Operator space embedding of Lq into Lp

The idea of replacing functions by linear operators, the process of quantization, goes back to the foundations of quantum mechanics and has a great impact in mathematics. This applies for instance to representation theory, operator algebra, noncommutative geometry, quantum and free probability or operator space theory. The quantization of measure theory leads to the theory of Lp spaces defined over general von Neumann algebras, so called noncommutative Lp spaces. This theory was initiated by Segal, Dixmier and Kunze in the fifties and continued years later by Haagerup, Fack, Kosaki and many others. We refer to the recent survey [39] for a complete exposition. In this paper we will investigate noncommutative Lp spaces in the language of noncommutative Banach spaces, so called operator spaces. The theory of operator spaces took off in 1988 with Ruan’s work [44]. Since then, it has been developed by Blecher/Paulsen, Effros/Ruan and Pisier as a noncommutative generalization of Banach space theory, see e.g. [4, 29, 34]. In his book [33] on vector valued noncommutative Lp spaces, Pisier considered a distinguished operator space structure on Lp. In fact, the right category when dealing with noncommutative Lp is in many aspects that of operator spaces. Indeed, this has become clear in the last years by recent results on noncommutative martingales and related topics. In this paper, we prove a fundamental structure theorem of Lp spaces in the category of operator spaces, solving a problem formulated by Gilles Pisier.