NONCOMMUTATIVE L p - SPACE AND OPERATOR SYSTEM

We show that noncommutative Lp-spaces satisfy the axioms of the (nonunital) operator system with a dominating constant 2 1 p . Therefore, noncommutative Lpspaces can be embedded into BpHq 2 1 p -completely isomorphically and complete order isomorphically. 1. Introducton A unital involutive subspace of BpHq had been abstractly characterized by Choi and Effros [CE]. Their axioms are based on observations of the relationship between the unit, the matrix order, and the matrix norm of the unital involutive subspace of BpHq. Indeed, the unit and the matrix order can be used to determine the matrix norm by applying }x} inftλ ¡ 0 : λI ¤ 0 x x 0 ¤ λIu for a unital involutive subspace X of BpHq and x P MnpXq. The abstract characterization of a nonunital involutive subspace of BpHq was completed by Werner [W]. The axioms are based on observations of the relationship between the matrix norm and the matrix order of the involutive subspace of BpHq. Since the unit may be absent, the above equality is replaced by }x} supt|φp 0 x x 0 q| : φ P M2npXq 1, u for an involutive subspace X of BpHq and x P MnpXq. We denote by M2npXq 1, the set of positive contractive functionals on M2npXq. A complex involutive vector space X is called a matrix ordered vector space if for each n P N, there is a set MnpXq € MnpXqsa such that (1) MnpXq X r MnpXq s t0u for all n P N, (2) MnpXq `MmpXq € Mn mpXq for all m,n P N, (3) γ MmpXq γ € MnpXq for each m,n P N and all γ P Mm.,npCq. One might infer from these conditions that MnpXq is actually a cone. An operator space X is called a matrix ordered operator space iff X is a matrix ordered vector space and for every n P N, (1) the -operation is an isometry on MnpXq and (2) the cones MnpXq are closed. 2000 Mathematics Subject Classification. 46L07, 46L52, 47L07.