## Banach Journal of Mathematical Analysis

### Nearly relatively compact projections in operator algebras

Lawrence G. Brown

#### Abstract

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

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 259-293.

Dates
Received: 8 December 2016
Accepted: 17 January 2017
First available in Project Euclid: 21 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1505959314

Digital Object Identifier
doi:10.1215/17358787-2017-0033

Mathematical Reviews number (MathSciNet)
MR3779714

Zentralblatt MATH identifier
06873501

Subjects
Primary: 46L05: General theory of $C^*$-algebras
Secondary: 47C15: Operators in $C^*$- or von Neumann algebras

#### Citation

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

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