Vol.40 (3) 2018 — 2 articles found.

Uniqueness of the Index Map in Banach Algebra K-theory, II

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (3) 2018, pp. 91-100
Vol.40 (3) 2018
George A. Elliott Details
(Received: 2018-09-01 , Revised: 2018-09-01 )
(Received: 2018-09-01 , Revised: 2018-09-01 )

George A. Elliott,Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that the index map in the theory of real Banach algebras is unique as a natural transformation, up to an integral multiple, and modulo a (unique) two-torsion “ghost” map arising from the order-two K\(_1\)-group of the Banach algebra \({\mathbb R}\) (of real numbers). (In the earlier paper this was shown for complex Banach algebras, of course without the “ghost” map, but in way—using Bott periodicity to pass to the opposite parity—that is not available for real Banach algebras. The present approach yields a new proof in the complex case.)

On démontre que l’application index dans la K-théorie des algèbres de Banach réelles (ou complexes) est essentiellment unique.

Keywords: K-theory, index theory

AMS Subject Classification: Index theory, K-theory and operator algebras -including cyclic theory 19K56, 46L80

PDF(click to download): Uniqueness of the Index Map in Banach Algebra K-theory, II

A Survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity

C. R. Math. Rep. Acad. Sci. Canada Vol. 40 (3) 2018, pp. 65-90
Vol.40 (3) 2018
Frederic Latremoliere Details
(Received: 2018-05-05 , Revised: 2018-05-05 )
(Received: 2018-05-05 , Revised: 2018-05-05 )

Frederic Latremoliere,Department of Mathematics, University of Denver, Denver CO 80208; e-mail: frederic@math.du.edu

Abstract/Résumé:

We survey the symmetry preserving properties for the dual propinquity, under natural non-degeneracy and equicontinuity conditions. These properties are best formulated using the notion of the covariant propinquity when the symmetries are encoded via the actions of proper monoids and groups. We explore the issue of convergence of Cauchy sequences for the covariant propinquity, which captures, via a compactness result, the fact that proper monoid actions can pass to the limit for the dual propinquity.

Nous étudions les propriétés de conservation des symmétries des espaces quantiques pour la proximité duale, sous des conditions naturelles d’équicontinuité et de non dégénérescence. Ces propriétés sont exprimées naturellement dans le language de la proximité covariante, qui permet de discuter la convergence d’actions de groupes et semigroupes sur les espaces quantiques. Nous explorons le problème de la convergence des suites de Cauchy pour la proximité covariante, qui capture, grâce a un théoreme de compacité, le fait que les actions de monoides propres passent à la limite pour la proximité duale.

Keywords: C*-dynamical systems, Gromov-Hausdor convergence, Gromov-Hausdor distance for proper monoids, Lip-norms, Monge- Kantorovich distance, Noncommutative metric geometry, proper monoids, quantum metric spaces

AMS Subject Classification: K-theory and operator algebras -including cyclic theory, Noncommutative geometry (__ la Connes) 46L80, 58B34

PDF(click to download): A Survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity