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 → B be a surjective continuous linear map which preserves twosided 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 → 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.2020 Mathematics Subject Classification. 43A20, 46H05, 46L05. Key words and phrases. Zero product determined Banach algebra, weakly amenable Banach algebra, group algebra of a locally compact group, C * -algebra, algebra of approximable operators, weighted Jordan homomorphism, two-sided zero products, linear preserver.