Title: Maps preserving two-sided zero products on Banach algebras
Abstract:Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $Φ:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $Φ(a)Φ(b)=Φ(b)Φ(a)=0$ whe...Let $A$ and $B$ be Banach algebras with bounded approximate identities and let $Φ:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $Φ(a)Φ(b)=Φ(b)Φ(a)=0$ whenever $ab=ba=0$). We show that $Φ$ 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 $Φ:A\to B$ that satisfy $Φ(a)Φ(b)+Φ(b)Φ(a)=0$ whenever $ab=ba=0$. We show in particular that if $Φ$ is surjective and $A$ is a $C^*$-algebra, then $Φ$ is a weighted Jordan homomorphism.Read More