Rosenthal operator spaces

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an Lp-space, then it is either a Lp-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non–Hilbertian complemented operator subspaces of non commutative Lp-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p <∞ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence σ we prove that most of these spaces are operator Lp-spaces, not completely isomorphic to previously known such spaces. However it turns out that some column and row versions of our spaces are not operator Lp-spaces and have a rather complicated local structure which implies that the Lindenstrauss–Rosenthal alternative does not carry over to the non-commutative case. Introduction In 1970 Rosenthal [26] constructed new examples of Lp–spaces for every 2 ≤ p < ∞ using probabilistic methods now famous as the Rosenthal inequalities. These methods were later used by Bourgain, Rosenthal and Schechtman [3] to construct an uncountable family of mutually non-isomorphic Lp–spaces. In the framework of operator spaces a theory of operator Lp-spaces, called OLp-spaces, is now being developed, see e.g. [4] and [14]. These are spaces where the operator space structure of the finite dimensional subspaces is determined by a system of finite dimensional non commutative Lp-spaces. If in a given space these Lp-spaces can be chosen to be completely complemented, the space is called a COLp-space. If they can be chosen to be S p ’s (Sp denotes the Schatten p-class), then the space is called an OSp-space and a COSp-space if the S p ’s can be chosen completely complemented. In the present paper we consider some operator space analogues of the Rosenthal sequence spaces, sequence spaces as well as matricial analogues. 02000 Mathematics Subject Classification: 46B20, 46L07, 46L52. ∗Supported by NSF grant DMS–0301116 and DMS 05-56120 †Supported by the Danish Natural Science Research Council, grant 21020436. ‡Supported by NSF grant DMS–0500957