Representations of Higman-Thompson groups from Cuntz algebras

Abstract

Every representation of the Cuntz algebra ${\mathcal{O}}_n$ leads to a unitary representation of the Higman-Thompson group $V_n$. We consider the family $\{{\pi }_x{\}}_{x\in [0,1[}$ of permutative representations of ${\mathcal{O}}_n$ that arise from the interval map $f(x)=nx$ (mod 1) acting on the Hilbert space that underlies each orbit, and then study the unitary equivalence and the irreducibility of the corresponding family $\{{\rho }_x{\}}_{x\in [0,1[}$ of representations of Higman-Thompson group $V_n$, showing that that these representations are indeed irreducible and moreover ${\rho }_x$ and ${\rho }_y$ are equivalent if and only if the orbits of $x$ and $y$ coincide.

Introduction expanded