Classification of sub-Cuntz states

Abstract

Let ${\cal O}_n$ denote the Cuntz algebra for $2\leq n<\infty$. With respect to a homogeneous embedding of ${\cal O}_{n^m}$ into ${\cal O}_n$, an extension of a Cuntz state on ${\cal O}_{n^m}$ to ${\cal O}_n$ is called a sub-Cuntz state, which was introduced by Bratteli and Jorgensen. We show (i) a necessary and sufficient condition of the uniqueness of the extension, (ii) the complete classification of pure sub-Cuntz states up to unitary equivalence of their GNS representations, and (iii) the decomposition formula of a mixing sub-Cuntz state into a convex hull of pure sub-Cuntz states. Invariants of GNS representations of pure sub-Cuntz states are realized as conjugacy classes of nonperiodic homogeneous unit vectors in a tensor-power vector space. It is shown that this state parameterization satisfies both the $U(n)$-covariance and the compatibility with a certain tensor product. For proofs of main theorems, matricizations of state parameters and properties of free semigroups are used.