TAILIEUCHUNG - Báo cáo toán học: "Interpolation spaces between a von Neumann algebra and its predual "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Không gian nội suy giữa một đại số von Neumann và predual. | Copyright by INCREST 1982 J. OPERATOR THEORY 8 1982 327-360 INTERPOLATION SPACES BETWEEN A VON NEUMANN ALGEBRA AND ITS PREDUAL MARIANNE TERP INTRODUCTION The theory of non-commutative Lp spaces the analogs of ordinary Lebesgue spaces LP X with a non-commutative von Neumann algebra playing the role of L X fj. was first developed for semifinite von Neumann algebras by J. Dix-mier 7 I. E. Segal 20 and R. A. Kunze 18 Much later u. Haagerup presented 12 cf. also 23 a theory of Lp spaces associated with not necessarily semifinite von Neumann algebras. Using 5 M. Hilsum 14 has given a spatial realization of these spaces as spaces of in general unbounded operators on a Hilbert space H on which the von Neumann algebra M acts. This realization depends on the choice of a . weight i r on the commutant M of M. Recently H. Kosaki 17 has shown that one may take still another point of view. Suppose that p is a normal faithful functional on M. Then one may inject M into M. via r - x- p. Now the theory of complex interpolation spaces 3 applies and provides interpolation spaces Cg M 0 0 1. Kosaki shows directly by interpolation theory that these spaces have all the properties that one usually requires for Lp spaces and that they are isomorphic to Haagerup s Lp spaces. fn the present paper we shall investigate this point of view in the case where cp is only supposed to be a weight normal faithful and semifinite . The first difficulty arising from the more general situation that we consider is this we have to find a suitable space in which M and are both continuously embedded. We shall find it convenient to start with the definition of the intersection L of Mand M . The subspace plays a key role in this construction and we have mv s L. We next inject M andM continuously into the dual L of L L being a Banach space when equipped with the maximum of the norms inherited from M and M . The Banach spaces M and M. . are now compatible in the sense of 2 Section so that we can .

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