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Vol.29 (1) 2007 — 3 results found.

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

A relative double commutant theorem for hereditary sub-C*-algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (1) 2007, pp. 22–27
George A. Elliott; Dan Kučerovský (Received: 2007/03/12)

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We prove a double commutant theorem for hereditary subalgebras of a large class of C*-algebras, partially resolving a problem posed by Pedersen. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu proved a C*-algebraic double commutant theorem for separable subalgebras of the Calkin algebra. We prove a similar result for hereditary subalgebras which holds for more general corona C*-algebras. (It is not clear how generally Voiculescu’s double commutant theorem holds.)

Nous démontrons un théorème de commutant double (d’après Voiculescu et von Neumann) pour les sous-C*-algèbres héréditaires d’une C*-algèbre corona, c’est-à-dire de l’algèbre \(M(A)/A\) pour une C*-algèbre \(A\). Les théorèmes de type commutant double ont commencé avec von Neumann, et son résultat séminal est maintenant la fondation de la théorie des algèbres de Neumann. Voiculescu a démontré un théorème de commutant double pour les sous-C*-algèbres séparables de l’algèbre \(B(H)/K(H)\). Nous démontrons un résultat semblable pour les sous-C*-algèbres héréditaires des algèbres \(M(A)/A\). Il n’est pas clair dans quel cadre le théorème de commutant double de Voiculescu est valable en général.

Jet schemes, arc spaces and the Nash problem
C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (1) 2007, pp. 1–21
Shihoko Ishii (Received: 2007/03/26)

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This paper is an introduction to the jet schemes and the arc space of an algebraic variety. We also introduce the Nash problem on arc families.

Ce papier constitue une introduction aux espaces de jets et à l’espace d’arcs d’une variété algébrique. Nous introduisons également le problème de Nash pour les familles d’arcs.

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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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