For a Banach algebra $A$, $A''$ is $(-1)$-Weakly amenable if $A'$ is a Banach $A''$-bimodule and $H^1(A'',A')={0}$. In this paper, among other things,  we study the relationships between the $(-1)$-Weakly amenability of $A''$ and the weak amenability of $A''$ or $A$. Moreover, we show that the second dual of every $C^ast$-algebra is $(-1)$-Weakly amenable.

Let A be a Banach algebra. A is called ideally amenable if for every closed ideal I of A, the first cohomology group of A with coefficients in I* is trivial. We investigate the closed ideals I for which H1 (A,I* )={0}, whenever A is weakly amenable or a biflat Banach algebra. Also we give some hereditary properties of ideal amenability.

In this paper among many other things we prove that the topological left amenability and left amenability of a weighted hypergroup (K, ?) are equivalent. For a normal subgroup H of K, we define a weight function ?? on KIH and obtain connection between left amenability of (K, ?) and (K|H, ??). Let H be a compact subhypergroup of K. We define the weight function on K||H and obtain connection...

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We give examples of non-amenable ICC groups Γ with the Haagerup property, weakly amenable with constant Λcb(Γ) = 1, for which we show that the associated II1 factors L(Γ) are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra P ⊂ L(Γ) generates an amenable von Neumann algebra. Nevertheless, for these examples of groups Γ, L(Γ) is not isomorphic to any interpolated free group...

We give examples of non-amenable ICC groups Γ with the Haagerup property, weakly amenable with constant Λcb(Γ) = 1, for which we show that the associated II1 factors L(Γ) are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra P ⊂ L(Γ) generates an amenable von Neumann algebra. Nevertheless, for these examples of groups Γ, L(Γ) is not isomorphic to any interpolated free group...

in this paper we define an order structure on the $p$-operator projective tensor product of herz algebras and we show that the canonical isometric isomorphism between $a_p(gtimes h)$ and $a_p(g)widehat{otimes}^p a_p(h)$ is an order isomorphism for amenable groups $g$ and $h$.

We prove that the first reduced cohomology with values in a mixing Lp-representation, 1 < p < ∞, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced lp-cohomology. As a byproduct, we prove a conjecture by Pansu. ...

Column and row operator spaces — which we denote by COL and ROW, respectively — over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p, p ∈ (1,∞) with 1 p + 1 p = 1, we use the operator space structure on CB(COL(L ′ (G))) to equip the Figà-Talamanca–Herz algebra Ap(G) with ...