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# Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

Published:2016-08-17
Printed: Mar 2017
• Michael Christ,
Department of Mathematics, University of California , Berkeley, CA 94720-3840
• Marc A. Rieffel,
Department of Mathematics, University of California , Berkeley, CA 94720-3840
 Format: LaTeX MathJax PDF

## Abstract

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
 MSC Classifications: 46L87 - Noncommutative differential geometry [See also 58B32, 58B34, 58J22] 20F65 - Geometric group theory [See also 05C25, 20E08, 57Mxx] 22D15 - Group algebras of locally compact groups 53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces 58B34 - Noncommutative geometry (a la Connes)

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