Double duals and Hilbert modules

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.