# Mathematical ReportsComptes rendus mathématiques

Stable rank of depth two inclusions of $C^*$-algebras

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (1) 2007, pp. 28–32

Hiroyuki Osaka, Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan; email: osaka@se.ritsumei.ac.jp

Tamotsu Teruya, College of Science and Engineering, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan; email: teruya@se.ritsumei.ac.jp

Abstract/Résumé:

Let $$1 \in A \subset B$$ be an inclusion of unital $$C^*$$-algebras of index-finite type and depth $$2$$. Suppose that $$A$$ is infinite dimensional, simple, with the property $$\operatorname{SP}$$. We prove that if $$\operatorname{tsr}(A) = 1$$, then $$\operatorname{tsr}(B) \leq 2$$. An interesting special case is $$B = A \rtimes_\alpha G$$, where $$\alpha$$ is an action of a finite group $$G$$ on $$\operatorname{Aut}(A)$$.

Soit $$1 \in A \subset B$$ une inclusion de $$C^*$$-algèbres unitals du type indice-fini et de profondeur $$2$$. On suppose que $$A$$ est de dimension infinie, simple, et que $$A$$ a la propriété $$\operatorname{SP}$$. On démontre que, si $$\operatorname{tsr}(A) = 1$$, donc $$\operatorname{tsr}(B) \leq 2$$. Un cas intéressant est $$B = A \rtimes_\alpha G$$, oú $$\alpha$$ est une action d’un groupe fini $$G$$ sur $$\operatorname{Aut}(A)$$.

Keywords: C*-algebra, property SP, stable rank

AMS Subject Classification: General theory of $C^*$-algebras 46L05

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