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Ideal Structure of Multiplier Algebras of Simple C*-algebras With Real Rank Zero

Published online by Cambridge University Press:  20 November 2018

Francesc Perera
Francesc Perera*
Affiliation:
Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona Spain, e-mail: perera@mat.uab.es
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Abstract

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We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma $-unital simple ${{C}^{*}}$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mathcal{M}(A)$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mathcal{M}(A)$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mathcal{M}(A)$ is reflected in the fact that $\mathcal{M}(A)$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

Keywords

46L0546L8006F05C*-algebramultiplier algebrareal rank zerostable rankrefinement monoid
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 3 , 01 June 2001 , pp. 592 - 630
Copyright
Copyright © Canadian Mathematical Society 2001

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