On Banach algebras defined by multipliers

Abstract

In this paper, we investigate a Banach algebra $A_T$, where $A$ is a Banach algebra and $T$ is a left (right) multiplier on $A$. We study some concepts on $A_T$ such as $n$-weak amenability, cyclic amenability, biflatness, biprojectivity and Arens regularity. For the group algebra $L^1(G) $ of an infinite compact group $G$, it is shown that there is a multiplier $T$ such that $L^1(G)_T$ has not a bounded approximate identity. For $\ell^1(S)$, where $S$ is a regular semigroup with a finite number of idempotents, we show that there is a multiplier $T$ such that Arens regularity of $\ell^1(S)_T$ implies that $S$ is compact.