Abstract view
Nilpotent Group C*algebras as Compact Quantum Metric Spaces


Published:20160817
Printed: Mar 2017
Michael Christ,
Department of Mathematics, University of California , Berkeley, CA 947203840
Marc A. Rieffel,
Department of Mathematics, University of California , Berkeley, CA 947203840
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 wordlength functions
on finitely generated nilpotentbyfinite groups.
MSC Classifications: 
46L87, 20F65, 22D15, 53C23, 58B34 show english descriptions
Noncommutative differential geometry [See also 58B32, 58B34, 58J22] Geometric group theory [See also 05C25, 20E08, 57Mxx] Group algebras of locally compact groups Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces Noncommutative geometry (a la Connes)
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)
