Nearly relatively compact projections in operator algebras
arXiv:1406.3651 · doi:10.1215/17358787-2017-0033
Abstract
Let A be a C*-algebra and A** its enveloping von Neumann algebra. C. Akemann suggested a kind of non-commutative topology in which certain projections in A** play the role of open sets. The adjectives "open", "closed", "compact", and "relatively compact" all can be applied to projections in A**. Two operator inequalities were used by Akemann in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in A**, 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. It turns out that 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. Since the subject concerns the relation between a projection and its closure, Tomita's concept of regularity of projections seems relevant, and some results and examples on regularity are also given. A few related results on semicontinuity are also included.
This paper was written in 1990. The paper was rejected by two journals in the early '90's,on the grounds that despite being original, non-trivial, and presumably correct, the subject was too specialized and technical. I abandoned further attempts to publish this paper and also refrained from submitting two other papers which were already typed, arXiv #0708.2290 and #1404.2897