An invariant of states on Cuntz algebras

TL;DR

This paper introduces a new invariant for states on Cuntz algebras, enabling classification through minimal states and providing applications to endomorphisms of operator algebras.

Contribution

It defines the invariant ppa, introduces minimal states for classification, and applies these concepts to endomorphisms of operator algebras.

Findings

ppa is well-defined and classifies states up to unitary equivalence.

Conditions for minimality of states are established.

A new invariant for certain endomorphisms is constructed.

Abstract

For an arbitrary state $\omega$ on a Cuntz algebra, we define a number $1\leq \kappa(\omega)\leq \infty$ such that if the GNS representations of $\omega$ and $\omega'$ are unitarily equivalent, then $\kappa(\omega)=\kappa(\omega')$. 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 ${\cal B}({\cal H})$ is given.