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Journal of Operator Theory

Volume 70, Issue 1, Summer 2013  pp. 53-73.

Module weak Banach--Saks and module Schur properties of Hilbert $C^*$-modules

Authors Michael Frank (1) and Alexander A. Pavlov (2)
Author institution: (1) Fakultat Informatik, Mathematik und Naturwissenschaften, Hochschule fur Technik, Wirtschaft und Kultur (HTWK) Leipzig, PF 301166, 04251 Leipzig, Germany
(2) Mathematical department, All-Russian Institute of Scientific and Technical Information, Russian Academy of Sciences (VINITI RAS), 125190 Moscow, Russia

Summary:  Continuing research on Banach--Saks and Schur properties started by C.-H.~Chu, M.~Kusuda, and the authors, we investigate analogous properties in the Banach $C^*$-module context. As an environment serves the class of Hilbert $C^*$-modules. Some properties of weak module topologies on Hilbert $C^*$-modules are described. Natural module analogues of the classical weak Banach--Saks and Schur properties are defined and studied. A number of useful characterizations of properties of Hilbert $C^*$-modules is obtained. In particular, some interrelations of these properties with the self-duality property of countably generated Hilbert $C^*$-modules are established.

Keywords:  $C^*$-algebras, Hilbert $C^*$-modules, module Banach-Saks properties, module Schur property, self-duality

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