Nearly relatively compact projections in operator algebras
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.
Cite
@article{arxiv.1406.3651,
title = {Nearly relatively compact projections in operator algebras},
author = {Lawrence G. Brown},
journal= {arXiv preprint arXiv:1406.3651},
year = {2018}
}
Comments
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