0.3. CONDITIONAL L p SPACES 5 inequality for other indices follows by convexity since K is the convex hull of ∂∞K, so that any other point in K is associated to an interpolation space between two spaces living in ∂∞K. Another technical difficulty is the fact that the intersection of two amalgamated Lp spaces is in general quite difficult to describe. Thus, any attempt to use a density argument meets this obstacle. The second step is to prove Theorem A for finite von Neumann algebras, where the intersections are easier to handle. Moreover, most of the factorization arguments (as Szeg¨ o’s theorem) a priori only apply in the finite setting. In the third step we consider general von Neumann algebras using Haagerup’s crossed product construction [12] to approximate σ-finite von Neumann algebras by direct limits of finite von Neumann algebras. Finally, we need a different argument for the case min(q0, q1) = ∞, which is out of the scope of Haagerup’s construction. The main technique here is a Grothendieck-Pietsch version of the Hahn-Banach theorem. Let us observe that in the hyperfinite case Pisier was able to establish many of his results using the Haagerup tensor product. Though similar in nature, we can not directly use tensor product formulas for our interpolation results due to the complicated structure of general von Neumann subalgebras. Theorem A will also be useful in understanding certain interpolation spaces in martingale theory. Let us mention some open problems, for partial results see Chapter 5 below. Problem 4. Let M be a von Neumann algebra and denote by Hp(M) r and Hc(M) p the row and column Hardy spaces of noncommutative martingales over M. Let us consider an interpolation parameter 0 θ 1. (a) Calculate the interpolation norms [Hp(M), r Hp(M)]θ.c (b) If x [Hr(M), 1 Hc(M)]θ, 1 the maximal function is in L1. 0.3. Conditional Lp spaces Once we know which amalgamated Lp spaces are Banach spaces it is natural to investigate their dual spaces. We assume as above that N is a von Neumann subalgebra of M and E : M N is a normal faithful conditional expectation. Let 1/s = 1/u + 1/p + 1/v 1. The conditional Lp space Lp (u,v) (M, E) is defined as the completion of Lp(M) with respect to the norm x Lp (u,v) (M,E) = sup axb Ls(M) a Lu(N ) , b Lv(N ) 1 . In our next result we show that amalgamated and conditional Lp are related by anti-linear duality. This will allow us to translate the interpolation identities in Theorem A in terms of conditional Lp spaces. In this context the correct set of parameters is given by K0 = (1/u, 1/v, 1/q) K 2 u, v ∞, 1 q ∞, 1/u + 1/q + 1/v 1 . Theorem B. Let 1 p given by 1/q = 1/u + 1/p + 1/v, where the indices (u, q, v) belong to the solid K0 and q is conjugate to q. Then, the following isometric isomorphisms hold via the anti-linear duality bracket x, y = tr(x∗y) ( Lu(N )Lq(M)Lv(N ) )∗ = Lp (u,v) (M, E),
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