Noncommutative Boyd interpolation theorems

We present a new, elementary proof of Boyd's interpolation theorem. Our approach naturally yields a noncommutative version of this result and even allows for the interpolation of certain operators on l^1-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob's maximal inequality and the dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces.


Introduction
Symmetric Banach function spaces play a pivotal role in many fields of mathematical analysis, especially probability theory, interpolation theory and harmonic analysis. A cornerstone result in the interpolation theory of these spaces is the Boyd interpolation theorem, named after D.W. Boyd. Together with the Calderón-Mitjagin theorem, which characterizes the symmetric Banach function spaces which are an exact interpolation space for the couple (L 1 (R + ), L ∞ (R + )), Boyd's theorem provides an invaluable tool for the analysis of symmetric spaces.
The history of Boyd's interpolation theorem begins with the announcement of Marcinkiewicz [28], shortly before his death, of an extension of the Riesz-Thorin theorem. Let us say that a sublinear operator T is of Marcinkiewicz weak type (p, p) if for any f ∈ L p (R + ), where d(·; T f ) denotes the distribution function of T f . Marcinkiewicz demonstrated that if a sublinear operator T is simultaneously of Marcinkiewicz weak types (p, p) and (q, q) for 1 ≤ p < q ≤ ∞, then T is bounded on L r (R + ), for any p < r < q. A full proof of this result was published years later by Zygmund [36], based on Marcinkiewicz' notes. Soon after it was observed by Stein and Weiss [33] that Marcinkiewicz' result is valid for the larger class of operators which are simultaneously of weak types (p, p) and (q, q). Here T is said to be of weak type (p, p) if for any f in the Lorentz space L p,1 (R + ), The class of sublinear operators which are simultaneously of weak types (p, p) and (q, q) was subsequently characterized by Calderón [6] as consisting of precisely those maps T which satisfy where µ(f ) denotes the decreasing rearrangement of f and S p,q is a linear integral operator which is nowadays known as Calderón's operator. Finally, in [5] Boyd introduced two indices p E and q E for any symmetric Banach function space E on R + and showed that the operator S p,q is bounded on E precisely when p < p E ≤ q E < q. Together with Calderón's characterization, this yields Boyd's interpolation theorem: every sublinear operator which is simultaneously of weak types (p, p) and (q, q) is bounded on E if and only if p < p E ≤ q E < q.
In this paper we are concerned with obtaining a generalization of Boyd's result to noncommutative and, to a limited extent, also noncommutative vector-valued symmetric Banach function spaces. As it turns out, the original approach sketched above remains feasible in the noncommutative setting (see the appendix of this paper), but becomes problematic for noncommutative vector-valued spaces. We develop a new, elementary approach to Boyd's interpolation theorem for the class of Marcinkiewicz weak type operators. Our approach consists of two observations, which are close in spirit to the original approach. Firstly, we characterize the sublinear operators of simultaneous Marcinkiewicz weak types (p, p) and (q, q) as being exactly those which for some α > 0 satisfy the inequality where Θ p,q is the linear operator defined by Secondly, we show that Θ p,q is bounded on E if p < p E ≤ q E < q. Our approach immediately extends to yield both a vector-valued and a noncommutative version of Boyd's result. Moreover, all results are valid for symmetric quasi-Banach function spaces. Thus we obtain Boyd's theorem and its extensions for the full scale of L p -spaces. Interestingly, our method even yields interpolation results for certain operators defined on noncommutative l 1 -and l 2 -valued L p -spaces in the sense of Pisier [31]. In particular, it allows for the interpolation of noncommutative probabilistic inequalities such as the dual Doob inequality, in the noncommutative setting due to Junge [17], and the 'upper' noncommutative Khintchine inequalities, originally due to Lust-Piquard [25,26]. In fact, our approach has its origins in the proof of the Khintchine inequalities for noncommutative symmetric spaces given in [7,8], which the author only later understood as Boyd-type interpolation results.
By adapting the duality argument in Junge's proof of the Doob maximal inequality for noncommutative L p -spaces, we can dualize our noncommutative l 1 -valued interpolation theorem to find an interpolation result for noncommutative maximal inequalities. In particular, we deduce a version of Doob's maximal inequality for a large class of noncommutative symmetric spaces. In the final section we utilize the latter inequality and its dual version to prove Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities, respectively, for noncommutative symmetric spaces. Our results extend the Burkholder-Gundy and Rosenthal inequalities established in [7], as well as the Burkholder-Rosenthal inequalities for noncommutative L p -spaces and Lorentz spaces obtained in [18] and [15], respectively.
During the writing of this manuscript we discovered that an interpolation result for noncommutative Φ-moment inequalities associated with Orlicz functions was proved recently in [1]. We discuss the connection of our work with this result and in fact show that many of our interpolation results have a 'Φ-moment version'.
The paper is organized so that the first part, up to the classical Boyd interpolation theorem, can be read without any knowledge of noncommutative analysis.

