Journal of Operator Theory

Volume 82, Issue 2, Fall 2019 pp. 307-355.

Purely infinite corona algebras

Authors: Victor Kaftal (1), P.W. Ng (2), Shuang Zhang (3)
Author institution:(1) Department of Mathematics, University of Cincinnati, P. O. Box 210025, Cincinnati, OH, 45221-0025, U.S.A.
(2) Department of Mathematics, Univ. of Louisiana, 217 Maxim D.~Doucet Hall, P.O. Box 43568, Lafayette, Louisiana, 70504-3568, U.S.A.
(3) Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, OH, 45221-0025, U.S.A.

Summary: Let $\mathcal{A}$ be a simple, $\sigma-$unital, non-unital C*-algebra, with metrizable tracial simplex $\mathcal{T}(\mathcal{A})$, projection surjectivity and injectivity, and strict comparison of positive elements by traces. Then the following are equivalent: $\newcommand{\nr}[1]{\langle#1\rangle}$ $\nr{i} \mathcal{A}$ has quasicontinuous scale; $\nr{ii} \mathcal{M}(\mathcal{A})$ has strict comparison of positive elements by traces; $\nr{iii} \mathcal{M}(\mathcal{A})/\mathcal{A}$ is purely infinite; $\nr{iii'} \mathcal{M}(\mathcal{A})/I_\mathrm{min}$ is purely infinite; $\nr{iv} \mathcal{M}(\mathcal{A})$ has finitely many ideals; $\nr{v} I_\mathrm{min}=I_\mathrm{fin}$. If furthermore $M_n(\mathcal{A})$ has projection surjectivity and injectivity for every $n$, then the above conditions are equivalent to: $\nr{vi} V(\mathcal{M}(\mathcal{A}))$ has finitely many order ideals.

DOI: http://dx.doi.org/10.7900/jot.2018may17.2218
Keywords: multiplier algebras, ideals in multiplier algebras, corona algebras, strict comparison

Contents Full-Text PDF