tailieunhanh - Báo cáo toán học: "Ergodic actions of compact abelian groups "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Ergodic hành động của các nhóm nhỏ gọn abelian. | J. OPERATOR THEORY 3 1980 237-269 Copyright by INCREST 1980 ERGODIC ACTIONS OF COMPACT ABELIAN GROUPS DORTE OLESEN GERT K. PEDERSEN and MASAMICH1 TAKESAKI 1. INTRODUCTION A few years ago Stormer raised the question whether a von Neumann algebra . is finite if it admits an ergodic representation sc G - Aut .y for some compact group G. He showed in 28 that the answer is yes if G is abelian. Well-aimed but unsuccessful attempts at a positive solution of the general problem cf. 10 and I made it clear that the question is hard and the answer may in fact be negative. By contrast the abelian case is relatively straightforward and as we shall show it is possible to give a complete description of the finite von Neumann algebras appearing in the ergodic representations a G - Aut y of some fixed compact abelian group. Given such a von Neumann algebra Ji one first observes that it has a complete set u p I pe G of unitary eigenoperators for the action a and that the eigenspaces are all one-dimensional and span Ji. This means that p - u p is a projective representation of G and we are led to consider the 2-cocyle m p q u p u q u p qY with values in the circle group T. Thus our algebra Ji is a crossed product of G and c over the 2-cocycle m as described by Zeller-Meier in 34 . The next observation is that the group of all unitary eigenoperators is an extension of T by G with the map p - u p as a cross-section for the quotient map of ÍA on G . Classifying group extensions is a well-known exercise in homological algebra the invariants being the elements in the second cohomology group H G T . Since G is discrete and T is a direct summand in any abelian locally compact group in which it is a closed subgroup one may describe H G T as the set X2 G T of symplectic bicharacters Z GxG- T. Alternatively X2 G T is the set of homomorphisms G - G for which z p p 1 for every p in G. Thus the pair Ji a is completely determined up to conjugacy by the corresponding symplectic bicharacter Za .