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, Q(lambda) for A E (0, 1) that are classified by their K groups. There is an action of the circle T with a unique KMS state on each Q(lambda). For lambda = 1/n, Q(1/n) congruent to O-n, with its usual T action and KMS state. For lambda = p/q, rational in lowest terms, Q(1/n) congruent to O-n (n=q-p+1) with UHF fixed point algebra of type (pq)(infinity). For any n > 1, Q(1/n) congruent to O-n for infinitely many A with distinct KMS states and UHF fixed-point algebras. For any lambda is an element of (0, 1), QA 0 O-infinity. For lambda irrational the fixed point algebras, are NOT AF and the QA are usually NOT Cuntz algebras. For lambda transcendental, K-1 (Q(lambda)) congruent to K-0(Q(lambda))L congruent to Z(infinity), so that Q(lambda) is Cuntz' Q(N) [Cuntz (2008) [16]]. If lambda and lambda(-1) are both algebraic integers, the only O-n which appear are those for which n equivalent to 3 (mod 4). For each lambda, the representation of Q(lambda) defined by the KMS state ip generates psi type III lambda 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]. (C) 2011 Elsevier Inc. All rights reserved.