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

  Published:2001-06-01
 Printed: Jun 2001
  • Francesc Perera
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Abstract

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 $C^\ast$-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 $\mul$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mul$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mul$ is reflected in the fact that $\mul$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.
Keywords: $C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid $C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid
MSC Classifications: 46L05, 46L80, 06F05 show english descriptions General theory of $C^*$-algebras
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Ordered semigroups and monoids [See also 20Mxx]
46L05 - General theory of $C^*$-algebras
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
06F05 - Ordered semigroups and monoids [See also 20Mxx]
 

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