Families of type III KMS states on a class of C ^*-algebras containing O _n and Q _{N{double-struck}}

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 (p q) ^∞. 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 O _n which appear are those for which n ≡ 3 (mod 4). 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].