Compact and “compact” operators on standard Hilbert modules over W∗-algebras

Abstract

We construct a topology on the standard Hilbert module $l^2\left(\mathcal{A}\right)$ over a unital $W^{\ast }$-algebra $\mathcal{A}$ such that any “compact” operator (i.e., any operator in the norm closure of the linear span of the operators of the form $x\mapsto z\langle y,x\rangle$) maps bounded sets into totally bounded sets.