tailieunhanh - Báo cáo toán học: "Commutant representations of completely bounded maps "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Commutant đại diện của các bản đồ hoàn toàn giới hạn. | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 87 101 COMMUTANT REPRESENTATIONS OF COMPLETELY BOUNDED MAPS VERN I. PAULSEN and CHING YUN SUEN 1. INTRODUCTION Let sd and Sd be c -algebras and let L sd - á be a bounded linear map. If for the maps L X ln sd X Mn - ỔỈ one has that sup L 1 II is finite n then L is called completely bounded and we let II L cb denote this supremum. The map L is called positive provided that p is positive whenever p is positive and is called completely positive if Z 1 is positive for all n. It is well-known that every completely positive map is completely bounded and that for such maps their ordinary norm and cb-norm coincide. Let sd be a c -algebra let be the bounded linear operators on a Hilbert space df and p sd - g dd be a completely positive map. Stinespring s representation theorem 7 asserts that given any such map there is a Hilbert space df a -homomorphism n sđ - d d and a bounded linear operator V de - - df with ợ K F such that p a V n 0 V for all a in sd. Furthermore if a certain minimality condition is imposed on the triple r V dd then this representation is unique up to unitary equivalence. The goal this paper is to attempt to generalize the above theory to the class of completely bounded maps. If L sd - d dd is a completely bounded map then we show in Section 2 that there exists a Hilbert space JT a -homomorphism n sd - S jT a bounded operator V de de and an operator T in the commutant of 7t X such that L a V Tn a V for all a in sd. We show that such representations are not particularly well-behaved with respect to the cb-norm. Indeed if one normalizes the above situation by requiring V to be an isometry then there exist completely bounded maps such that for any such representation r V T dd of L one has ỊT L cb. Analogously if one requires instead that L ob F K then one can find completely bounded maps such that for any such representation r V T dd of L one has II I ll 1. Also it is possible to construct two representa