σ -Approximately module amenable Banach algebras

For $$p\in [1,\infty )$$ we study representations of a locally compact group $$G$$ on $$L^p$$ -spaces and $$\textit{QSL}^p$$ -spaces. The universal completions $$F^p(G)$$ and $$F^p_{\mathrm {QS}}(G)$$ of $$L^1(G)$$ with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group $$C^{*}$$ -algebra of $$G$$ (which is the case $$p=2$$ ). We study these completions of $$L^1(G)$$ in relation to the algebra $$F^p_\lambda (G)$$ of $$p$$ -pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, $$G$$ is amenable if and only if $$F^p_{\mathrm {QS}}(G)=F^p(G)=F^p_\lambda (G)$$ . One of our main results is that for $$1\le p< q\le 2$$ , there is a canonical map $$\gamma _{p,q}:F^p(G)\rightarrow F^q(G)$$ which is contractive and has dense range. When $$G$$ is amenable, $$\gamma _{p,q}$$ is injective, and it is never surjective unless $$G$$ is finite. We use the maps $$\gamma _{p,q}$$ to show that when $$G$$ is discrete, all (or one) of the universal completions of $$L^1(G)$$ are amenable as a Banach algebras if and only if $$G$$ is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to $$L^p$$ -operator crossed products of topological spaces.

Approximate and pseudo-amenability of various classes of Banach algebras

Amenability of any of the algebras described in the title is known to force\nthem to be finite-dimensional. The analogous problems for \\emph{approximate}\namenability have been open for some years now. In this article we give a\ncomplete solution for the first two classes, using a new criterion for showing\nthat certain Banach algebras without bounded approximate identities cannot be\napproximately amenable. The method also provides a unified approach to existing\nnon-approximate amenability results, and is applied to the study of certain\ncommutative Segal algebras.\n Using different techniques, we prove that \\emph{bounded} approximate\namenability of the second dual of a Fourier algebra implies that it is\nfinite-dimensional. Some other results for related algebras are obtained.\n

In this paper we define the module topological center of the second dual \(\mathcal{A}^{**}\) of a Banach algebra \(\mathcal{A}\) which is a Banach \(\mathfrak{A}\)-module with compatible actions on another Banach algebra \(\mathfrak{A}\). We calculate the module topological center of ℓ 1(S)**, as an ℓ 1(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ℓ 1(S)** is ℓ 1(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.

Approximate amenability and pseudo-amenability in Banach algebras

Let A, B be Banach A-modules with compatible actions and M be a left Banach A- A-module and a right Banach B- A-module. In the current paper, we study module amenability, n-weak module amenability and module Arens regularity of the triangular Banach algebra - . We employ these results to prove that for an inverse semigroup S with subsemigroup E of idempotents, the triangular Banach algebra is permanently weakly module amenable (as an . As an example, we show that T0 is T0-module Arens regular if and only if the maximal group homomorphic image GS of S is finite.

Let A be a Banach algebra and ' be a character on A. The notions of approximate essential '-amenability and approximate essential character '-amenability of A are introduced and some properties of such algebras are investigated. Then by means of some examples the distinctions between this new notions and essentially -amenability for A are shown.

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigroup $E(S)$ of idempotents. In symmetric connes amenability, we have proved that $M(G)$ is symmetric connes amenable if and only if $G$ is amenable.

We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.