# 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.

**Keywords:**$C^*$-index theory, {\rm SP}-property, conditional expectation

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