Double duals and Hilbert modules

2023/11/27 by Lin, Huaxin
#46L05 #46L08 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2311.15462

Abstract

Let A be a C^*-algebra, H be a Hilbert A-module and K(H) be the closure of the set of finite rank module maps. We show that the W^*-algebra of all bounded A^**-module maps on the smallest self-dual Hilbert A^**-module containing H is isomorphic to K(H)^** as W^*-algebras. We also show that the unit ball of H is closed in H^\sharp, the dual of H, in an A-weak topology of H^\sharp as well as dense in the unit ball of H^\sharp in a weak*-topology and some versions of Kaplansky density theorem for Hilbert C^*-modules.

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