Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một sự bất bình đẳng trừu tượng Kato cho máy phát điện của semigroups tích cực đến các nhà khai thác Banach lattices. | J. OPERATOR THEORY 6 1981 125-133 Copyright by INCREST 1981 K-THEORY FOR ACTIONS OF THE CIRCLE GROUP ON C -ALGEBRAS WILLIAM L. PASCHKE 0. INTRODUCTION M. Pimsner and D. Voiculescu 12 see also 5 3 have recently obtained the following cyclic exact sequence for the K-groups of a c -algebra A and its crossed product B by a single automorphism O . . 0 id _ _ _ Ớ id _ I . ơ 0 1 . Our project here is to extend this important result to the situation in which B is a c -algebra equipped with an action p of the circle group whose spectral subspaces are large see 2 and A is the fixed-point algebra for p. The prototypical case is when B C A 0 and p is the action dual to Ỡ as in 15 . In this more general setting with the additional technical assumption that B has a strictly positive element we show that there is an exact sequence of the form 1 where 0 is a certain automorphism of A K. Here K is as usual the c -algebra of compact operators on separable infinite-dimensional Hilbert space. The automorphism 0 is not of course unique. One way to concoct a suitable 0 is via the spectral subspace Ey x e B pẢ x Vz which we assume to be large in the sense that E E1 and EyE are dense in A. We may use 2 Thm. to associate to E viewed as an A A equivalence bimodule with the obvious left and right A-vaỉueả inner products an automorphism of A K which turns out to have the right effect on K A . Another approach invokes Theorem 2 of 10 to see that A K is isomorphic to the crossed product of B K by p ids whereupon Takai s duality theorem 15 gives B K as the crossed product of A K by the automorphism corresponding to the generator of the action dual to p idK. These two constructions which we show induce the same map onA t t are non-formulaic and involve several arbitrary choices. Computation of Ớ K A thus becomes a problem in spe- cific examples. We determine 0 when .6 is unital in as explicit a fashion as may be WILLIAM L. PASCHKE 126 hoped for and as a consequence obtain the sequence . -
