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

Charles J. K. Griffin, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4, e-mail: charlie.griffin@mail.utoronto.ca

Abstract/Résumé:

A new proof following a suggestion of Kaplansky to use a result of Dixmier, and in this way avoid unbounded nets of operators, is given of the Kaplansky density theorem.

Une nouvelle démonstration, suivant une suggestion de Kaplansky, d’utiliser un résultat de Dixmier, et dans cette façon d’éviter les familles d’opérateurs non bornées, est donnée du théorème de densité de Kaplansky.

Keywords:

AMS Subject Classification: General theory of $C^*$-algebras, General theory of von Neumann algebras 46L05, 46L10

PDF(click to download): On a Question of Kaplansky Concerning his Density Theorem

Chris Bruce, School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom; e-mail: Chris.Bruce@glasgow.ac.uk

Charles Starling, Carleton University, School of Mathematics and Statistics, 4302 Herzberg Laboratories, Ottawa, ON K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We correct the proof of Theorem 4.1 from [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112].

Nous corrigeons la démonstration du théorème 4.1 dans l’article [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112].

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): Corrigendum to ``A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup'' [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88--112]

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

Mohammad Rouzbehani, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran; e-mail: rouzbehani.m.math@gmail.com

Abstract/Résumé:

We show that a C*-algebra with topological dimension zero is purely infinite if it is weakly purely infinite (a question of Kirchberg and Rørdam). We give an application of this result.

On démontre qu’une C*-algèbre de dimension topologique égale à zéro est purement infinie si elle est faiblement purement infinie (une question de Kirchberg et de Rørdam). On donne une application de ce résultat.

Keywords: (weak) pure infiniteness, C*-algebra, topological dimension zero

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

PDF(click to download): Weakly Purely Infinite C*-algebras with Topological Dimension Zero are Purely Infinite

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

Qingzhai Fan , Department of Mathematics, Shanghai Maritime University, Shangha, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang, Department of Mathematics, Tongji University, Shanghai, China 200092; e-mail: xfang@tongji.edu.cn

Abstract/Résumé:

In this paper, we introduce some classes of generalized tracial approximation C*-algebras. Consider the class of unital C*-algebras which are tracially 𝒵-absorbing (or have tracial nuclear dimension at most n, or have the property SP, or are m-almost divisible). Then A is tracially 𝒵-absorbing (respectively, has tracial nuclear dimension at most n, has the property SP, is weakly (n, m)-almost divisible) for any simple unital C*-algebra A in the corresponding class of generalized tracial approximation C*-algebras. As an application, let A be an infinite-dimensional unital simple C*-algebra, and let B be a centrally large subalgebra of A. If B is tracially 𝒵-absorbing, then A is tracially 𝒵-absorbing. This result was obtained by Archey, Buck, and Phillips in Archey et al. (2018).

On introduit la notion d’approximation traciale généralisée d’une C*-algèbre par des C*-algèbres dans une class donnée. Cette notion généralise la notion de Lin d’approximation triviale simple, et aussi la notion d’Archey et de Phillips de centralement grande sousalgèbre, deux notions qui se sont démontrées très importantes.

Keywords: Cuntz Semigroup, C∗-algebras, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Generalized Tracially Approximated C*-algebras

Charles Starling, Carleton University, School of Mathematics and Statistics. 4302 Herzberg Laboratories, Ottawa ON, K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We prove a new uniqueness theorem for the tight C*-algebras of an inverse semigroup by generalising the uniqueness theorem given for étale groupoid C*-algebras by Brown, Nagy, Reznikoff, Sims, and Williams. We use this to show that in the nuclear and Hausdorff case, a *-homomorphism from the boundary quotient C*-algebra of a right LCM monoid is injective if and only if it is injective on the subalgebra generated by the core submonoid. We also use our result to clarify the identity of the tight C*-algebra of an inverse semigroup we previously associated to a subshift and erroneously identified as the Carlsen-Matsumoto algebra.

Nous prouvons un nouveau thèoréme d’unicité pour les C*-algèbres serrées d’un semi-groupe inverse en généralisant le théorème d’unicité donné pour les C*-algèbres groupoides étales par Brown, Nagy, Reznikoff, Sims et Williams. Nous utilisons ceci pour montrer que dans le cas nucléaire et de Hausdorff, un *-homomorphisme de l’algèbre C* du quotient aux limites d’un monoïde LCM droit est injectif si et seulement s’il est injectif sur la sous-algèbre générée par le sous-monoide de noyau. Nous utilisons également notre résultat pour clarifier l’identité de l’algèbre C* serrée d’un semi-groupe inverse que nous avons précédemment associé à un sous-décalage et identifié à tort comme l’algèbre de Carlsen-Matsumoto.

