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Double duals and Hilbert modules (2311.15462v1)

Published 27 Nov 2023 in math.OA

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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