Symmetric quasi-Banach function spaces
In this preliminary section we introduce symmetric quasi-Banach function spaces and discuss their most important properties. The results presented below are all well known for Banach function spaces, but not easy to find for quasi-Banach function spaces. We shall need the following well-known result due to T. Aoki and S. Rolewicz, which states that every quasi-normed vector space can be equipped with an equivalent r-norm (see e.g. [21] for a proof).
Theorem 2.1. (Aoki-Rolewicz) Let X be a quasi-normed vector space. Then there is a C > 0 and 0 < r ≤ 1 such that for any x 1 , . . . , x n ∈ X, LetS(R + ) be the linear space of all measurable, a.e. finite functions f on R + . For any f ∈S(R + ) we define its distribution function by for all t > 0.
A (quasi-)normed linear subspace E of S(R + ) is called a (quasi-)Banach function space on R + if it is complete and if for f ∈ S(R + ) and g ∈ E with |f | ≤ |g| we have f ∈ E and f E ≤ g E . A (quasi-)Banach function space E on R + is called symmetric if for f ∈ S(R + ) and g ∈ E with µ(f ) ≤ µ(g) we have f ∈ E and f E ≤ g E . It is called fully symmetric if, in addition, for f ∈ S(R + ) and g ∈ E with f ≺≺ g it follows that f ∈ E and f E ≤ g E .
A symmetric (quasi-)Banach function space is said to have a Fatou (quasi-)norm if for every net The space E is said to have the Fatou property if for every net (f β ) in E satisfying 0 ≤ f β ↑ and sup β f β E < ∞ the supremum f = sup β f β exists in E and f β E ↑ f E . We say that E has order continuous (quasi-)norm if for every net (f β ) in E such that f β ↓ 0 we have f β E ↓ 0. In the literature, a symmetric (quasi-)Banach function space is often called rearrangement invariant if it has order continuous (quasi-)norm or the Fatou property. We shall not use this terminology.
Let us finally discuss some results specific for symmetric Banach function spaces. The Köthe dual of a symmetric Banach function space E is the Banach function space E × given by The space E × is fully symmetric and has the Fatou property. It is isometrically isomorphic to a closed subspace of E * via the map A symmetric Banach function space on R + has a Fatou norm if and only if E embeds isometrically into its second Köthe dual E ×× = (E × ) × . It has the Fatou property if and only if E = E ×× isometrically. It has order continuous norm if and only if it is separable, which is also equivalent to the statement E * = E × . Moreover, a symmetric Banach function space which is separable or has the Fatou property is automatically fully symmetric. For proofs of these facts and more details we refer to [4,23,24].
2.1. Boyd indices. We now discuss the Boyd indices, which were introduced by D.W. Boyd in [5]. For any 0 < a < ∞ we define the dilation operator D a on S(R + ) by (D a f )(s) = f (as) (s ∈ R + ). The following lemma is well known for symmetric Banach function spaces (cf. [23]). Lemma 2.2. Let E be a symmetric quasi-Banach function space on R + . Then, for every 0 < a < ∞, D a defines a bounded linear operator on E. Moreover, a → D a is a decreasing, submultiplicative function on R + .

