Abstract

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 Kgroups. 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
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For any n > 0, Q>. ~ On for infinitely many ,\ with distinct KMS states and UHF
fixedpoint 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].