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

2024/06/05 by Arnaud Brothier, Brothier, Arnaud, Dilshan Wijesena +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2406.02967

Abstract

We continue to study Pythagorean unitary representation of Richard Thompson's groups F,T,V and their extension to the Cuntz(-Dixmier) algebra. 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.

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