Countable Amenable Identity Excluding Groups

[摘要] A discrete group $G$ is called emph{identity excluding/} if the only irreducibleunitary representation of $G$ which weakly contains the $1$-dimensional identityrepresentation is the $1$-dimensional identity representation itself. Given aunitary representation $pi$ of $G$ and a probability measure $mu$ on $G$, let$P_mu$ denote the $mu$-average $intpi(g) mu(dg)$. The goal of this articleis twofold: (1)~to study the asymptotic behaviour of the powers $P_mu^n$, and(2)~to provide a characterization of countable amenable identity excluding groups.We prove that for every adapted probability measure $mu$ on an identity excludinggroup and every unitary representation $pi$ there exists and orthogonal projection$E_mu$ onto a $pi$-invariant subspace such that $s$-$lim_{noinfty}igl(P_mu^n-pi(a)^nE_muigr)=0$ for every $ainsuppmu$. This also remains true for suitablydefined identity excluding locally compact groups. We show that the class of countableamenable identity excluding groups coincides with the class of $FC$-hypercentralgroups; in the finitely generated case this is precisely the class of groups ofpolynomial growth. We also establish that every adapted random walk on a countableamenable identity excluding group is ergodic.

[学科分类] 数学（综合）

[关键词] congruence lattice;congruence-preserving extension