Journal of Operator Theory

Volume 42, Issue 1, Summer 1999 pp. 121-144.

States of Toeplitz-Cuntz algebras

Authors: Neal J. Fowler
Author institution: Department of Mathematics, University of Newcastle, Callaghan, NSW 2308, Australia

Summary: We characterize the state space of a Toeplitz-Cuntz algebra $\mathcal{TO}_n$ in terms of positive operator matrices $\Omega$ on Fock space which satisfy ${\rm} sl\Omega \le \Omega$, where ${\rm sl}\Omega$ is the operator matrix obtained from $\Omega$ by taking the trace in the last variable. Essential states correspond to those matrices $\Omega$ which 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 $\mathcal{TO}_n$.

Keywords: Cuntz algebras, Fock space, product states

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