Compact and "compact" operators on the standard Hilbert module over a $W^*$ algebra

TL;DR

This paper introduces a new topology on the standard Hilbert module over a unital W*-algebra, ensuring that 'compact' operators map bounded sets into totally bounded sets, enhancing understanding of operator behavior in this context.

Contribution

It constructs a topology on the Hilbert module over a W*-algebra that guarantees 'compact' operators have desirable mapping properties, advancing operator theory in this setting.

Findings

'Compact' operators map bounded sets into totally bounded sets under the new topology

The topology is constructed specifically to analyze the behavior of 'compact' operators

Provides a framework for studying operator compactness in Hilbert modules over W*-algebras

Abstract

We construct a topology on the standard Hilbert module $l^2(\mathcal A)$ over a unital $W^*$-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 $x\mapsto\left<y,x\right>z$) maps bounded sets into totally bounded sets.