On $(p, q)-$centralizers of certain Banach algebras

TL;DR

This paper investigates $(p, q)$-centralizers in Banach algebras with right identities, proving their properties, boundedness in normed cases, and characterizing their behavior in group algebras, especially for $L^1(G)$.

Contribution

It establishes that all $(p, q)$-centralizers are two-sided and bounded in normed algebras, and characterizes their existence in group algebras like $L^1(G)$.

Findings

Every $(p, q)$-centralizer is a two-sided centralizer.

$(p, q)$-centralizers are bounded in normed algebras.

Existence of nonzero weakly compact $(p, q)$-centralizer in $L^1(G)$ iff $G$ is compact.

Abstract

Let $A$ be an algebra with a right identity. In this paper, we study $(p, q)-$centralizers of $A$ and show that every $(p, q)-$centralizer of $A$ is a two-sided centralizer. In the case where, $A$ is normed algebra, we also prove that $(p, q)-$centralizers of $A$ are bounded. Then, we apply the results for some group algebras and verify that $L^1(G)$ has a nonzero weakly compact $(p, q)-$centralizer if and only if $G$ is compact and the center of $L^1(G)$ is non-zero. Finally, we investigate $(p, q)-$Jordan centralizers of $A$ and determine them.