Generalized Measures of Non-precompactness Relative to von Neumann Algebras

Abstract

Suppose M is a von Neumann algebra on a Hilbert space H and J is any norm closed ideal in M. We will characterise J in terms of a generalised measure of non-precompactness on H which, in the case where M is the algebra of all bounded operators on H and J is the ideal of compact operators, reduces to the classical ball measure of non-compactness. Now suppose M is a semifinite von Neumann algebra, τ a faithful semifinite normal trace on M, and M̃ the space of τ-measurable operators. We find generalised measures of non-precompactness on H which are parametrised by the positive half-line and which can be used to characterise the ideal of τ-compact operators.