Nearly relatively compact projections in operator algebras

Abstract

Let $A$ be a $C^{\ast }$-algebra, and let $A^{\ast \ast }$ be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in $A^{\ast \ast }$ play the role of open sets, and he used two operator inequalities in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in $A^{\ast \ast }$, but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. The study of all three inequalities can be unified by considering a numerical invariant which is equivalent to the distance of a projection from the set of relatively compact projections. Tomita’s concept of regularity of projections seems relevant, and so we give some results and examples on regularity. We also include a few related results on semicontinuity.