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Vol.40 (4) 2018 — 2 results found.

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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.

Uniqueness of the Index Map in Real K-theory
C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (4) 2018, pp. 101-103
Ralf Meyer (Received: 2019/03/15, Revised: 2019/04/01)

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The index map in topological K-theory for real Banach algebra extensions is a natural transformation from the first K-theory of the quotient to the zeroth K-theory of the ideal. We show that any such natural transformation is an integer multiple of the index map.

L’application index dans la K-théorie des extensions d’algèbres de Banach réelles est une transformation naturelle entre le premier K-groupe de l’algèbre quotient et le zéroième K-groupe de l’idéal. On démontre qu’une telle transformation naturelle doit être un multiple intégral de l’application index.

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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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