Lawrence G. Brown — Page 2

arXiv2007

Limits and C*-algebras of low rank or dimension

We explore various limit constructions for C*-algebras, such as composition series and inverse limits, in relation to the notions of real rank, stable rank, and extremal richness. We also consider extensions and pullbacks. We identify some conditions under which the constructions preserve low rank for the C*-algebras or their multiplier algebras. We also discuss the version of topological dimension theory appropriate for primitive ideal spaces of C*-algebras and provide an analogue for rank of the countable sum theorem of dimension theory. As an illustration of how the main results can be applied, we show that a CCR algebra has stable rank one if and only if it has topological dimension zero or one, and we characterize those sigma-unital CCR algebras whose multiplier algebras have stable rank one or extremal richness. (The real rank zero case was already known.)

math.OA↗

arXiv2007

MASA's and certain type I closed faces of C*-algebras

A result of Akemann, Anderson, and Pedersen states that if a sequence of pure states of a C*-algebra A approaches infinity in a certain sense, then there is a MASA B such that each of the states has the unique extension property with respect to B. We generalize this in two ways: We prove that B can be required to contain an approximate identity of A, and we show that the discrete space which underlies the result cited can be replaced with a totally disconnected space. We consider two special kinds of type I closed faces, both related to the above, atomic closed faces and closed faces with nearly closed extreme boundary. One specific question is whether an atomic closed face always has an "isolated point". We give a counterexample for this and also show that the answer is yes if the the atomic face has nearly closed extreme boundary. We prove a complement to Glimm's theorem on type I C*-algebras which arises from the theory of type I closed faces. One of our examples is a type I closed face which is isomorphic to a closed face of every non-type I separable C*-algebra and which is not isomorphic to a closed face of any type I C*-algebra.

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