Atomic representations of R. Thompson’s groups and Cuntz’s algebra

Abstract

We continue to study Pythagorean unitary representations of Richard Thompson’s groups $F,T,V$ and their extension to the Cuntz(–Dixmier) algebra $\mathcal{O}$. Any linear isometry from a Hilbert space to its direct sum square produces such. We focus on those arising from a finite-dimensional Hilbert space. We show that they decompose as a direct sum of a so-called diffuse part and an atomic part. We previously proved that the diffuse part is Ind-mixing: it does not contain induced representations of finite-dimensional ones. In this article, we fully describe the atomic part it is a finite direct sum of irreducible monomial representations arising from a precise family of parabolic subgroups.