Monic representations of the Cuntz algebra and Markov measures

Abstract: We study representations of the Cuntz algebras $\O_N$. While, for fixed $N$, the set of equivalence classes of representations of $\O_N$ is known not to have a Borel cross section, there are various subclasses of representations which can be classified. We study monic representations of $\O_N$, that have a cyclic vector for the canonical abelian subalgebra. We show that $\O_N$ has a certain universal representation which contains all positive monic representations. A large class of examples of monic representations is based on Markov measures. We classify them and as a consequence we obtain that different parameters yield mutually singular Markov measure, extending the classical result of Kakutani. The monic representations based on the Kakutani measures are exactly the ones that have a one-dimensional cyclic $S_i^*$-invariant space.