Classification of Sub-Cuntz States

Abstract

Let \(\mathcal {O}_{n}\) denote the Cuntz algebra for \(2\leq n<\infty \). With respect to a homogeneous embedding of \(\mathcal {O}_{n^{m}}\) into \(\mathcal {O}_{n}\), an extension of a Cuntz state on \(\mathcal {O}_{n^{m}}\) to \(\mathcal {O}_{n}\) 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.