Weak Amenability Research Articles

Amenability properties of Banach algebra valued continuous functions

Let X be a compact Hausdorff space and A a Banach algebra. We investigate amenability properties of the algebra C(X,A) of all A-valued continuous functions. We show that C(X,A) has a bounded approximate diagonal if and only if A has a bounded approximate diagonal; if A has a compactly central approximate diagonal (unbounded) then C(X,A) has a compactly approximate diagonal. Weak amenability of C(X,A) for commutative A is also considered.

The Weak Haagerup Property II: Examples

The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda. In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups the weak Haagerup constant coincides with the weak amenability constant, although this is not true for locally compact groups in general. It is also shown that the semidirect product of R^2 by SL(2,R) does not have the weak Haagerup property.

Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1, 1)

We study weak amenability for locally compact quantum groups in the sense of Kustermans and Vaes. In particular, we focus on non-discrete examples. We prove that a coamenable quantum group is weakly amenable if there exists a net of positive, scaling invariant elements in the Fourier algebra A(G) whose representing multipliers form an approximate identity in C_0(G) that is bounded in the M_0A(G) norm; the bound being an upper estimate for the associated Cowling-Haagerup constant. As an application, we find the appropriate approximation properties of the extended quantum SU(1,1) group and its dual. That is, we prove that it is weakly amenable and coamenable. Furthermore, it has the Haagerup property in the quantum group sense, introduced by Daws, Fima, Skalski and White.

Weak amenability of commutative Beurling algebras

For a locally compact Abelian group G and a continuous weight function ! on G we show that the Beurling algebra L 1 (G,!) is weakly amenable if and only if there is no nontrivial continuous group homomorphism �: G ! C such that supt2G |�(t)| !(t)!(t 1) 0 such that liminfn!1 !(t n)b(t n) nm for all t 2 G.

A generalized notion of $n$-weak amenability

A generalized notion of $n$-weak amenability

Semigroups of Herz–Schur multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz–Schur multipliers generated by a proper function (see Theorem 1.2). It is then shown that a (not necessarily proper) generator of a semigroup of Herz–Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the second half of the paper we study semigroups of radial Herz–Schur multipliers on free groups. We prove that a generator of such a semigroup is linearly bounded by the word length function (see Theorem 1.6).

A semilattice structure for Arens products

Let (A,⋅) be a Banach algebra. Then for n∈ℕ, A (2n) has 2 n Arens products. In this paper we study the relations between the Arens products on A (2n). Moreover, if P n (A) denotes the set of all Arens products on A (2n), for n∈ℕ, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)−{⋅} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.

Weak connes amenability of the even duals of a Banach algebra

Let A be a Banach algebra. In this paper, we introduce the concept of weak Connes amenability of A. Next, by some conditions, we show that how weak amenability of A (2n) implies weak Connes amenability of A (2n+2) . Finally, we relate this notion to the factorization property of odd duals of A.

TERNARY WEAKLY AMENABLE C*-ALGEBRAS AND JB*-TRIPLES

A well-known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan–Banach triples, focussing on commutative JB*-triples and some Cartan factors.

A Note on weak amenability for free products of discrete quantum groups

We prove that the Cowling–Haagerup constant of a reduced free product of weakly amenable discrete quantum groups with Cowling–Haagerup constant equal to 1 is again equal to 1.

Examples of groups which are not weakly amenable

We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non–weak amenability results of Haagerup (1988) and Ozawa and Popa (2010). A von Neumann algebra analogue is also obtained.

Weak module amenability of triangular Banach algebras

Abstract Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra $\mathcal{T} = \left[ {_B^{AM} } \right]$ and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When $\mathfrak{A}$ is a Banach algebra and A and B are Banach $\mathfrak{A}$-module with compatible actions, and M is a commutative left Banach $\mathfrak{A}$-A-module and right Banach $\mathfrak{A}$-B-module, we show that A and B are weakly $\mathfrak{A}$-module amenable if and only if triangular Banach algebra T is weakly $\mathfrak{T}$-module amenable, where $\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} $.

Beurling–Fourier algebras, operator amenability and Arens regularity

We introduce the class of Beurling–Fourier algebras on locally compact groups and show that they are non-commutative analogs of classical Beurling algebras. We obtain various results with regard to the operator amenability, operator weak amenability and Arens regularity of Beurling–Fourier algebras on compact groups and show that they behave very similarly to the classical Beurling algebras of discrete groups. We then apply our results to study explicitly the Beurling–Fourier algebras on SU ( 2 ) , the 2 × 2 special unitary group. We demonstrate that how Beurling–Fourier algebras are closely connected to the amenability of the Fourier algebra of SU ( 2 ) . Another major consequence of our results is that our investigation allows us to construct families of unital infinite-dimensional closed Arens regular subalgebras of the Fourier algebra of certain products of SU ( 2 ) .

Approximate weak amenability of Banach algebras

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate $n$-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs. Our main result is that under some mild conditions on a given Banach algebra $\mathcal{A}$, if its second dual $\mathcal{A}^{**}$ is $(2n-1)$-weakly [respectively approximately/ approximately weakly/ approximately $n$-weakly] amenable, then so is $\mathcal{A}$. Also if $\mathcal{A}$ is approximately $(n+2)$-weakly amenable, then it is approximately $n$-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability.

Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra l1(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but l1(S) is weakly amenable. Examples are then given.

DERIVATIONS ON MATRIX ALGEBRAS WITH APPLICATIONS TO HARMONIC ANALYSIS

In this paper, the derivations between ideals of the Banach algebra $\frak {E}$$_\infty(I)$ are characterized. Necessary and sufficient conditions for weak amenability of Banach algebras $\frak {E}$$_p(I)$, $1\leq p\leq\infty$, are found. Also, some applications to compact groups and hypergroups are given.

Approximate weak amenability of abstract Segal algebras

In this paper the approximate weak amenability of abstract Segal algebras is investigated. Applications to compact groups are given. Also an open problem raised by Ghahramani and Lau is answered negatively.

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1 < p < ∞ . We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on L p -spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figà–Talamanca–Herz algebra A p ( G ) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗-algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p-completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.

WEAK AMENABILITY OF CONVOLUTION BANACH ALGEBRAS ON COMPACT HYPERGROUPS

In this paper we find necessary and sufficient conditions for weak amenability of the convolution Banach algebras A(K) and L2(K) for a compact hypergroup K, together with their applications to convolution Banach algebras Lp(K) (2 ≤ p < ∞). It will further be shown that the convolution Banach algebra A(G) for a compact group G is weakly amenable if and only if G has a closed abelian subgroup of finite index.

Approximate and weak amenability of certain Banach algebras

The notions of approximate amenability and weak amenability in Banach algebras are formally stronger than that of approximate weak amenability. We demonstrate an example confirming that approximate weak amenability is indeed actually weaker than either ap