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup

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

Abstract/Résumé:

It is shown that for a unital C*-algebra, what is sometimes referred to as the Elliott invariant—loosely speaking, K-theory and traces— i.e., the order-unit K\(_0\)-group, the K\(_1\)-group, and the trace simplex, paired in the natural way with K\(_0\), can be expressed purely in terms of K-theory, with the trace simplex and its pairing with K\(_0\) recoverable in a simple way (using polar decomposition) from algebraic K\(_1\), defined as in the purely algebraic context using invertible elements rather than just unitaries.

L’invariant naïf d’Elliott, qui est à la base de la classification complète récente d’une énorme classe de C*-algèbres simples (celles qui sont de dimension nucléaire finie, qui sont séparables, et qui satisfont à l’UCT), peut s’exprimer entièrement dans le cadre de K-théorie algébrique.

Keywords: Algebraic K1-group of a C*-algebra encodes bounded traces

AMS Subject Classification: None of the above; but in this section, $K_0$ as an ordered group; traces, General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 19B99, 19K14, 46L05, 46L35, 46L80

PDF(click to download): K-Theory and Traces

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

Klaus Thomsen, Department of Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark; e-mail: matkt@math.au.dk

Abstract/Résumé:

It is shown that, for any unital simple infinite-dimensional AF algebra, the KMS-state bundle for a one-parameter automorphism group is isomorphic to an arbitrary proper simplex bundle over the real line with (as is necessary) fibre at (inverse temperature) zero isomorphic to the trace simplex.

On démontre que, pour toute C*-algèbre AF simple à élément unité et à dimension infinie, le faisceau d’états KMS pour un grouped’automorphismes à un paramètre est isomorphe à un faisceau de simplices propre arbitraire sur la ligne réelle tel que (nécessairement) le fibre sur la température inverse zéro est isomorphe au simplex tracial.

Keywords: One-parameter automorphism group of an AF algebra, bundle of equilibrium (KMS) states arbitrary

AMS Subject Classification: General theory of $C^*$-algebras, Noncommutative dynamical systems 46L05, 46L55

PDF(click to download): The Bundle of KMS State Spaces for Flows on a Unital AF C*-algebra

Guihua Gong, Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050016, China
and Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00936, USA; e-mail: ghgong@gmail.com

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai 200062, China and
(Current) Department of Mathematics, University of Oregon, Eugene, Oregon, 97402, USA; e-mail: hlin@uoregon.edu

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, WY, USA, 82071; e-mail: zniu@uwyo.edu,

Abstract/Résumé:

A classification theorem is obtained for a class of unital simple separable amenable \({\cal Z}\)-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable \({\cal Z}\)-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank.

Dans cet article et le précédent on donne une classification complète, au moyen de l’invariant d’Elliott, d’une sous-classe de la classe des C*-algèbres simples, moyennables, séparables, à élément unité, absorbant l’algèbre de Jiang-Su, et satisfaisant au UCT, qui épuise l’ensemble des valeurs possibles de l’invariant pour cette class. La partie I réalise une grande partie de ce projet, et la partie II l’achève.

Keywords: Classication of Simple C*-algebras

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): A Classification of Finite Simple Amenable Z-stable C*-algebras, II: C*-algebras with Rational Generalized Tracial Rank One

Guihua Gong, Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050016, China and Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00936, USA; e-mail: ghgong@gmail.com

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai 200062, China and (Current) Department of Mathematics, University of Oregon, Eugene, Oregon, 97402, USA; e-mail: hlin@uoregon.edu

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, WY, USA, 82071; e-mail: zniu@uwyo.edu

Abstract/Résumé:

A class of C*-algebras, to be called those of generalized tracial rank one, is introduced. A second class of unital simple separable amenable C*-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable \({\cal Z}\)-stable C*-algebras. A number of results toward the classification of the second class are presented including an isomorphism theorem for a special sub-class of the first class, leading to the general classification of all unital simple s with rational generalized tracial rank one in Part II.

Dans cet article et le prochain, on donne une classification complète, au moyen de l’invariant d’Elliott, d’une sous-classe de la classe des C*-algèbres simples, moyennables, séparables, à élément unité, absorbant l’algèbre de Jiang-Su, et satisfaisant au UCT, qui épuise l’ensemble des valeurs possibles de l’invariant pour cette class. La partie I réalise une grande partie de ce projet, et la partie II l’achève.

