Compact and “compact” operators on standard Hilbert modules over $C^*$-algebras

Abstract

We construct a topology on the standard Hilbert module $H_{\mathcal{A}}$ over a unital $C^*$-algebra and topology on $H_ \mathcal {A} ^{＃}$ (the extension of the module $H_{\mathcal{A}}$ by the 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 $z\mapsto x \langle y,z \rangle$, $x,y\in H_{\mathcal{A}}$ (or $x,y \in H_ \mathcal {A} ^{＃}$)) maps bounded sets into totally bounded sets.