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Search: MSC category 58B34 ( Noncommutative geometry (a la Connes) )

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1. CMB Online first

Christ, Michael; Rieffel, Marc A.
Nilpotent group C*-algebras as compact quantum metric spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

2. CMB 2015 (vol 58 pp. 846)

Sundar, S.
A Computation with the Connes-Thom Isomorphism
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
Categories:46L80, 58B34

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