Keywords: Classification of simple C*-algebras

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): A Classification of Finite Simple Amenable Z-stable C*-algebras, I: C*-algebras with Generalized Tracial Rank One

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

Cristian Ivanescu.Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada T5J 4S2; e-mail: IvanescuC@macewan.ca

Dan Kurcerovsky.Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada E38 5A3; e-mail: dkucerov@unb.ca

Abstract/Résumé:

We calculate the Cuntz semigroup of the tensor product of two C\(^*\)-algebras, restricting attention to the case that the Cuntz semigroup, both for the given algebras and for the tensor product, is given by affine functions. We show that the answer is the universal Cuntz category tensor product of Antoine et al. (2018).

On démontre que, dans certains cas, le semigroupe de Cuntz du produit tensoriel de deux C\(^*\)-algèbres est le produit tensoriel dans la catégorie de Cuntz.

Keywords: C*-algebra tensor product, Cuntz Semigroup, tracial cone

AMS Subject Classification: Convex sets in topological linear spaces; Choquet theory, General theory of $C^*$-algebras 46A55, 46L05

PDF(click to download): The Cuntz Semigroup of the Tensor Product of C*-algebras

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

Qingzhai Fan,Department of Mathematics, Shanghai Maritime University, Shanghai, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang,Department of Mathematics, Tongji University, Shangha, China 200092; e-mail: xfang@mail.tongji.edu.cn

Abstract/Résumé:

We show that the following properties of the \({\rm C^*}\)-algebras in a class \(\Omega\) are inherited by simple unital \({\rm C^*}\)-algebras in the class \({\rm TA}\Omega\): \((1)\) \(\beta\)-comparison (\(1\leq \beta < \infty\)), \((2)\) \(n\)-comparison, \((3)\) trace \(\mathcal{Z}\)– absorption, \((4)\) \(m\)-almost divisibility, \((5)\) \((n,m) ~(m\neq 0)\) comparison, and \((6)\) tracial approximate divisibility. As an application, every unital simple \({\rm C^*}\)-algebra with tracial topological rank at most \(k\) has the property of \(k\)-comparison. Also as an application, let \(A\) be an infinite-dimensional simple unital \({\rm C^*}\)-algebra such that \(A\) has one of the above-listed properties. Suppose that \(\alpha: G\to {\rm Aut}(A)\) is an action of a finite group \(G\) on \(A\) which has the tracial Rokhlin property. Then the crossed product \({\rm C^*}\)-algebra \({\rm C^*}( G, A,\alpha)\) also has the property under consideration.

On considère plusieurs propriétés d’une C*-algèbre simple à élément unité qui sont héritées par approximation traciale. Comme application on démontre que ces propriétés sont aussi héritées par la C*-algèbre produit croisé associée à une action d’un groupe fini qui possède la propriété de Rokhlin traciale.

Keywords: C*-algebra, Cuntz Semigroup, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Certain Properties of Tracial Approximation ${C^*}$-Algebras

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

Dickson Wong,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: dickson.wong@mail.utoronto.ca

Abstract/Résumé:

An elementary groupoid construction is shown to underlie Rieffel’s Hilbert module construction of a non-trivial projection in the irrational rotation C*-algebra.

Une construction élémentaire de groupoïde se révèle à la base de la construction de Rieffel à module de Hilbert d’un projecteur non-trivial dans la C*-algèbre d’une rotation irrationnelle.

Keywords: Irrational rotation algebra, Rieffel projection, groupoid construction

AMS Subject Classification: General theory of $C^*$-algebras, K-theory and operator algebras -including cyclic theory, Noncommutative topology 46L05, 46L80, 46L85

PDF(click to download): The Rieffel Projection Via Groupoids

Uffe Haagerup, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark; e-mail: haagerup@math.ku.dk

Abstract/Résumé:

It is shown that all 2-quasitraces on a unital exact \(C^*\)-algebra are traces. As consequences one gets: (1) Every stably finite exact unital \(C^*\)-algebra has a tracial state, and (2) if an \(AW^*\)-factor of type \(II_1\) is generated (as an \(AW^*\)-algebra) by an exact \(C^*\)-subalgebra, then it is a von Neumann \(II_1\)-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and Rørdam to prove that \(RR(A)=0\) for every simple non-commutative torus of any dimension.

On démontre que toute 2-quasitrace sur une C*-algèbre exacte à élément unité est une trace. On en déduit les deux conséquences suivantes: (1) Toute C*-algèbre stablement finie et exacte possède un état tracial, et (2) si un AW*-facteur de type \(II_1\) est engendré (comme AW*-algèbre) par une sous-C*-algèbre exacte, il est une algèbre de von Neumann. Ceci est une solution partielle à un problème bien connu de Kaplansky. Le résultat principal a été utilisé par Blackadar, Kumjian, et Rørdam pour démontrer que \(RR(A) = 0\) pour tout tore non-commutatif simple de dimension quelconque.

