Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: K-lý thuyết đối với một số C *- đại số. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 101-108 K-THEORY FOR CERTAIN C -ALGEBRAS. II JOACHIM CUNTZ In a remarkable recent paper 7 M. Pimsner and D. Voiculescu established an exact sequence for the K-groups and Ext-groups of crossed product c -algebras that permits to compute these groups for many important c -algebras. The method they used is similar to the one used before in 4 to treat the special case of the algebras ữ . In the present paper we follow more closely the ideas of 4 to give a very simple proof of a result that is somewhat more general than the one of Pimsner and Voiculescu. We consider twisted tensor products by the c -algebras ữ . For n 2 d is the c -algebra generated by n isometries 5 . Sn such that JjSiS 1 cf. 3 . For n 1 we define as the c -algebra generated by a single unitary whose spectrum is the whole unit circle. If sể c ffXlff iff some Hilbert space is a unital c -algebra see for the non-unital case and qi t j . t7 is a family of pairwise commuting unitaries in ff lff . implementing automorphisms a Adĩ7 of si the twisted tensor product si X is the c -subalgebra of X note that is nuclear generated by sđ X 1 together with ƯỊ X sv . Un s . Essentially repeating the proof given in 3 it can be shown that the algebra sđ x n does not depend on the choice of the unitaries ƯỊ implementing the automorphisms a . For the case n 1 this reduces to the ordinary crossed product of sđ by the single automorphism cq. While the crossed product is in some sense a universal c -algebra for which the automorphisms induced by oq on the K-groups becomes trivial the algebra SỈ xv 9n with Uỵ . Un is the universal algebra for which n times these automorphisms become trivial. We obtain an exact sequence for the K-groups of sá X that generalizes the exact sequence of 7 The proof we give uses a homotopy argument from 4 to reduce the problem to a trivial exercise in homological algebra. It should be pointed out that the additional generality gained in .
