Maps preserving two-sided zero products on Banach algebras

Abstract

Let A and B be Banach algebras with bounded approximate identities and let F: A→B be a surjective continuous linear map which preserves twosided zero products (i.e., F(a)F(b) = F(b)F(a) = 0 whenever ab = ba = 0). We show that F 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 L1(G) with G any locally compact group. We also study a more general type of continuous linear maps F : A → B that satisfy F(a)F(b)+F(b)F(a) = 0 whenever ab = ba = 0. We show in particular that if F is surjective and A is a C∗-algebra, then F is a weighted Jordan homomorphism.