An invariant of states on Cuntz algebras

Abstract

For an arbitrary state $\omega$ on a Cuntz algebra, we define a number $1\le \kappa (\omega )\le \infty$ such that if the GNS representations of $\omega$ and ${\omega }^{\prime }$ are unitarily equivalent, then $\kappa (\omega )=\kappa ({\omega }^{\prime })$. By using $\kappa$, we define minimal states and it is shown that the classification problem of states is reduced to that of minimal states. By using results of Dutkay, Haussermann, and Jorgensen, we give a sufficient condition of the minimality of a state. Properties of $\kappa$ and examples are shown. As an application, a new invariant of a certain class of endomorphisms of $\mathcal{B}(\mathcal{H})$ is given.

27 pages

References in corpus (1)

• Endomorphisms of B(H), extensions of pure states, and a class of representations of O_n