Journal of Operator Theory

Volume 46, Issue 1, Summer 2001 pp. 159-171.

SP-property for a pair of

$C^*$ -algebras

Authors: Hiroyuki Osaka
Author institution: Department of Mathematical Sciencees, Ritsumeikan University, Kusatsu, Shiga 525--8577, Japan

Summary: Recall that a

$C^*$ -algebra

$A$ has the {\rm SP}-property if every non-zero hereditary

$C^*$ -subalgebra of

$A$ has a non-zero projection. Let

$1 \in A \subset B$ be a pair of unital

$C^*$ -algebras. In this paper we investigate a sufficient condition for

$B$ to have the {\rm SP}-property, given that

$A$ has it. In particular, if there exists a faithful conditional expectation

$E$ from

$B$ to

$A$ of index-finite type in the sense of Watatani, then

$B$ has the {\rm SP}-property under the condition that

$A$ is simple with the {\rm SP}-property. As an application, we have the structure theory of purely infinite simple

$C^*$ -algebras.

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