Maps preserving two-sided zero products on Banach algebras
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Zero product determined Banach algebraWeakly amenable Banach algebraGroup algebra of a locally compact groupC∗-algebraAlgebra of approximable operatorsWeighted Jordan homomorphismTwo-sided zero productsLinear preserver
Published version: M. Brešar, M.L.C. Godoy, A.R. Villena, Maps preserving two-sided zero products on Banach algebras, Journal of Mathematical Analysis and Applications, Volume 515, Issue 1, 2022, 126372, ISSN 0022-247X, [https://doi.org/10.1016/j.jmaa.2022.126372]
SponsorshipSlovenian Research Agency - Slovenia P1-0288; MCIU/AEI/FEDER Grant PGC2018-093794-B-I00; Junta de Andalucia FQM-185 P20_00255 MIU FPU18/00419 EST19/00466; Proyectos I+D+i del programa operativo FEDER-Andalucia Grant A-FQM-484-UGR18
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.
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