Keywords: Quasitraces, classification of C*-algebras, exact C*-algebras

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

PDF(click to download): Quasitraces on Exact C*-algebras are Traces

Jan Cameron, Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, USA; e-mail: jacameron@vassar.edu

Abstract/Résumé:

In this note we show that Kadison’s similarity problem for $C^*$-algebras is equivalent to a problem in perturbation theory: must close $C^*$-algebras have close commutants?

Dans cette note, nous montrons que le problème de similarité de Kadison est équivalent à la question suivante en théorie de la perturbation: les commutants de deux $C^*$-algèbres proches sont-ils nécessairement proches?

Keywords:

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

PDF(click to download): A Remark on the Similarity and Perturbation Problems

Ilijas Farah, Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3, and Matematicki Institut, Kneza Mihaila 34, Belgrade, Serbia; e-mail: ifarah@mathstat.yorku.ca URL: http://www.math.yorku.ca/∼ifarah

Abstract/Résumé:

We present unified proofs of several properties of the corona of $\sigma\$-unital C*-algebras such as AA-CRISP, SAW*, being sub-$\sigma\$-Stonean in the sense of Kirchberg, and the conclusion of Kasparov’s Technical Theorem. Although our results were obtained by considering C*-algebras as models of the logic for metric structures, the reader is not required to have any knowledge of model theory of metric structures (or model theory, or logic in general). The proofs involve analysis of the extent of model-theoretic saturation of corona algebras.

Nous présentons des démonstrations unifiées de plusieurs propriétés de la corona des C*-algèbres $\sigma\$-unitales tel qu’AA-CRISP, SAW*, étant sous-$\sigma\$-Stonean au sens de Kirchberg, et la conclusion du théorème technique de Kasparov. Bien que nos résultats aient été obtenus en considérant les C*-algèbres comme modèles de la logique pour les structures métriques, le lecteur n’est pas requis d’avoir aucune connaissance de la théorie des modèles des structures métriques (ou la théorie des modèles, ou de la logique en général). Les démonstrations impliquent l’analyse de l’ampleur de la saturation modèle-théorétique des algèbres de corona.

Keywords:

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

PDF(click to download): Countable Saturation of Corona Algebras

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

Brian Skinner, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA; e-mail: bskinner@caltech.edu

Abstract/Résumé:

It is shown that the universal enveloping abelian semigroup of the semigroup of Fredholm operators on an infinite-dimensional Hilbert space is the group of integers.

On démontre que l’index de Fredholm est le morphisme universel du semigroupe d’opérateurs de Fredholm sur un espace de Hilbert de dimension infinie dans un semigroupe abélien.

Keywords: Fredholm index, Fredholm operators, Fredholm semigroup relative to a stable C*-algebra, abelian semi-group, semigroup homomorphisms

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

PDF(click to download): Homomorphisms from the Fredholm Semigroup to Abelian Semigroups

Nathanial P. Brown, Department of Mathematics, Penn State University, State College, PA 16802, USA; e-mail: nbrown@math.psu.edu

Wilhelm Winter, School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK; e-mail: wilhelm.winter@nottingham.ac.uk

Abstract/Résumé:

Uffe Haagerup proved that quasitraces on unital exact \(C^*\)-algebras are traces. We give a short proof under the stronger hypothesis of locally finite nuclear dimension; our result generalizes to the case of lower semicontinuous extended quasitraces on nonunital \(C^*\)-algebras.

Uffe Haagerup a démontré qu’une quasi-trace sur une \(C^*\)-algèbre exacte à élément unité est une trace. Nous donnons une courte démonstration sous l’hypothèse plus forte de dimension nucléaire localement finie; ce résultat se généralise jusqu’au cas d’une quasi-trace étendue semicontinue inférieurement sur une \(C^*\)-algèbre sans élément unité.

