EMBEDDINGS OF NON - COMMUTATIVE Lp - SPACES INTO NON - COMMUTATIVE L 1 - SPACES , 1 < p < 2 Marius Junge

It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomorphic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative Lp(N, τ)-spaces defined by a finite trace on a finite von Neumann algebra. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting. Introduction and Notation The theory of p-stable processes is a classical tool in probability and analysis and in particular in the theory of Banach spaces. The intention of this paper is to present non-commutative versions of a p-stable process. It is a general phenomenon in non-commutative functional analysis that point sets disappear after quantization. We proceed in a similar way by constructing linear isomorphic or isometric embeddings of a non-commutative Lp-space in the predual of a suitable von Neumann algebra which looks like the integral against a p-stable process when restricted to an arbitrary commutative subalgebra. The theory of embeddings of classical Lp-spaces started with the work of Bretagnolle, Dacuhna-Castelle and Krivine [BrDK] and later Bretagnolle and Dacuhna-Castelle [BrD] between 1966 and 1969. In particular, they found embeddings of Lq-spaces and also Orlicz spaces in Lp-spaces based on the Lévy-Khintchine representation of infinite divisible random variables. Finite dimensional results were obtained with combinatorical tools by Kwapien and Schütt and extended by Schütt and Raynaud [KS1,2], [RS]. All these results motivated the general problem of determining the set of those p’s such that the spaces `p are uniformly represented in a given Banach space. Indeed, due to the fundamental work of Maurey-Pisier [MaP], This research is partially supported by procope 1997/1998.