CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

A Computation with the Connes-Thom Isomorphism

  Published:2015-07-20
 Printed: Dec 2015
  • S. Sundar,
    Chennai Mathematical Institute, H1 Sipcot IT Park, , Siruseri, Padur, 603103, Tamilnadu, India
Format:   LaTeX   MathJax   PDF  

Abstract

Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider the semi-direct product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where the action of $\mathbb{Z}$ on $\mathbb{R}^{n}$ is induced by the left multiplication by $A$. Let $(\alpha,\tau)$ be a strongly continuous action of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a $C^{*}$-algebra $B$ where $\alpha$ is a strongly continuous action of $\mathbb{R}^{n}$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\widetilde{\tau}$ on $B \rtimes_{\alpha} \mathbb{R}^{n}$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes-Thom map if $\det(A)\gt 0$ and anticommutes if $\det(A)\lt 0$. As an application, we recompute the $K$-groups of the Cuntz-Li algebra associated to an integer dilation matrix.
Keywords: K-theory, Connes-Thom isomorphism, Cuntz-Li algebras K-theory, Connes-Thom isomorphism, Cuntz-Li algebras
MSC Classifications: 46L80, 58B34 show english descriptions $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Noncommutative geometry (a la Connes)
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
58B34 - Noncommutative geometry (a la Connes)
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/