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46L05, General theory of $C^*$-algebras — 22 results found.

      
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Countable Saturation of Corona Algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (2) 2013, pp. 35–56
Ilijas Farah (Received: 2011/12/16, Revised: 2013/04/15)

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We present unified proofs of several properties of the corona of -unital C*-algebras such as AA-CRISP, SAW*, being sub--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 -unitales tel qu’AA-CRISP, SAW*, étant sous--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.

Homomorphisms from the Fredholm Semigroup to Abelian Semigroups
C. R. Math. Rep. Acad. Sci. Canada Vol. 34 (1) 2012, pp. 1–8
George A. Elliott; Brian Skinner (Received: 2010/04/18, Revised: 2011/08/15)

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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.

Quasitraces are traces: A short proof of the finite-nuclear-dimension case
C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (2) 2011, pp. 44–49
Nathanial P. Brown; Wilhelm Winter (Received: 2010/05/12, Revised: 2010/10/21)

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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.

Nuclearity through absorbing extensions
C. R. Math. Rep. Acad. Sci. Canada Vol. 32 (4) 2010, pp. 106–119
Dan Kučerovský; P.W. Ng (Received: 2009/09/11, Revised: 2010/04/15)

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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.

Morita equivalent subalgebras of irrational rotation algebras and real quadratic fields
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (3) 2009, pp. 87–96
Norio Nawata (Received: 2008/10/16)

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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.

AF Embedding of Crossed Products of Certain Graph ${\rm C}^*$-Algebras by Quasi-free Actions
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (3) 2009, pp. 76–86
Xiaochun Fang (Received: 2008/10/24)

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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.

A remark on orthogonality of elements of a C*-algebra
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (3) 2009, pp. 72–75
George A. Elliott (Received: 2008/12/27)

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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.

On AF embeddability of continuous fields
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (1) 2009, pp. 16–19
Marius Dadarlat (Received: 2008/12/18)

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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.

Hilbert modules over a $C^*$-algebra of stable rank one
C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (2) 2007, pp. 48–51
George A. Elliott (Received: 2007/06/29)

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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.

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; Tamotsu Teruya (Received: 2006/07/11)

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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 \(\operatorname{Aut}(A)\).

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

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05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F11 11F66 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K14 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F25 39B72 42C05 43A07 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 83C05

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