Type III KMS States on a Class of C*-Algebras Containing O-n and Q(N) and Their Modular Index
We construct a family of purely infinite, simple, separable, nuclear C* -algebras, Q\ for,\ E (0, 1). These algebras are also in the class SJlnuc and therefore by results of E. Kirchberg and N. C. Phillips they are classified by their K-groups. There is an action of the circle 1I' with a unique KMS state 'If; on each Q>.. For ,\ = 1/n, Q1fn ~On, with its usual 1I' action and KMS state. For,\= p/q, rational in lowest terms, Q>. ~ On (n = q - p + 1) with UHF fixed point algebra of type (pq) 00 • For any n > 0, Q>. ~ On for infinitely many ,\ with distinct KMS states and UHF fixed-point algebras. However, none of the Q>. is isomorphic to 0 00 . For,\ irrational the fixed point algebras are not AF and the Q>. are usually not Cuntz algebras. For ,\transcendental, K1 (Q>.) ~ K0 (Q>.) ~ Z00 , so that Q>. is Cuntz' QN, [Cul]. If,\ is algebraic (and not rational), then K1 (Q>.) and K0 (Q>.) have the same finite rank: K1 0 Ql ~ Ko 0 Ql ~ Qlk. If ,\ and >..-1 are both algebraic integers, the only On which appear are those for which n = 3(mod4). For each>.., the representation of Q>. defined by the KMS state 'If; generates a type III>. factor. These algebras fit into the framework of the modular index theory/twisted cyclic theory of [CPR2, CRT] and [CNNR].