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

摘要

We give a description of the monoid of Murray-von Neumann equivalenceclasses of projections for multiplier algebras of a wide class of$sigma$-unital simple $C^ast$-algebras $A$ with real rank zero and stablerank one. The lattice of ideals of this monoid, which is known to becrucial for understanding the ideal structure of the multiplieralgebra $mul$, is therefore analyzed. In important cases it is shownthat, if $A$ has finite scale then the quotient of $mul$ modulo anyclosed ideal $I$ that properly contains $A$ has stable rank one. Theintricacy of the ideal structure of $mul$ is reflected in the factthat $mul$ can have uncountably many different quotients, each onehaving uncountably many closed ideals forming a chain with respect toinclusion.