Author: Hongbing Su
Publisher: American Mathematical Soc.
ISBN: 0821826077
Category : Mathematics
Languages : en
Pages : 98

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In this paper a [italic capital]K-theoretic classification is given of the real rank zero [italic capital]C*-algebras that can be expressed as inductive limits of sequences of finite direct sums of matrix algebras over finite connected graphs (possibly with multiple vertices). The special case that the graphs are circles is due to Elliott.

Author: Hongbing Su
Publisher: National Library of Canada = Bibliothèque nationale du Canada
ISBN: 9780315787506
Category :
Languages : en
Pages :

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Author: Hongbing Su
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Author: Liangqing Li
Publisher: American Mathematical Soc.
ISBN: 0821805967
Category : Mathematics
Languages : en
Pages : 138

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In this paper, it is shown that the simple unital C*-algebras arising as inductive limits of sequences of finite direct sums of matrix algebras over [italic capital]C([italic capital]X[subscript italic]i), where [italic capital]X[subscript italic]i are arbitrary variable trees, are classified by K-theoretical and tracial data. This result generalizes the result of George Elliott of the case of [italic capital]X[subscript italic]i = [0, 1]. The added generality is useful in the classification of more general inductive limit C*-algebras.

Author: Huaxin Lin
Publisher: World Scientific
ISBN: 9814490660
Category : Mathematics
Languages : en
Pages : 333

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The theory and applications of C∗-algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C∗-algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C∗-algebras (up to isomorphism) by their K-theoretical data. It started with the classification of AT-algebras with real rank zero. Since then great efforts have been made to classify amenable C∗-algebras, a class of C∗-algebras that arises most naturally. For example, a large class of simple amenable C∗-algebras is discovered to be classifiable. The application of these results to dynamical systems has been established.This book introduces the recent development of the theory of the classification of amenable C∗-algebras — the first such attempt. The first three chapters present the basics of the theory of C∗-algebras which are particularly important to the theory of the classification of amenable C∗-algebras. Chapter 4 otters the classification of the so-called AT-algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C∗-algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH-algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C∗-algebras. Besides being as an introduction to the theory of the classification of amenable C∗-algebras, it is a comprehensive reference for those more familiar with the subject.

Author: Hongbing Su
Publisher:
ISBN:
Category :
Languages : en
Pages : 83

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Author: K. R. Goodearl
Publisher: American Mathematical Soc.
ISBN: 082182435X
Category : Mathematics
Languages : en
Pages : 161

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Motivated by (i) Elliott's classification of direct limits of countable sequences of finite-dimensional semisimple complex algebras and complex AF C*-algebras, (ii) classical results classifying involutions on finite-dimensional semisimple complex algebras, and (iii) the classification by Handelman and Rossmann of automorphisms of period two on the algebras appearing in (i) we study the real algebras described above and completely classify them, up to isomorphism, Morita equivalence, or stable isomorphism. We also show how our classification easily distinguishes various types of algebras within the given classes, and we partially solve the problem of determining exactly which values are attained by the invariants used in classifying these algebras.

Author: M. Rordam
Publisher: Springer Science & Business Media
ISBN: 3662048256
Category : Mathematics
Languages : en
Pages : 206

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to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. Afactor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II and III. C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum. Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics.

Author: Joan Bosa
Publisher: American Mathematical Soc.
ISBN: 1470434709
Category : C*-algebras
Languages : en
Pages : 97

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The authors introduce the concept of finitely coloured equivalence for unital -homomorphisms between -algebras, for which unitary equivalence is the -coloured case. They use this notion to classify -homomorphisms from separable, unital, nuclear -algebras into ultrapowers of simple, unital, nuclear, -stable -algebras with compact extremal trace space up to -coloured equivalence by their behaviour on traces; this is based on a -coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application the authors calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, -stable -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, the authors derive a “homotopy equivalence implies isomorphism” result for large classes of -algebras with finite nuclear dimension.

Author: Cristian Ivanescu
Publisher: Library and Archives Canada = Bibliothèque et Archives Canada
ISBN: 9780612944046
Category :
Languages : en
Pages : 226

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A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely, the class of simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0, 1] or to a finite disjoint union of closed intervals. In particular, a classification of those stably AI algebras which are inductive limits of hereditary sub-C*-algebras of interval algebras is obtained. Also, the range of the invariant is calculated.