Double duals and Hilbert modules

Abstract

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