Measures of noncompactness on the standard hilbert C*-module

Abstract

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