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Báo cáo toán học: "On K_*(C*(SL_2(Z))). (Appendix to "K-theory for certain group C*-algebras" by E. C. Lance) "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: On K_*(C*(SL_2(Z))). (Appendix to "K-theory for certain group C*-algebras" by E. C. Lance) . | J. OPERATOR THEORY 13 1985 103-118 Copyright by INCREST 1985 ON K C SL2 Z APPENDIX TO K-THEORY FOR CERTAIN GROUP C -ALGEBRAS by E. c LANCE TOSHIKAZU NATSUME The purpose of this note is to generalize the result obtained in 4 By J. Cuntz s approach to KK-theory the structure of the proof becomes much clearer. In particular we calculate the K-groups KM. C SL2 Z of the group c -algebra of SL2 Z . 1. PROLOGUE Let G be a countable discrete group and let if be a subgroup of G. Let Ấ denote the unitary representation of G on ỈHGỊH induced from the left multiplication. Definition. The pair G H has property A if there exists a one-parameter family z of unitary representations of G on GfH such that i Ẳo I ii Lfg 5- 0- for every g e G iii Ấ considered as a one-parameter family of representations of C G is a K-homotopy that is for each X e C G Ấ x is a continuous path in Bự GỊHf and Ấ x - I x e J r G H iv Ảt h Ấ í for every h e H. In particular G has property A if G e has property A 4 . Our main result is the following Theorem al Let r G H s be the amalgamated product of countable discrete groups G and s along a subgroup H. Assume that G H has property A. Then for every C -dynamical system A a r there exists a six-term cyclic exact 104 TOSHIKAZU NATSƯME sequence Kou X ỈỈ K0 Ắ X G KaC4 X S Ko X ar r sịỊ e. xỊ xỉ KjG4 xerr KjU Xar G KiM xar S - - KJA x r If where it1 resp. it1 is a natural inclusion of A X r H into A Xjr G resp. A 7. S and E1 resp. is a natural inclusion of A X sr G resp. A xa s into A X .r r. In the case H e Theorem Al coincides with Lance s resuk 4 Theorem 5.4 . It is easy to see that if H is a norma subgroup of G and the quotient group GỊH has property A then G H has property A. Since any countable amenable group has property A 4 Theorem 2.1 if ỈĨ is a normal subgroup of G and GỊH is amenable then G H has property A. It is well-known that SL2 Z s z4 7 Z6. Hence we can apply Theorem AỈ to the group SL2 Z . Since SL2 Z is K-amenable the natural map C SL2 Z C SL3