Families of Type {\rm III KMS} States on a Class of $C^{\ast }$-Algebras containing $O_n$ and ${\mathcal{Q}}_{\mathbb{N}}$

Abstract

We construct a family of purely infinite $C^{\ast }$-algebras, ${\mathcal{Q}}^{\hat{I}}»$ for $\hat{I}»\in (0,1)$ that are classified by their $K$-groups. There is an action of the circle $\T$ with a unique $\mathrm{KMS}$ state $\ddot{I}ˆ$ on each ${\mathcal{Q}}^{\hat{I}}».$ For $\hat{I}»=1/n,$ ${\mathcal{Q}}^{1/n}\cong O_n$, with its usual $\T$ action and $\mathrm{KMS}$ state. For $\hat{I}»=p/q,$ rational in lowest terms, ${\mathcal{Q}}^{\hat{I}}»\cong O_n$ ($n=q-p+1$) with UHF fixed point algebra of type $(pq{)}^{\infty }.$ For any $n>0,$ ${\mathcal{Q}}^{\hat{I}}»\cong O_n$ for infinitely many $\hat{I}»$ with distinct KMS states and UHF fixed-point algebras. For any $\hat{I}»\in (0,1),$ ${\mathcal{Q}}^{\hat{I}}»\ne O_{\infty }.$ For $\hat{I}»$ irrational the fixed point algebras, are NOT AF and the ${\mathcal{Q}}^{\hat{I}}»$ are usually NOT Cuntz algebras. For $\hat{I}»$ transcendental, $K_1\cong K_0\cong {\mathbb{Z}}^{\infty }$, so that ${\mathcal{Q}}^{\hat{I}}»$ is Cuntz' ${\mathcal{Q}}_{\mathbb{N}}$, \cite{Cu1}. If $\hat{I}{»}^{\pm 1}$ are both algebraic integers, the {\bf only} $O_n$ which appear satisfy $n\equiv 3(mod4).$ For each $\hat{I}»$, the representation of ${\mathcal{Q}}^{\hat{I}}»$ defined by the KMS state $\ddot{I}ˆ$ generates a type ${\mathrm{III}}_{\hat{I}}»$ factor. These algebras fit into the framework of modular index (twisted cyclic) theory of \cite{CPR2,CRT} and \cite{CNNR}.