. Let F = Z(I). Since I is non-zero, there exists an x_{0} ∈ G \ Z(I). By translating if necessary, we may assume that x_{0} ∈ G_{1}. Then I_{1} = 1_{G_{1}}I can be viewed as a closed ideal in A(G_{1}) which is nonzero and has a bounded approximate identity. It follows from

[15, Proposition 3.5] that G_{1} is amenable.

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Theorem 8.2:

A(G) has a nonzero closed ideal which is operator amenable if and only if G has an open amenable subgroup.

Proof:

Assume that G_{1} is open and amenable. Then I(G \ G_{1}) is a closed ideal of A(G) which is completely isometrically isomorphic to A(G_{1}). Since G_{1} is amenable, I(G \ G_{1}) is operator amenable. Conversely, assume that I is a non-zero closed ideal of A(G) which is operator amenable. Then by [36], I has a bounded approximate identity. It follows from Lemma 8.1 that G has an open amenable subgroup.

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Corollary 8.3:

If A(G) has a non-zero ideal I which is operator amenable, then A(G) is operator weakly amenable.

Proof:

Since G has an open amenable subgroup, this follows immediately from Theorem 5.4, and Theorem 8.3.

It was observed in [15], that the Fourier algebra of SL(2, ℝ) did not have any amenable closed ideals. Theorem 8.2 allows us to ex-

tend this result to the category of operator spaces.

Corollary 8.4:

A(SL(2, ℝ)) has no non-zero closed operator amenable ideals.

Let R(G) denote the coset ring of G. R(G) is the ring of subsets of G generated by the cosets of an open subgroup of G. Let R_{c}(G) denote those subsets of G which are closed in G and which are contained in R(G_{d}), the coset ring of G when viewed as a discrete group. In [21], it was shown that if a closed ideal I in A(G) posses a bounded approximate identity, then there exists some F ∈ R_{c}(G) such that I = I(F). Clearly, these are the only possible candidates for operator amenable ideals. To determine whether or

not these ideals are in fact operator amenable, we will need the following analog of [28, Proposition 5.1]

Theorem 8.5:

Let A be an operator amenable completely contractive Banach algebra, Let I be a closed ideal of A. Then I is operator amenable if and only if I has a bounded approximate identity.

Proof:

Let {e_{α}} be a bounded approximate identity in I with ∥e_{α}∥ ≤ M for each α. Let X be an operator I-module. Since I has a bounded approximate, in order to show that each completely bounded derivation from A

into X^{∗} is inner, we may assume just as in the bounded case that X is neounital. As such X^{∗} becomes an operator A module with respect to the induced action of A on X^{∗}. That is if Γ = aΓ_{1}b where a, b ∈ I and Γ_{1} ∈ X^{∗}, then for any u ∈ A, uΓ = (ua)Γ_{1}b and Γu = aΓ_{1}(bu). Let D : I ↦ X^{∗} be a completely bounded derivation. Then D lifts to a derivation D̃ : A ↦ X^{∗}, with D̃(u) = lim_{α} D(e_{α}u) for each

u ∈ A [28]. Let [u_{ij}] ∈ 𝕄_{n}(A). Then

∥D̃^{(n)}([u_{ij}])∥_{n} = lim_{α} ∥D(e_{α}u_{ij})∥_{n}
≤ lim_{α} ∥D∥_{cb}∥[e_{α}u_{ij}]∥_{n}
≤ sup_{α} ∥D∥_{cb}∥∥e_{α}∥∥[u_{ij}]∥_{n}
≤ ∥D∥_{cb}M∥[u_{ij}]∥_{n}

It follows that D̃ is a completely bounded extension of D to A. However, since A is operator amenable D̃ is inner, and hence D is inner. Thus I is also operator amenable. The converse is obvious since each operator amenable completely contractive Banach algebra has a bounded approximate identity.

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We close with the following complete characterization of those ideals in the Fourier algebra of an amenable [SIN] group which are operator amenable.

Theorem 8.6:

Let G be an amenable [SIN] group. Let I be a closed ideal in A(G). Then I is operator amenable if and only if Z(I) ∈ R_{c}(G)

Proof:

By Theorem 8.5, we know that I is operator amenable if and only if it posses a bounded approximate identity. However in [21] it

was shown that an ideal in a [SIN] group posses a bounded approximate identity if and only if Z(I) ∈ R_{c}(G).

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Pure Math. Dept., U. Waterloo, Waterloo, Ont. N2L–3G1, CANADA

E-mail address: pwood@barrow.uwaterloo.ca
E-mail address: beforrest@math.uwaterloo.ca