Exotic C*-algebras of geometric groups

TL;DR

This paper introduces new classes of potentially exotic group C*-algebras, establishes their relationships, and applies these results to unitary representation theory, including proofs of classical theorems and partial solutions to longstanding conjectures.

Contribution

It defines and relates new classes of exotic group C*-algebras, generalizes previous results on their distinctness, and applies these findings to unitary representation theory, including proofs of key theorems.

Findings

$C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p o (2,\, ext{infinity})$

$C^*_{L^p}(G)$ are pairwise distinct for certain nonamenable groups

Provides a short proof of Cowling-Haagerup-Howe Theorem and a near solution to Cowling's 1978 conjecture.

Abstract

We consider a new class of potentially exotic group C*-algebras $C^*_{PF_p^*}(G)$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*_{PF_p^*}(G)$. This greatly generalizes earlier results of Okayasu and the second author on the pairwise distinctness of $C^*_{L^p}(G)$ for $2<p<\infty$ when $G$ is either a noncommutative free group or the group $SL(2,\mathbb R)$, respectively. As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group $G$. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of $G$ in the left regular representation of $G$. Also we give a near solution to a 1978 conjecture of Cowling. This conjecture of Cowling states if $G$ is a Kunze-Stein group and $\pi$ is a unitary representation of $G$ with cyclic vector $\xi$ such that the map $$G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$$ belongs to $L^p(G)$ for some $2< p <\infty$, then $A_\pi\subseteq L^p(G)$. We show $B_\pi\subseteq L^{p+\epsilon}(G)$ for every $\epsilon>0$ (recall $A_\pi\subseteq B_\pi$).