Nilpotent group C*-algebras as compact quantum metric spaces

Abstract

Let $L$ be a length function on a group $G$, and let $M_L$ denote the operator of pointwise multiplication by $L$ on ${\ell }^2(G)$. Following Connes, $M_L$ can be used as a "Dirac" operator for the reduced group C*-algebra $C_r^{\ast }(G)$. It defines a Lipschitz seminorm on $C_r^{\ast }(G)$, which defines a metric on the state space of $C_r^{\ast }(G)$. We show that for any length function of a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$\ast$ 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.

References in corpus (1)

• Matricial bridges for "Matrix algebras converge to the sphere"

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