Abstract

We construct a family of purely infinite C¤algebras, Q¸ for ¸ 2 (0, 1) that are classified by their
Kgroups. There is an action of the circle T with a unique KMS state Ã on each Q¸. For ¸ = 1/n,
Q1/n »= On, with its usual T 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)1. For any n > 1, Q¸ »= On for infinitely
many ¸ with distinct KMS states and UHF fixedpoint algebras. For any ¸ 2 (0, 1), Q¸ 6= O1. For
¸ irrational the fixed point algebras, are NOT AF and the Q¸ are usually NOT Cuntz algebras. For
¸ transcendental, K1(Q¸) »= K0(Q¸) »= Z1, so that Q¸ is Cuntz’ QN, [Cu1]. If ¸ and ¸−1 are both
algebraic integers, the only On which appear are those for which n ´ 3(mod 4). For each ¸, the
representation of Q¸ defined by the KMS state Ã generates a type III¸ factor. These algebras fit into
the framework of modular index theory / twisted cyclic theory of [CPR2, CRT] and [CNNR].