process

214 J. CuntzProposition 7.4. The K-groups of the algebra F 0 D F Ì Z=2 are given by K 0 .F 0 / DQ ˚ Z and K 1 .F 0 / D 0.Proof. This follows from the fact that F 0 is the inductive limit of the algebras A 0 n andthe description of the maps K 0 .A 0 n / ! K 0.Apn 0 / given in Lemma 7.3 (c). The elementŒf C C Œuf C Œ1 is invariant under the map K 0 .A 0 n / ! K 0.Apn 0 / and maps to thegenerator of Z in the inductive limit.Remark 7.5. Since F 0 is a simple C -algebra with unique trace in a classifiableclass (in the sense of the classification program), this computation of its K-theory incombination with the classification result in [6] shows that F 0 is an AF -algebra. Thuswe obtain an example of a simple AF -algebra F 0 admitting an action by Z=2 suchthat the fixed point algebra is the Bunce–Deddens algebra F and thus not AF . Thisexample had been considered before in [7].In analogy to the case over N we denote by B 0 n the C -subalgebra of Q Z generatedby u; f; s p1 ;:::;s pn . We now immediately deduceTheorem 7.6. We haveandK 0 .B 0 n / D Z2n 1 ; K 1 .B 0 n / D Z2n 1K 0 .Q Z / D Z 1 ; K 1 .Q Z / D Z 1 :Note added in proof. The arguments in the computation of the K-theory are notcomplete. This problem will be addressed in a subsequent paper.References[1] J.-B. Bost and A. Connes, Hecke algebras, type III factors and phase transitions with spontaneoussymmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), 411–457.[2] Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics,Vol. 2, Equilibrium states. Models in quantum statistical mechanics, sec. ed., Texts Monogr.Phys., Springer-Verlag, Berlin 1997.[3] John W. Bunce and James A. Deddens, A family of simple C -algebras related to weightedshift operators, J. Functional Analysis 19(1975), 13–24.[4] Joachim Cuntz, Simple C -algebras generated by isometries, Comm. Math. Phys. 57 (1977),173–185.[5] Ilan Hirshberg, On C -algebras associated to certain endomorphisms of discrete groups,New York J. Math. 8 (2002), 99–109.[6] Xinhui Jiang and Hongbing Su, A classification of simple limits of splitting interval algebras,J. Funct. Anal. 151 (1997), 50–76.[7] A. Kumjian, An involutive automorphism of the Bunce-Deddens algebra, C. R. Math. Rep.Acad. Sci. Canada 10 (1988), 217–218.