Banach Journal of Mathematical Analysis
- Banach J. Math. Anal.
- Volume 12, Number 2 (2018), 259-293.
Nearly relatively compact projections in operator algebras
Let be a -algebra, and let be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in 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 , 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.
Banach J. Math. Anal., Volume 12, Number 2 (2018), 259-293.
Received: 8 December 2016
Accepted: 17 January 2017
First available in Project Euclid: 21 September 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 47C15: Operators in $C^*$- or von Neumann algebras
Brown, Lawrence G. Nearly relatively compact projections in operator algebras. Banach J. Math. Anal. 12 (2018), no. 2, 259--293. doi:10.1215/17358787-2017-0033. https://projecteuclid.org/euclid.bjma/1505959314