Jordan left $\alpha$-centralizer on certain algebras

TL;DR

This paper explores the properties of Jordan left -centralizers on various algebras, establishing conditions under which they coincide with left -centralizers and examining their behavior on Banach and group algebras.

Contribution

It proves that Jordan left -centralizers are equivalent to left -centralizers on algebras with a right identity and extends this to Banach algebras with bounded approximate identities, including group algebras.

Findings

Jordan left -centralizers are left -centralizers on algebras with right identity

Extension of results to Banach algebras with bounded approximate identities

Analysis of Jordan left -centralizers on group algebra L^1(G)

Abstract

In this paper, we investigate Jordan left $\alpha$-centralizer on algebras. We show that every Jordan left $\alpha$-centralizer on an algebra with a right identity is a left $\alpha$-centralizer. We also investigate this result for Banach algebras with a bounded approximate identity. Finally, we study Jordan left $\alpha$-centralizer on group algebra $L^1(G)$.