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Cuntz Semigroup — 3 results found.

      
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The Cuntz Semigroup of the Tensor Product of C*-algebras
C. R. Math. Rep. Acad. Sci. Canada Vol. 41 (2) 2019, pp. 32-44
George A. Elliott; Cristian Ivanescu; Dan Kucerovsky (Received: 2014/12/08, Revised: 2019/10/24)

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We calculate the Cuntz semigroup of the tensor product of two C\(^*\)-algebras, restricting attention to the case that the Cuntz semigroup, both for the given algebras and for the tensor product, is given by affine functions. We show that the answer is the universal Cuntz category tensor product of Antoine et al. (2018).

On démontre que, dans certains cas, le semigroupe de Cuntz du produit tensoriel de deux C\(^*\)-algèbres est le produit tensoriel dans la catégorie de Cuntz.

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.

The Cuntz Semigroup of Some Spaces of Dimension at Most Two
C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (1) 2013, pp. 22–32
Leonel Robert (Received: 2012/09/12, Revised: 2013/03/26)

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It is shown that the Cuntz semigroup of a space with dimension at most two, and with second cohomology of its compact subsets equal to zero, is isomorphic to the ordered semigroup of lower semicontinuous functions on the space with values in the natural numbers with the infinity adjoined. This computation is then used to obtain the Cuntz semigroup of all compact surfaces. A converse to the first computation is also proven: if the Cuntz semigroup of a separable C*-algebra is isomorphic

Il est montré que le semi-groupe de Cuntz d’un espace de dimension au plus deux, et avec cohomologie deuxième de ses sous-ensembles compacts égales à zéro, est isomorphe au semi-groupe ordonné de fonctions semi-continue inférieurement sur l’espace de baisse avec des valeurs au entiers naturels augmentée à l’infini. Ce calcul est ensuite utilisé pour obtenir le semi-groupe de Cuntz de toutes les surfaces compacts. Un inverse du premier calcul est également prouvé: si le semi-groupe de Cuntz d’un C*-algèbre séparable est isomorphe aux fonctions semi-continue inférieurement de une space topoligique à valeurs dans les entiers naturels augmentée, alors la C*-algébre est commutative à stabilisation près, et son spectrum satisfait aux conditions dimensionnelles et cohomologique mentionné ci-dessus.

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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

Most used AMS

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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