Narrow Results

In this paper, we introduce the concept of configuration graph and show how one can use this notion to simplify the theorem proved by Rejali and Yousofzadeh [Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin18 (2011) 157–172].

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. We show that (super) amenability of ${𝒜}⋊𝔄$ implies (super) module amenability of ${𝒜}$ and (super) amenability $𝔄$. We investigate biprojectivity and biflatness of ${𝒜}⋊𝔄$ in the some especial cases. We also give some results related to module biprojectivity and module biflatness of ${𝒜}$, when ${𝒜}⋊𝔄$ is biprojective or biflat.

Let $\mathrm{\Lambda }$ be a direct set, ${\{A_{\alpha }\}}_{\alpha \in \mathrm{\Lambda }}$ be a family of Banach algebras with bounded approximate identity (or unital) and $I$ be a set. We consider the Banach algebra ${\ell }^{\infty }({\ell }_{A_{\beta }}(I,A_{\alpha }))$. We show that this algebra has a bounded approximate identity (or is unital) if and only if $I$ is finite. We also characterize the left multipliers of these algebras and investigate their amenability of them. Moreover, we characterize the character spaces (Gelfand spaces) of these algebras in a special case.