Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

Canadian mathematical bulletin

Marc A. Rieffel¹  Michael Christ¹ 

[1] Department of Mathematics, University of California , Berkeley, CA 94720-3840

关键词: group C*-algebra;    Dirac operator;    quantum metric space;    discrete nilpotent group;    polynomial growth;   

DOI  :  10.4153/CMB-2016-040-6

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $mathbb{L}$ be a length function on a group $G$, and let $M_mathbb{L}$denote theoperator of pointwise multiplication by $mathbb{L}$ on $lt(G)$. Following Connes,$M_mathbb{L}$ can be used as a ``Dirac'' operator for the reducedgroup C*-algebra $C_r^*(G)$. It defines aLipschitz seminorm on $C_r^*(G)$, which defines a metric on thestate space of$C_r^*(G)$. We show that for any length function satisfying a strong form of polynomialgrowth on a discrete group,the topology from this metric coincides with theweak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functionson finitely generated nilpotent-by-finite groups.

【 授权许可】

Unknown