Measures of noncompactness on the standard Hilbert $C^{\ast }$-module

Abstract

We define a measure of noncompactness $\lambda$ on the standard Hilbert $C^{\ast }$-module $l^2(\mathcal{A})$ over a unital $C^{\ast }$-algebra, such that $\lambda (E)=0$ if and only if $E$ is $\mathcal{A}$-precompact (i.e.\ it is $\epsilon$-close to a finitely generated projective submodule for any $\epsilon >0$) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istrăţescu 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 $\lambda$.

15 pages