The Banach algebra induced by a double centralizer

Abstract

Given a Banach algebra $A$, R. Larsen defined, in his book “An introduction to the theory of multipliers", a Banach algebra $A_{T}$ by means of a multiplier $T$ on $A$, and essentially used it in the case of a commutative semisimple Banach algebra $A$ to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra ${\mathcal A}_{R}$ induced by a bounded double centralizer $\langle L , R \rangle$ of a Banach algebra $A$. Then, our main concern is devoted to the relationships between $L$, $R$, and the algebras of bounded double centralizers ${\mathcal W}(A)$ and ${\mathcal W}({\mathcal A}_{R})$ of $A$ and ${\mathcal A}_{R}$, respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.