Banach Journal of Mathematical Analysis

Nearly relatively compact projections in operator algebras

Lawrence G. Brown


Let A be a C-algebra, and let A be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in A 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, 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.

Article information

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

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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

compact projection regular projection semicontinuous operator


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.

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