Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Sản phẩm đánh giá trạng thái của các đại số CAR bất biến và bất biến luân phiên. | J. OPERATOR THEORY 10 1983 365-393 Copyright by INCREST 1983 PRODUCT STATES OF THE GAUGE INVARIANT AND ROTATIONALLY INVARIANT CAR ALGEBRAS B. M. BAKER and R. T. POWERS 1. INTRODUCTION In this paper we study the restriction of product states of the CAR algebra 21 to the gauge invariant algebra 2Ir and the rotationally invariant algebra 2lG c 2lr. We obtain necessary and sufficient conditions for such states to be factor states classify the factors according to type and obtain necessary and sufficient conditions for the quasi-equivalence of two such states. We also obtain results for the restriction of general factor states of 21 to 2I7 and 2tG. This paper is a continuation of 4 although it can be read with no knowledge of our earlier paper. The main new idea of this paper emerged while we were talking with Vaughan Jones and we wish to acknowledge our debt to him. The idea is this. Suppose 7t is a -representation of 21 induced by a factor state co. Then if co has certain properties the weak closures 7t 2ír and 7t 2IG may be bigger than one would expect. In fact one may have 7i 2l7 7ĩ 2I or 7t 2lG 7r 2lr see Theorems and . Then the analysis of representations of 2IG and 21 reduces to analysis of representations of 21 about which much more is known. This is a great simplification which among other things allows US to classify with regards to type the restriction of product states to 2lr and 2IG see Theorems and . 2. NOTATION AND DEFINITIONS Let á 0 M 2 C be the algebra of all complex 2 x2 -matrices. Set isomorphic to ăỉữ for all p e N 1 2 3 . and 2Im ổỉp isomorphic to p-1 M 2m C for all me N. We denote by 21 the c -algebra obtained from the norm completion of the inductive limit of the 2Im . 21 _J 2Ĩm. This c -algebra m sometimes called the CAR algebra is a UHF algebra of type 2 see 7 . 366 B. M. BAKER and R. T. POWERS Let G SU 2 be the group of all complex unitary 2 X 2 -matrices of determinant one and T be the subgroup of all diagonal .
