Maps preserving two-sided zero products on Banach algebras

TL;DR

This paper characterizes surjective linear maps between Banach algebras that preserve two-sided zero products, showing they are weighted Jordan homomorphisms under certain conditions, with applications to group algebras and $C^*$-algebras.

Contribution

It establishes that such zero product preserving maps are weighted Jordan homomorphisms when the domain is zero product determined and weakly amenable, extending to $C^*$-algebras.

Findings

Surjective maps preserving two-sided zero products are weighted Jordan homomorphisms.

Conditions like zero product determination and weak amenability are sufficient.

Results apply to group algebras $L^1(G)$ and $C^*$-algebras.

Abstract

Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $\Phi:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $\Phi(a)\Phi(b)=\Phi(b)\Phi(a)=0$ whenever $ab=ba=0$). We show that $\Phi$ is a weighted Jordan homomorphism provided that $A$ is zero product determined and weakly amenable. These conditions are in particular fulfilled when $A$ is the group algebra $L^1(G)$ with $G$ any locally compact group. We also study a more general type of continuous linear maps $\Phi:A\to B$ that satisfy $\Phi(a)\Phi(b)+\Phi(b)\Phi(a)=0$ whenever $ab=ba=0$. We show in particular that if $\Phi$ is surjective and $A$ is a $C^*$-algebra, then $\Phi$ is a weighted Jordan homomorphism.