Banach-compact operators, $\mathcal A$-precompactness, and frames in Hilbert $C^*$-modules

TL;DR

This paper explores different notions of compactness and rank-1 operators in Hilbert $C^*$-modules, providing geometric characterizations and linking these concepts to frames in such modules.

Contribution

It introduces a geometric characterization of Banach-compact operators in Hilbert $C^*$-modules, extending previous notions of $ ext{A}$-compactness and relating them to frames.

Findings

Banach-compact operators are characterized by $ ext{A}$-precompact images of the unit ball.

Total boundedness in a specific uniform structure characterizes these operators.

The work connects operator classes with the concept of frames in Hilbert $C^*$-modules.

Abstract

For a couple $\mathcal M$, $\mathcal N$ of Hilbert $C^*$-modules over a $C^*$-algebra $\mathcal A$, one has two notions of ``$\mathcal A$-rank 1 operators'': $\theta_{x,y}:\mathcal M\to\mathcal N$, $\theta_{x,y}(z)=x\langle y,z\rangle$, where $y,z\in\mathcal M$, $x\in\mathcal N$, (called elementary $\mathcal A$-compact, or elementary Kasparov, operators) and $\theta_{x,f}:\mathcal M\to\mathcal N$, $\theta_{x,f}(z)=xf(z)$, where $z\in\mathcal M$, $x\in\mathcal N$, and $f$ is a bounded $\mathcal A$-functional on $\mathcal M$ (introduced by Manuilov). They generate a $C^*$-bimodule ${\mathbf{K}}(\mathcal M,\mathcal N)$ ($\mathcal A$-compact operators) over the $C^*$-algebras of adjointable operators and a Banach bimodule ${\mathbf{BK}}(\mathcal M,\mathcal N)$ (Banach-compact operators) over the algebras of all bounded morphisms, respectively. In order to give a geometrical characterization of these classes of operators, we introduce the notion of $\mathcal A$-compactness (developing the one introduced by Manuilov). Banach-compact operators can be characterized as those with $\mathcal A$-precompact image of the unit ball. Another obtained characterization is in terms of total boundedness of this set relatively the uniform structure introduced by one of us previously. The constructions and proofs turn out to be closely related to the concept of frame in a Hilbert $C^*$-module.