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

Abstract

For a pair of Hilbert $C^{\ast }$-modules $\mathcal{M},\mathcal{N}$ over a $C^{\ast }$-algebra $\mathcal{A}$, there are two notions of ``rank-one operators'': ${\theta }_{x,y}:\mathcal{M}\to \mathcal{N}$, ${\theta }_{x,y}(z)=x⟨y,z⟩$, where $y,z\in \mathcal{M}$, $x\in \mathcal{N}$ (these operators are 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 geometric characterization of these classes of operators, we consider the notion of \(\mathcal{A}\)-precompactness (developing the one introduced by Manuilov). Banach-compact operators can be characterized as those with \(\mathcal{A}\)-precompact image of the unit ball. Another characterization obtained here is in terms of total boundedness of this set with respect to the uniform structure introduced by one of us previously. The constructions and proofs turn out to be closely related to the concept of frames in a Hilbert $C^{\ast }$-module.