Measures of noncompactness on the standard Hilbert $C^*$-module

Abstract

We define a measure of noncompactness $\lambda$ on the standard Hilbert $C^*$-module $l^2(\mathcal A)$ over a unital $C^*$-algebra, such that $\lambda(E)=0$ if and only if $E$ is $\mathcal A$-precompact (i.e.\ it is $\varepsilon$-close to a finitely generated projective submodule for any $\varepsilon>0$) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istr\u{a}\c{t}escu measure of noncomapctnes on $l^2(\mathcal A)$ regarded as a locally convex space with respect to a suitable topology, and obtain their properties as well as some relationship between them and introduced measure of noncompactness $\lambda$.