An invariant of states on Cuntz algebras

Abstract

For an arbitrary state $ω$ on a Cuntz algebra, we define a number $1\leq κ(ω)\leq \infty$ such that if the GNS representations of $ω$ and $ω'$ are unitarily equivalent, then $κ(ω)=κ(ω')$. By using $κ$, 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 $κ$ and examples are shown. As an application, a new invariant of a certain class of endomorphisms of ${\cal B}({\cal H})$ is given.