# 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:
\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

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