Families of type III KMS states on a class of C⁎-algebras containing On and QN

We construct a family of purely infinite C⁎-algebras, Qλ for λ∈(0,1) that are classified by their K-groups. There is an action of the circle T with a unique KMS state ψ on each Qλ. For λ=1/n, Q1/n≅On, with its usual T action and KMS state. For λ=p/q, rational in lowest terms, Qλ≅On (n=q−p+1) with UHF fixed point algebra of type ∞(pq). For any n>1, Qλ≅On for infinitely many λ with distinct KMS states and UHF fixed-point algebras. For any λ∈(0,1), Qλ≠O∞. For λ irrational the fixed point algebras, are NOT AF and the Qλ are usually NOT Cuntz algebras. For λ transcendental, K1(Qλ)≅K0(Qλ)≅Z∞, so that Qλ is Cuntz' QN [Cuntz (2008) [16], ]. If λ and λ−1 are both algebraic integers, the only On which appear are those for which . For each λ, the representation of Qλ defined by the KMS state ψ generates a type IIIλ factor. These algebras fit into the framework of modular index theory/twisted cyclic theory of Carey et al. (2010) [8], , Carey et al. (2009) [12], , Carey et al. (in press) [5].