tailieunhanh - Báo cáo toán học: "Noncommutative integration for states on von Neumann algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Noncommutative hội nhập cho các nước trên đại số von Neumann. | ễ. OPERATOR THEORY 15 1986 217-237 g Copyright by INCREST 1986 N0NC0MMUTAT1VE INTEGRATION FOR STATES ON VON NEUMANN ALGEBRAS CARLO CECCH1N1 1. INTRODUCTION The general theory of noncommutative integration has its model in the theory developed for the particular case in which the von Neumann algebra yz acting on a Hilbert space yp playing the role of the space L Q ỉ p in the abelian case is semifinite while the role of the measure n is given to a normal faithful semifinite . trace tr on . . It has been developed by Segal 13 Kunze 10 Dye 6 Stinespring 16 Nelson 11 and it is remarkably close to the ordinary measure and integration theory. In particular it is possible to represent Ji the predual of. Z as the Banach space called I Ji tr of closed densely defined in general unbounded linear operators on Jf affiliated with . . It is also possible to define interpolation spaces between JI L Jt tr and l Ji tr of closed densely defined operators on. yp affiliated with . .which are called Lp Ji tr for 1 y p -Ị- e . . tr 1 ỵ p the Lp Jl tr norm is given by tr ÍU 1 P. For X 6 tr and y e . tr with- r - 1 their duality coupling F p is given by X Y tr X Y . Among all the other features which can be carried over from the classical case to this situation it is worthwhile to note that LPi X tr n ZA . Z tr is norm dense in lAf. tr z - 1 2 for 1 c Pi ooand weak- dense in LX Ji tr if Pi -j-oo. It would be desirable to obtain a theory as close as possible to this model for the general case of a von Neumann algebra Ji on which a . weight a is defined. The problems however start at the very first step . the representation of the elements of JI the positive part of JI as positive closed densely defined 218 CARLO CECCHINI operators on Ji which in general is not possible. Various theories for the general noncommutative integration theory have been therefore developed whose main difference from our point of view lies in the choice of the features of the .