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46L35, Classifications of $C^*$-algebras; factors — 15 results found.

      
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K-Theory and Traces
C. R. Math. Rep. Acad. Sci. Canada Vol. 44 (1) 2022, pp. 1-15
George A. Elliott (Received: 2021/09/16, Revised: 2021/12/19)

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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.

A Classification of Finite Simple Amenable Z-stable C*-algebras, II: C*-algebras with Rational Generalized Tracial Rank One
C. R. Math. Rep. Acad. Sci. Canada Vol. 42 (4) 2020, pp. 451-539
Guihua Gong; Huaxin Lin; Zhuang Niu (Received: 2020/09/20, Revised: 2021/01/31)

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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.

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.

Certain Properties of Tracial Approximation ${\rm C^*}$-Algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (4) 2018, pp. 104-133
George A. Elliott, FRSC; Qingzhai Fan; Xiaochun Fang (Received: 2019/04/07)

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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.

Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (4) 2015, pp. 121-130
Sam Walters (Received: 2014/10/10, Revised: 2014/10/10)

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We prove that for locally compact, compactly generated self-dual Abelian groups \(G\), there are canonical unitary integral operators on \(L^2(G)\) analogous to the Fourier transform but which have orders 3 and 6. To do this, we establish the existence of a certain projective character on \(G\) whose phase multiplication with the FT gives rise to the Cubic transform (of order 3). (Thus, although the Fourier transform has order 4, one can “make it” have order 3 (or 6) by means of a phase factor!)

Soit \(G\) un groupe localement compact, engendré par un sousensemble compact, et isomorphe à son groupe dual. On construit des operateurs intégrals unitaires canoniques qui sont analogues à la transformée de Fourier, mais qui sont d’ordres trois et six.

Periodic Integral Transforms and Associated Noncommutative Orbifold Projections
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 114-120
Sam Walters (Received: 2014/11/02, Revised: 2015/02/04)

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We report on recent results on the existence of Cubic and Hexic integral transforms on self-dual locally compact groups (orders 3 and 6 analogues of the classical Fourier transform) and their application in constructing a canonical continuous section of smooth projections \(\mathcal E(t)\) of the continuous field of rotation C*-algebras \(\{A_t\}_{0 \le t \le 1}\) that is invariant under the noncommutative Hexic transform automorphism. This leads to invariant matrix (point) projections of the irrational noncommutative tori \(A_\theta\). We also present a quick method for computing the (quantized) topological invariants of such projections using techniques from classical Theta function theory.

On décrit des résultats récents sur l’existence d’une transformation intégrale d’ordre trois (ou d’ordre six) sur un groupe localement compact abélien self-dual. On étudie l’application possible à la construction d’un champs continu de projecteurs invariants sous l’automorphisme associé du champs de C*-algèbres de rotation. On calcule certains invariants topologiques de ces projecteurs.

Topological Obstruction to Approximating the Irrational Rotation C*-algebra by Certain Fourier Invariant C*-subalgebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (3) 2015, pp. 94-99
Sam Walters (Received: 2014/07/10, Revised: 2014/07/10)

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We demonstrate, in a rather quantitative manner, the existence of topological obstructions to approximating the irrational rotation C*-algebra \(A_\theta\) by Fourier invariant unital C*-subalgebras of either of the forms \[M \oplus B \oplus \sigma(B), \qquad M \oplus N \oplus D \oplus \sigma(D) \oplus \sigma^2(D) \oplus \sigma^3(D),\] where \(M, N\) are Fourier invariant matrix algebras (over \(\mathbb C\)), \(B\) is a C*-subalgebra whose unit projection is flip invariant and orthogonal to its Fourier transform, and \(D\) is a C*-subalgebra whose unit projection is orthogonal to its orbit under the Fourier transform. Here, \(\sigma\) is the noncommutative Fourier transform automorphism of \(A_\theta\) defined by \(\sigma(U) = V^{-1},\ \sigma(V)=U\) on the canonical unitary generators \(U,V\) obeying the unitary Heisenberg commutation relation \(VU = e^{2\pi i\theta}UV\).

On montre l’existence d’obstructions topologiques à l’approximation du tore non-commutatif par sous-algèbres de certains types qui sont invariantes sous l’automorphisme de Fourier.

A Classification of Tracially Approximate Splitting Interval Algebras. III. Uniqueness Theorem and Isomorphism Theorem
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (2) 2015, pp. 41-75
Zhuang Niu (Received: 2012/06/26, Revised: 2013/03/26)

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Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

A Classification of Tracially Approximate Splitting Interval Algebras. II. Existence Theorem
C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (1) 2015, pp. 1–32
Zhuang Niu (Received: 2012/01/26, Revised: 2013/03/26)

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Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu ("splitting interval"), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

A Classification of Tracially Approximate Splitting Interval Algebras~~I. The Building Blocks and the Limit Algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 36 (2-3) 2014, pp. 33–66
Zhuang Niu (Received: 2012/06/26, Revised: 2013/03/26)

Show AbstractHide Abstract

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

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Most used Keywords

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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