tailieunhanh - Báo cáo toán học: "Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trình tự chính xác cho K-nhóm và Ext nhóm sản phẩm qua một số C *- đại số. | Copyright by INCREST 1980 J. OPERATOR THEORY 4 1980 93-118 EXACT SEQUENCES FOR A-G ROUPS AND EAT-GROUPS OF CERTAIN CROSS-PRODUCT C -ALGEBRAS M. PIMSNER and D. VOICULESCU Recently M. A. Rieffel began studying Ko and Ext for the irrational rotation c -algebras Ag . the crossed product of the continuous functions on the circle by the automorphism corresponding to a rotation of angle 2n0 where 0 is an irrational number. Also Ko of such algebras appeared as the range of an index map in the context of A. Connes work 8 on operator algebras associated with foliations the irrational rotation algebras corresponding to an extremely simple case the Kro-necker flows on the 2-torus. The irrational rotation algebras have a unique trace state and the results of M. A. Rieflel 28 and of the present authors 26 taken together provided a determination of the range of the homomorphism induced by the trace from Ko into R. On the other hand s. Popa and M. A. Rieffel 27 solved the problem of computing Ext for these algebras. In the present paper we study A-groups and Ext-groups for c -algebras which are crossed products by a single automorphism . the case of an automorphic action of the rational integers. We obtain for the A-groups and Ext-groups six terms exact sequences involving only the groups for the initial algebra and the cross-product algebra. These exact sequences are derived from the cyclic six terms exact sequences of K-theory and respectively Ext-theory applied to what we shall call the Toeplitz-extension associated with a crossed product. For the irrational rotation algebras these results have as an immediate consequence the fact that the homomorphism from Ao into R given by the trace is injective which confirms a fact conjectured by M. A. Rieffel and provides the missing part in the computation of Ao. Also for the same algebras our general results immediately answer the problem of computing A and describe its generators. We show in an Appendix that our present results