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 Kgroups. 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 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]. © 2011 Elsevier Inc. All rights reserved