# Mathematical ReportsComptes rendus mathématiques

### property SP — 2 articles found.

On the Property SP of Certain AH Algebras

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (3) 2007, pp. 81–86
Vol.29 (3) 2007
Toan M. Ho Details

Toan M. Ho, Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3; e-mail: toan@mathstat.yorku.ca

Abstract/Résumé:

A certain non-zero projection in a simple AH algebra with diagonal morphisms between the building blocks in its inductive limit decomposition is constructed and used to prove that this algebra has the property SP.

On construit une projection convenable dans une certaine algèbre AH simple, et on l’utilise pour montrer que cette algèbre a la propriété SP.

Keywords: AH algebra, inductive limit, property SP

AMS Subject Classification: None of the above; but in this section 46L99

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

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (1) 2007, pp. 28–32
Vol.29 (1) 2007
Hiroyuki Osaka; Tamotsu Teruya Details

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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