Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

Abstract

AbstractLetbe a length function on a group G, and let Mdenote the operator of pointwise multiplication byon l2(G). Following Connes, M𝕃can be used as a “Dirac” operator for the reduced group C*-algebra(G). It deûnes a Lipschitz seminorm on(G), which defines a metric on the state space of(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 ûnitely generated nilpotent-by-finite groups.