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arXiv · 1406.3651

Nearly relatively compact projections in operator algebras

Lawrence G. Brown
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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.

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Lawrence G. Brown. 2014-06-13. Nearly relatively compact projections in operator algebras. https://doi.org/10.1215/17358787-2017-0033

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