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Vol.42 (3) 2020 — 4 results found.

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A Classification of Finite Simple Amenable Z-stable C*-algebras, I: C*-algebras with Generalized Tracial Rank One
C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (3) 2020, pp. 63-450
Guihua Gong; Huaxin Lin; Zhuang Niu (Received: 2020/09/20, Revised: 2021/01/31)

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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.

The Atiyah-Bott Lefschetz Formula Applied to the Based Loops on SU(2)
C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (3) 2020, pp. 42-62
Jack Ding (Received: 2020/07/23, Revised: 2020/10/01)

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The Atiyah-Bott-Lefschetz Formula is a well-known formula for computing the equivariant index of an elliptic operator on a compact smooth manifold. We provide an analogue of this formula for the based loop group \(\Omega SU(2)\) with respect to the natural \((T \times S^1)\)-action. From this result we also derive an effective formula for computing characters of certain Demazure modules.

La formule d’Atiyah-Bott-Lefschetz est une formule bien connue pour l’indice équivariante d’un opérateur elliptique sur une variété lisse compacte. Nous donnons une analogue de cette formule pour le groupe de lacets basés \(\Omega SU(2)\) par rapport à l’action naturelle de \(T \times S^1\). Avec ce résultat nous démontrons aussi une formule effective pour les caractères de certains modules de Demazure.

The Surprising Power of Averaging over Groups
C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (3) 2020, pp. 38-41
James Hogan; Samuel Li (Received: 2020/09/20)

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We highlight the surprising power of averaging via a few illuminating examples. Two of these problems involve characterizations of Hilbert space, and the third is a fundamental result in noncommutative geometry.

Nous soulignons le pouvoir surprenant de la moyenne par quelques exemples éclairants. Deux de ces problèmes concernent la caractérisation de l’espace de Hilbert, et le troisième est un résultat fondamental en géométrie non commutative.

Gaussian Primes
C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (3) 2020, pp. 30-37
J. B. Friedlander (Received: 2020/09/30)

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We survey some of the many interesting questions and results that accrue to the prime numbers which are the sum of two squares.

On survole quelques-uns des plusieurs questions et résultats intéressants qui s’accroissent aux nombres premiers qui sont la somme de deux carrés.

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup cycles of ideals elliptic curves fixed point Fourier transform function fields. 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 noninterlacing numerical range orthogonal polynomials Predual space prime number property SP quadratic forms Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces Weak Markov set Whitney problems

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05C05 11A07 11A55 11B37 11B68 11D09 11D25 11D41 11E04 11F67 11G05 11R09 11R11 13B25 14J26 14M25 14P10 17B37 17B67 19K56 26A51 30C15 30H05 35B 37E10 37E20 37F20 37F25 39B72 42C05 43A07 43A62 46B20 46L05 46L35 46L40 46L55 46L80 47H10 53B25 53C55 54C60 60F10 60J75 83C05

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