Keywords:

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

PDF(click to download): Quasitraces are traces: A short proof of the finite-nuclear-dimension case

Dan Kučerovský, Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3; email: dan@math.unb.ca

P.W. Ng, Department of Mathematics, University of Louisiana at Lafayette, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, LA, 70504–1010 USA; email: png@louisiana.edu

Abstract/Résumé:

Let \(\mathcal{A}\) be a unital, separable, simple \(C^*\)-algebra. Denote by \(G := U \bigl( \mathcal{M}(\mathcal{A} \otimes \mathcal{K}) \bigr)\) the unitary group of the multiplier algebra of \(\mathcal{A} \otimes \mathcal{K}\), given the strict topology. Then the following conditions are equivalent:

(1) \(\mathcal{A}\) is a nuclear \(C^*\)-algebra.
(2) \(G\) is an amenable topological group.
(3) \(G\) is an extremely amenable topological group.
(4) The Kasparov extension of \(\mathcal{A} \otimes \mathcal{K}\) is absorbing.
(5) The Lin and Kasparov extensions of \(\mathcal{A} \otimes \mathcal{K}\) are approximately unitarily equivalent (with unitaries coming from \(\mathcal{M}( \mathcal{A} \otimes \mathcal{K})\)).
(6) The Kasparov extension of \(S \mathcal{A} \otimes \mathcal{K}\) is absorbing.
(7) The suspended Lin extension and the Kasparov extension, of \(S \mathcal{A} \otimes \mathcal{K}\), are approximately unitarily equivalent (with unitaries coming from \(\mathcal{M}(S \mathcal{A} \otimes \mathcal{K})\)).
(8) Every purely large extension of \(\mathcal{A} \otimes \mathcal{K}\) is absorbing.
(9) Every properly purely large extension of \(\mathcal{A} \otimes \mathcal{K}\) is absorbing.

Soit \(\mathcal{A}\) une \(C^*\)-algèbre unifère, séparable et simple et soit \(\mathcal{M}(\mathcal{A} \otimes \mathcal{K})\) l’algèbre des multiplicateurs de \(\mathcal{A} \otimes \mathcal{K}\). Dénotons par \(G\) le groupe unitaire de \(\mathcal{M}(\mathcal{A} \otimes \mathcal{K})\) muni de la topologie stricte. Nous démontrons plusieurs caractérisations équivalentes de la nucléarité de \(\mathcal{A}\). En particulier, nous prouvons l’équivalence des conditions suivantes:

(1) \(\mathcal{A}\) est une \(C^*\)-algèbre nucléaire.
(2) \(G\) est un groupe topologique moyennable.
(3) L’extension de Kasparov de \(\mathcal{A} \otimes \mathcal{K}\) est absorbante.

Keywords:

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

PDF(click to download): Nuclearity through absorbing extensions

Norio Nawata, Graduate School of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan; email: n-nawata@math.kyushu-u.ac.jp

Abstract/Résumé:

We determine the isomorphic classes of Morita equivalent subalgebras of irrational rotation algebras. It is based on the solution of the quadratic Diophantine equations. We determine the irrational rotation algebras that have locally trivial inclusions. We compute the index of the locally trivial inclusions of irrational rotation algebras.

Nous déterminons les classes isomorphe de sous-algébres d’algébres de la rotation irrationnelle qui sont Morita-équivalente à l’algébre ambiante. Il est basé sur la solution des équations diophantienne du second degré. Nous déterminons les algébres de la rotation irrationnelle qui ont des inclusions localement triviaux. Nous calculons l’indices des inclusions localement triviaux d’algébres de la rotation irrationnelle.

Keywords: C∗-index theory, Morita equivalence, irrational rotation algebras, real quadratic fields

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

PDF(click to download): Morita equivalent subalgebras of irrational rotation algebras and real quadratic fields

Xiaochun Fang, Department of Mathematics, Tongji University, Shanghai 200092, China; email: xfang@tongji.edu.cn

Abstract/Résumé:

We introduce the notion of quasi-free action of a locally compact abelian group on a graph \({\rm C}^*\)-algebra of a row-finite directed graph, with respect to a labeling of the edges of the graph by elements of the dual group, which we shall call a labeling map. A sufficient condition for AF embedding is given: if the row-finite directed graph is constructed by possibly attaching 1-loops to a row-finite directed graph each weakly connected component of which is a rooted (possibly infinite) directed tree, and the labeling map is almost proper, then the crossed product can be embedded into an AF algebra.

On introduit la notion d’action quasi-libre d’un groupe localement compact abélien sur la \({\rm C}^*\)-algèbre d’un graphe dirigé dont les rangs sont finis, par rapport à un choix d’étiquettes pour les bords du graphe par éléments du groupe dual, qu’on appellera une application d’étiquette. Une condition suffissante pour que la \({\rm C}^*\)-algèbre soit enfoncée dans une \({\rm C}\)-algèbre AF (c’est-à-dire, limite de \({\rm C}^*\)-algèbres de dimesion finie), est donnée, dans laquelle interviennent et le graphe lui-même et l’application d’étiquette.

Keywords: AF embedding, crossed products, graph C∗-algebras

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

PDF(click to download): AF Embedding of Crossed Products of Certain Graph ${rm C}^*$-Algebras by Quasi-free Actions