Measures of noncompactness on the standard hilbert C*-module

Abstract

We define a measure of noncompactness λ on the standard Hilbert C∗-module l2 (A) over a unital C∗-algebra, such that λ(E) = 0 if and only if E is A-precompact (i.e. it is ε-close to a finitely generated projective submodule for any ε > 0) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istrăţescu measure of noncompactnes on l2 (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 λ.