States of Toeplitz-Cuntz algebras

Abstract

We characterize the state space of a Toeplitz-Cuntz algebra TO_n in terms of positive operator matrices $Ω$ on Fock space which satisfy sl(Ω) \le Ω, where sl(Ω) is the operator matrix obtained from Ωby taking the trace in the last variable. Essential states correspond to those matrices Ωwhich are slice-invariant. As an application we show that a pure essential product state of the fixed-point algebra for the action of the gauge group has precisely a circle of pure extensions to TO_n.