Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Cyclic cohomology của đại số nhóm của các nhóm tự do. | J. OPERATOR THEORY 15 1986 345-357 Copyright by INCREST 1986 CYCLIC COHOMOLOGY OF THE GROUP ALGEBRA OF FREE GROUPS TETSUYA MASUD A Dedicated to Professor o. Takenouchi on his sixtieth birthday 1. INTRODUCTION Let C F- be the reduced group c -algebra of the free group FN with N generators. We have an obvious group homomorphism FN - ZAr so that there is a co-action of Z v on C FN which is T v action. The c -algebra C FN has a canonical normalized trace T which is easily seen to be invariant under the above TA action. The method of A. Connes 2 3 yields a family of 7-traces out of commuting derivations 5j . ỖN and the invariant trace T. In particular T itself is an element of 7 d and pJ j 1 . N are elements of where A is a suitable dense -subalgebra of C FW and fl1 T floỗy ữ1 a0 a1 e A j 1 . N. In view of the fact that the classifying space BFN of the group FN is homotopic to 7V-wedge sum of s the topological projective dimension of the algebra is expected to be one. So we can expect that the spectral sequence associated with the exact couple of A. Connes will be stable after taking Fi-terms. In this paper we shall compute the cyclic cohomology of the group algebra A C FW in terms of explicitly constructed projective resolution. We show 7Fven X CN with the generators above. In Section 2 we give an explicit description of projective resolution. We compute the cyclic cohomology in Section 3. Section 4 and the Appendix will be devoted to the arguments with Frechet topologies. However our discussion does not apply for the ordinary Schwartz group algebra which is known to have the same K-theory as C F v by p. Jolissaint. In Section 5 we shall give a brief discussion. This research is supported by the Educational Project for Japanese Mathematical Scientists. The author would like to express his thanks to Professor H. Araki 346 TETSUYA MASUDA Professor A. Connes Professor M. Takesaki and Dr. T. Natsume for several discussions. This work was carried out during the .
