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

Abstract

We construct a family of purely infinite $C^*$-algebras, $\mathcal{Q}^λ$ for $λ\in (0,1)$ that are classified by their $K$-groups. There is an action of the circle $\T$ with a unique ${\rm KMS}$ state $ψ$ on each $\mathcal{Q}^λ.$ For $λ=1/n,$ $\mathcal{Q}^{1/n}\cong O_n$, with its usual $\T$ action and ${\rm KMS}$ state. For $λ=p/q,$ rational in lowest terms, $\mathcal{Q}^λ\cong O_n$ ($n=q-p+1$) with UHF fixed point algebra of type $(pq)^\infty.$ For any $n>0,$ $\mathcal{Q}^λ\cong O_n$ for infinitely many $λ$ with distinct KMS states and UHF fixed-point algebras. For any $λ\in (0,1),$ $\mathcal{Q}^λ\neq O_\infty.$ For $λ$ irrational the fixed point algebras, are NOT AF and the $\mathcal{Q}^λ$ are usually NOT Cuntz algebras. For $λ$ transcendental, $K_1\cong K_0\cong\Z^\infty$, so that $\mathcal{Q}^λ$ is Cuntz' $\mathcal Q_{\N}$, \cite{Cu1}. If $λ^{\pm 1}$ are both algebraic integers, the {\bf only} $O_n$ which appear satisfy $n\equiv 3(mod 4).$ For each $λ$, the representation of $\mathcal{Q}^λ$ defined by the KMS state $ψ$ generates a type ${\rm III}_λ$ factor. These algebras fit into the framework of modular index (twisted cyclic) theory of \cite{CPR2,CRT} and \cite{CNNR}.