Jordan Left $\alpha$-centralizers on Algebras with Applications to Group Algebras

TL;DR

This paper proves that Jordan left $ ext{ extalpha}$-centralizers are actual left $ ext{ extalpha}$-centralizers in certain algebras, and characterizes their properties in group algebras, linking algebraic structure to group properties.

Contribution

It establishes that Jordan left $ ext{ extalpha}$-centralizers are homomorphisms under specific conditions and characterizes weakly compact Jordan centralizers in group algebras.

Findings

Jordan left $ ext{ extalpha}$-centralizers are homomorphisms in algebras with right identity.

Weakly compact Jordan left $ ext{ extalpha}$-centralizers in $L^1(G)$ are characterized when $ ext{ extalpha}$ is continuous and surjective.

Existence of non-zero $ ext{ extalpha}$-derivations on $L^1(G)$ implies $G$ is compact and non-abelian.

Abstract

We prove that every Jordan left $\alpha$-centralizer from an algebra $A$ with a right identity into an arbitrary algebra $B$ is a left $\alpha$-centralizer. This implies all Jordan homomorphisms between such algebras are homomorphisms. We extend this result to continuous Jordan left $\alpha$-centralizers when $A$ has a bounded left approximate identity. For the group algebra $L^1(G)$, we characterize weakly compact Jordan left $\alpha$-centralizers when $\alpha$ is continuous and surjective, showing $L^1(G)$ admits a weakly compact epimorphism if and only if $G$ is finite. Consequently, the existence of a non-zero $\alpha$-derivation on $L^1(G)$ is equivalent to $G$ being compact and non-abelian.