Operator Spaces and Araki - Woods Factors

We show that the operator Hilbert space OH introduced by Pisier embeds into the predual of the hyerfinite III1 factor. The main new tool is a Khintchine type inequality for the generators of the CAR algebra with respect to a quasi-free state. Our approach yields a Khintchine type inequality for the q-gaussian variables for all values −1 ≤ q ≤ 1. These results are closely related to recent results of Pisier and Shlyakhtenko in the free case. Probabilistic methods play an important role in the theory of operator algebras and Banach spaces. It is not surprising that a quantized theory of Banach spaces will require tools from quantum probability. This connection between noncommutative probability and the recent theory of operator spaces (sometimes called quantized Banach spaces) is well-established through the work of Haagerup, Pisier [HP93] and the general theory of Khintchine type inequalities by Lust-Piquard [LP86], Lust-Piquard and Pisier [LPP91]. The importance of type III von Neumann algebras in this line of research was discovered only recently through the work of Pisier/Shlyahtenko [PS] on Grothendieck's theorem for operator spaces and in [J3]. Both papers make essential use of the theory of free probability. It is well-known that in free probability theory some probabilistic estimates, classically only valid for p < ∞, now hold even for p = ∞ (see e.g. the work of Biane/Speicher [BS98] on stochastic process and Voiculescu's inequality for sums of free random variables [Voi98]). In this paper we prove norm estimates for the sum of independent copies in noncommutative L 1 spaces in a quite general setting. This includes free random variables as in [J3] and also classical commuting or anti-commuting random variables. Using a central limit procedure, similar as in [J3], we derive Khintchine type inequalities for the classical Araki-Wood factors. Although our results are motivated by the theory of operator spaces, the techniques used in the proof are (quantum) probabilistic in nature. Let us fix the notation required to state the main inequality of this paper. Let us assume that M ⊂ M ⊂ N are inclusions of von Neumann algebras and that there is a normal faithful conditional expectation E : N → M. In addition, we will assume that we have automorphisms