0.2. AMALGAMATED L p SPACES 3 i) If p ≤ q, the norm of x ∈ Lp ( N1 Lq(N2) ) is given by inf α L2r(N1) y Lq(N1 ¯ 2 ) β L2r(N1) x = αyβ . ii) If p ≥ q, the norm of x ∈ Lp ( N1 Lq(N2) ) is given by sup αxβ Lq(N1 ¯ 2 ) α, β ∈ BL 2r (N1) . On the other hand, the row and column subspaces of Lp are defined as follows Lp(M Rn) p = n k=1 xk ⊗ e1k xk ∈ Lp(M) ⊂ Lp ( M ¯ ( 2 ) ) , Lp(M Cp n ) = n k=1 xk ⊗ ek1 xk ∈ Lp(M) ⊂ Lp ( M ¯ ( 2 ) ) , where (eij) denotes the unit vector basis of B( 2 ). These spaces are crucial in the noncommutative Khintchine/Rosenthal type inequalities [26, 37, 40] and in noncommutative martingale inequalities [28, 51, 53], where the row and column spaces are traditionally denoted by Lp(M r 2 ) and Lp(M c 2 ). Now, considering a von Neumann subalgebra N of M with a normal faithful conditional expectation E : M → N , we may define Lp norms of the conditional square functions n k=1 E(xkxk) ∗ 1 2 and n k=1 E(x∗xk) k 1 2 . The expressions E(xkxk) ∗ and E(x∗xk) k have to be defined properly for 1 ≤ p ≤ 2, see [16] or Chapter 1 below. Note that the resulting spaces coincide with the row and column spaces defined above when N is M itself. When n = 1 we recover the spaces Lr(M, p E) and Lc(M, p E), which have been instrumental in proving Doob’s inequality [16], see also [21, 29] for more applications. 0.2. Amalgamated Lp spaces The definition of amalgamated Lp spaces is algebraic. We recall that by H¨ older’s inequality Lu(M)Lq(M)Lv(M) is contractively included in Lp(M) when 1/p = 1/u + 1/q + 1/v. Let us now assume that N is a von Neumann subalgebra of M with a normal faithful conditional expectation E : M → N . Then we have natural isometric inclusions Ls(N ) ⊂ Ls(M) for 0 s ≤ ∞ and we may consider the amalgamated Lp space Lu(N )Lq(M)Lv(N ) as the subset of elements x in Lp(M) which factorize as x = αyβ with α ∈ Lu(N ), y ∈ Lq(M) and β ∈ Lv(N ). The natural “norm” is then given by the following expression x u·q·v = inf α Lu(N ) y Lq(M) β Lv(N ) x = αyβ . However, the triangle inequality for the homogeneous expression u·q·v is by no means trivial. Moreover, it is not clear a priori that this subset of Lp(M) is indeed a linear space. Before explaining these diﬃculties in some detail, let us consider some examples. We fix an integer n ≥ 1 and the subalgebra N embedded in the

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