Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: K-nhóm các sản phẩm giảm vượt qua của các nhóm tự do. | J. OPERATOR THEORY 8 1982 131-156 Copyright by INCREST 1982 K-GROUPS OF REDUCED CROSSED PRODUCTS BY FREE GROUPS M. P1MSNER and D. VOICULESCU The starting point for the present paper was a problem of R. V. Kadison about projections in the reduced c -algebra of a free group see 18 . As conjectured by R. V. Kadison we prove that there are no non-trivial projections in the reduced c -algebra of a free group. Our results are in fact more general. For a c -algebra A endowed with an automorphic action of the free group onw generators Fn we obtain a six terms cyclic exact sequence which can be used for the computation of the K-groups of the reduced crossed product of A by F . In particular for C the reduced C -algebra of F we have KoiCT CF z and K1 Cr f Z . Now J. Cohen has proved in 5 see also 4 that there are no non-trivial projections in the full c -algebra of the free group and recently J. Cuntz in 8 has proved that K and Kj of the full c -al-gebra of F are z and respectively Z . Thus our results show that the full group c -algebra and the reduced c -algebra of F have the same K-theory. The main result of the present paper generalizes the exact sequence for crossed products by z which we obtained in 17 A modified proof of the exact sequence for crossed products by z discovered by J. Cuntz 7 has the important feature that it avoids a certain argument about spectral projections in the initial proof. This feature of the proof in 7 turned out to be extremely useful for the generalization to the case of actions of free groups. Though it won t be explicit in our proofs let US also mention that the natural framework for some of the constructions we use in this paper is the general theory of extensions of A of G. G. Kasparov 14 . The paper has three sections. Section 1 contains preliminaries about the Toe-plitz extension. Section 2 contains the main technical facts on which the proof of the exact sequence is based. In Section 3 the main result is obtained by putting together .
