Jordan left $\hat{I}\pm$-centralizer on certain algebras

Abstract

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