Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trong nhóm các phần mở rộng liên quan đến một yếu tố semifinite. | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 255-263 ON THE GROUP OF EXTENSIONS RELATIVE TO A SEMIFINITE FACTOR GEORGES SKANDALIS 0. INTRODUCTION In G. G. Kasparov s work on extensions 7 8 an important hypothesis is that the ideal has a countable approximate unit. However extensions of the form A - where TV is a Iloo factor and is the ideal of compact operators of TV seem quite interesting cf. 14 6 2 11 4 . Such extensions give rise to a semi-group Extw 4 . If -4 is separable and nuclear this semi-group is a group 3 . If not consider instead the group Extfl d -1. A natural question is then Does this functor have all the nice properties of the Kasparov Ext X B functor For instance homotopy invariance Bott periodicity Thom isomorphism cf. 8 see also 9 and 5 . Let M be a IIX factor such that N M H where H is a separable Hilbert space . We prove that ExtwC4 -1 Ext y4 M -1 and hence has all these properties. We conclude by some remarks on the KK functor based on the technique used in the proof of our main result. I would like to thank George A. Elliott who aroused my interest in this question. I would also like to thank the people of the Department of Mathematics and Statistics of Queen s University for their warm hospitality especially T. Giordano M Khoshkam E. J. Woods. 1. definitions and notations Let TV be a countably decomposable IIoo factor and let be the closed ideal of compact operators of TV. . We are interested in extensions of the form 0 - - E A - 0. Such an extension of A by is exactly equivalent to a homomorphism A Nj fi. 256 GEORGES SKANDALfS . Such an extension is called trivial if there exists a splitting which is a homomorphism E - A. This is equivalent to saying that the map cp A -A NỊ tỊ admits a lifting 7T which is a -homomorphism A N. . The sum of extensions p p A -A is the extension cp pr A -A- A AÍ2 jV Xa. s V M2 stands for 2 by 2 matrices . . The extensions p and p are said to be unitarily equivalent one writes co p1 .
