tailieunhanh - Báo cáo toán học: "The failure of the slice map criterion "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Sự thất bại của các tiêu chí bản đồ lát. | J. OPERATOR THEORY 8 1982 91-94 Copyright by INCREST 1982 THE FAILURE OF THE SLICE MAP CRITERION B. E. JOHNSON and s. WASSERMANN 1. INTRODUCTION Let A B be von Neumann algebras on Hilbert spaces yr yr- yr g the Hilbert space tensor product A 0 Bc L yr X Jf the algebraic tensor product A B its norm closure and A xj B its weak closure. For each ự e B . idx i z a X b - - ý ỉ a defines a ơ-weakly continuous map A X B - A with a continuous extension A g B A. For p e A the generalized right slice map R A gi B - B is then defined by R x py x e A B Ị e B see 1 . For c 6 A B the map rc p - A c is a map from A the unit ball of y4 with the w -topology into B with the norm topology. If c e A g B then rc is continuous and if at least one of A and B is a finite type I algebra the converse holds. Tomiyama 2 2 has suggested that continuity of rc might be a sufficient condition for c to lie in A B in general and has shown that such a criterion would be extremely convenient. In this note we show that unfortunately the criterion is not generally valid. Specifically we construct an element c of L yr g yr L J X L j with yr 2 N for which rc is continuous but such that c ệ L yT L yr . This construction actually applies to any pair A B of von Neumann algebras neither of which is finite of type I. 2. THE CONSTRUCTION a For neN let yr n be a Hilbert space of dimension n with orthonormal basis 1 . in and let Cjj- 1 z j be matrix units in L such that eijS k 1 i j n . Tr will denote the usual trace on L yr . Let _2- x n 2 E. Cii ii e L yrn yrnY ì X Then x .v eu en so that x 1. 92 B. E. JOHNSON and s. WASSERMANN Let f be the linear functional on L 3 s L 5 B given by - TrBc A- z . Then - ỊẸ 4 i i x ệ i - ij zn 2 ỵ Cl Cl 1 Cl Cl so that sỉ 1. As B xB 1 II II 1. It is well-known and easy to prove that for each g in L 3 y there is an a e such that g y - Tr ay and I gll -- Tr ứ . It follows that the injective cross-norm z on J n 0 Lụ n is given by . z sup i g 11 z g h e L f sup iTrna z a ố ị a b 6 .