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

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (1) 2007, pp. 28–32
(Received: 2006-07-11 )

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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