2021/12/16 by Brešar, M., Godoy, M. L. C., Villena, A. R. · 1 citation
#43A20 #46H05 #46L05 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2112.08809
Let A and B be Banach algebras with bounded approximate identities and let Φ:A→ 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 L1(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.