Abstract. We survey some recent studies of linear zero product or orthogonality preservers between $C^*$/$W^*$-algebras, their dual or predual spaces, and holomorphic disjointness preservers of $C^*$-algebras. Such maps are expected to provide algebra or linear Jordan ($*$-) homomorphisms between the underlying operator algebras. We also study orthogonality preservers between Hilbert $C^*$-modules and Fourier algebras. A few open problems are stated.

DOI: 10.14232/actasm-018-267-7

AMS Subject Classification (1991): 46L40, 46L10, 46H40

Keyword(s): zero product preservers, orthogonality preservers, Jordan homomorphisms, Fourier algebras, Hilbert $C^*$-modules, holomorphic maps of $C^*$-algebras

Received February 5, 2018 and in final form April 12, 2018. (Registered under 17/2018.)