Compact and ``compact'' ‎‎operators ‎on ‎standard ‎Hilbert ‎modules ‎over ‎‎$‎C^*‎$‎-algebras

Document Type: Original Article


Brace Kovac 66, Ulaz 3, stan 7


‎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 $z\mapsto x\left<y,z\right>$‎, ‎$x,y\in H_{\mathcal{A}}$ (or $x,y\in H_{\mathcal{A}}^{\#}$)) maps bounded sets into totally bounded sets.


Main Subjects