George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

A remark on orthogonality of elements of a C*-algebra Resume/Abstract: It is shown that any two non-zero hereditary sub-C*-algebras of a C*-algebra that has no minimal projections have approximately orthogonal elements of norm one. (The question of exact orthogonality is left open.)

On démontre que, dans une C*-algèbre sans projecteur minimal, deux sous-C*-algèbres héréditaires qui ne sont pas égales à zéro possèdent des éléments de norme un qui sont approximativement orthogonals.

Keywords: C*-algebra, approximate orthogonality, hereditary subalgebras, orthogonality

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

PDF(click to download): A remark on orthogonality of elements of a C*-algebra

Marius Dadarlat, Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, 47907-2067, U.S.A.; email: mdd@math.purdue.edu

Abstract/Résumé:

Let \(A\) be a separable and exact \(C^*\)-algebra which is a continuous field of \(C^*\)-algebras over a connected, locally connected, compact metrizable space. If at least one of the fibers of \(A\) is AF embeddable, then so is \(A\). As an application we show that if \(G\) is a central extension of an amenable and residually finite discrete group by \(\mathbb{Z}^n\), then the \(C^*\)-algebra of \(G\) is AF embeddable.

Soit A une \(C^*\)-algèbre séparable et exacte qui est un champ continu de \(C^*\)-algèbres sur un espace connexe, localement connexe, compact et metrizable. Si au moins l’une des fibres de \(A\) est embeddable dans une AF algèbre donc la \(C^*\)-algèbre \(A\) est aussi. Comme application, nous montrons que si \(G\) est une extension centrale d’un groupe discret amenable et résiduellement fini par le groupe \(\mathbb{Z}^n\), alors la \(C^*\)-algèbre de \(G\) est embeddable dans une AF algèbre.

Keywords: AF algebras, amenable groups, continuous fields of C*-algebras

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

PDF(click to download): On AF embeddability of continuous fields

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that for countably generated Hilbert C\(^*\)-modules over a C\(^*\)-algebra of stable rank one (i.e., a C\(^*\)-algebra in which the invertible elements are dense) the relation of compact inclusion up to isomorphism is cancellative, in a certain weak but natural sense. This generalizes the well-known fact that cancellation is valid in the abelian semigroup of isomorphism classes of finitely generated projective modules over such a C\(^*\)-algebra.

Il est démontré que la relation d’inclusion compacte entre modules de Hilbert dénombrablement engendrés sur une C\(^*\)-algèbre de rang stable égal à un est cancellative, dans un sens faible mais naturel. Ceci généralise un résultat bien connu pour le cas des modules projectifs finiment engendrés.

Keywords:

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

PDF(click to download): Hilbert modules over a $C^*$-algebra of stable rank one

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

PDF(click to download): Stable rank of depth two inclusions of $C^*$-algebras

Ping Wong Ng, The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario M5T 3J1; email: pwn@erdos.math.unb.ca

Wilhelm Winter, Mathematisches Institut der Universitat Munster, Einsteinstr. 62, D-48149 Munster, Germany; email: wwinter@math.uni-uenster.de

Abstract/Résumé:

We show that finitely generated subhomogeneous \(C^*\)-algebras have finite decomposition rank. As a consequence, any separable ASH \(C^*\)-algebra can be written as an inductive limit of subhomogeneous \(C^*\)-algebras each of which has finite decomposition rank.

It then follows from work of H. Lin and the second author that the class of simple unital ASH algebras which have real rank zero and absorb the Jiang-Su algebra tensorially satisfies the Elliott conjecture.

Nous établissons qu’une \(C^*\)-algèbre sous-homogène engendrée par un nombre fini d’éléments est de rang de décomposition fini. Par conséquence toute ASH \(C^*\)-algèbre peut être décrite comme limite inductive de \(C^*\)-algèbres sous-homogènes chacune desquelles est de rang de dècomposition fini.

Les traveaux de H. Lin et du deuxième auteur nous permettent d’en déduire que la classe d’AHS algèbres à élément unité qui sont simple, de rang réel zéro et qui absorbent tensoriellement l’algèbre de Jiang et Su, satisfait à la conjecture d’Elliott.

Keywords: approximately subhomogeneous C∗-algebras, covering dimension

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

PDF(click to download): A note on subhomogeneous $C^*$-algebras

G.A. Elliott / G. Gong / L. Li

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

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

PDF(click to download): Shape equivalence of AH inductive limit systems: Cutting down by projections