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

Perera, Francesc

Canadian journal of mathematics, Tome 53 (2001) no. 3, pp. 592-630

Résumé

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.

Détail

DOI : 10.4153/CJM-2001-025-2

Mots-clés : 46L05, 46L80, 06F05, C*-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid

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