Representations of Cuntz algebras associated to quasi-invariant Markov measures

TL;DR

This paper investigates when two quasi-stationary Markov measures on infinite words lead to equivalent or singular representations of Cuntz algebras, linking measure theory with operator algebra representations.

Contribution

It establishes a criterion for the equivalence of Cuntz algebra representations based on the unitary equivalence of associated monic representations derived from Markov measures.

Findings

Criteria for measure equivalence and singularity

Connection between Markov measures and Cuntz algebra representations

Characterization of when representations are unitarily equivalent

Abstract

In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, and the corresponding question of equivalence of associated Cuntz algebra $\O_N$ representations. We do this by associating certain monic representations of $\O_N$ to quasi-stationary Markov measures, and then proving that equivalence for pairs of measures is decided by unitary equivalence of the corresponding pair of representations.