Proof.
Since µ(f ) is decreasing, we have for any a ≤ b, Hence, if D a is bounded on E, then D b is bounded on E as well and D b ≤ D a . In particular, D a is bounded on E if a ≥ 1 and D a ≤ 1. Moreover, it suffices to show that D 1 n is bounded on E for every n ∈ N. Fix n ∈ N, let f ∈ E + and let f i , 1 ≤ i ≤ n, be mutually disjoint functions having the same distribution function as f . Then D 1 n f and The least constant M (q) for which this inequality holds is called the q-concavity constant of E. It is clear that every quasi-Banach function space is ∞-concave with M (∞) = 1 and any Banach function space is 1-convex with M (1) = 1. For 1 ≤ r < ∞, let the r-concavification and r-convexification of E be defined by respectively. As is shown in [24] (p. 53), if E is a Banach function space, then E (r) is a Banach function space. In general, E (r) is only a quasi-Banach function space. Using that µ(|f | s ) = µ(f ) s for any f ∈ S(R + ) and 0 < s < ∞, one sees that E (r) and E (r) are symmetric if E is symmetric. From the definitions one easily shows that if E is p-convex and q-concave for 0 < p ≤ q ≤ ∞, then E (r) is pr-convex and qr-concave and E (r) is p r -convex and q r -concave. It is also clear from the definitions that We conclude this section by discussing two concrete classes of symmetric quasi-Banach function spaces in more detail.
Example 2.1. (Lorentz spaces L p,q ) Let 0 < p, q ≤ ∞. The Lorentz space L p,q is the subspace of all f in S(R + ) such that is finite. If 1 ≤ q ≤ p < ∞ or p = q = ∞, then L p,q is a fully symmetric Banach function space. If 1 < p < ∞ and p ≤ q ≤ ∞ then L p,q can be equivalently renormed to become a fully symmetric Banach function space ( [4], Theorem 4.6). However, in general L p,q is only a symmetric quasi-Banach function space [20]. By the monotone convergence theorem, L p,q has the Fatou property. Its Boyd indices are determined by the first exponent, p L p,q = q L p,q = p. The Lorentz space L p,p coincides with the Lebesgue space L p . The spaces L p,∞ are referred to as weak L p -spaces.
If we equip L Φ with the Luxemburg norm then L Φ is a symmetric Banach function space with the Fatou property [4,24]. The Boyd indices of L Φ can be computed in terms of Φ. Indeed, let and define the Matuszewska-Orlicz indices by One can show that p Φ = p LΦ and q Φ = q LΦ , see e.g. the proof of [27], Theorem 4.2. For our discussion of Φ-moment inequalities we will need the following results on Orlicz functions. We say that an Orlicz function satisfies the global ∆ 2 -condition if for some constant C > 0, Under this condition we have, for any α ≥ 0, One can show ( [27], Theorem 3.2(b)) that (6) is equivalent to the assumption q Φ < ∞, which in turn holds if and only if Finally, we shall use the following characterization of Boyd's indices for Orlicz spaces ( [27], Theorem 6.4): We refer to [4,23,24] for many more concrete examples of symmetric quasi-Banach function spaces.

Characterization of Marcinkiewicz weak type operators
In this section we establish a key observation, which essentially reduces the proof of Boyd's theorem and its noncommutative extensions to proving a certain inequality for distribution functions, which is stated in Lemma 3.7 below. This observation moreover leads to a characterization of the subconvex operators which are simultaneously of weak types (p, p) and (q, q), see Theorem 3.8.
For 0 < p, q ≤ ∞ we define the functions φ q , ψ p , θ p,q : Here it is understood that φ ∞ = χ (0,1) . Corresponding to these functions we define The following observation is a reformulation of [7], Lemma 4.3.
The corresponding result for the lower Boyd index reads as follows. In the proof and later on, we use χ A to denote the indicator of a set A.
Lemma 3.2. Let E be a symmetric quasi-Banach function space on R + and let Proof. Fix p < p 0 < p E . It clearly suffices to prove , where C and 0 < r ≤ 1 are as in (1). By (4), there is some constant C p0 > 0 such that For the second assertion, notice first that µ(D s (f )) = D s µ(f ) for all 0 < s < ∞ and f ∈ E. Therefore, it suffices to show that there is a constant c > 0 such that for all 0 < s ≤ 1 and Since this holds for any 1 ≤ a < ∞, we conclude that p ≤ p E .
As a result of Lemmas 3.1 and 3.2 we find the following novel expressions for Boyd's indices: Moreover, we have the following result.
The bound for the operator norm of Θ p,q given in the proof of Corollary 3.3 can be improved in specific situations. For example, if 0 < p < r < q ≤ ∞, then one easily calculates that We now compute the distribution function of Φ q (f ), Ψ p (f ) and Θ p,q (f ). The first was already done in [7], Lemma 4.4.
Proof. Using a change of variable, which gives the conclusion. (10) and (11) follow immediately from Lemmas 3.4 and 3.5. On the other hand, for any v > 0 Hence, if d(v; f ) = ∞, then both sides of (10) and (11) are equal to ∞.
Lemma 3.7. Let E be a symmetric quasi-Banach function space on R + . Let α > 0 and f ∈ E + . Suppose that either p < p E ≤ q E < q < ∞ or p < p E and q = ∞ and g ∈ S(R + ) satisfies .
Proof. We take right continuous inverses in (12) to obtain As E is symmetric, it follows from Corollary 3.3 that g ∈ E and moreover, The following result is reminiscent of Calderón's characterization of weak type operators (see Theorem A.1 for a noncommutative extension). Recall that if D is a convex set in S(R + ), then an operator T : D → S(R + ) is called subconvex if for any f, g ∈ D and t ∈ [0, 1] we have if and only if there is some α > 0 such that for all f ∈ S(R + ), Proof. Suppose that (13) holds and fix v > 0. We may assume that d(v; Θ p,q (f )) < ∞, for otherwise there is nothing to prove. By Corollary 3.6 it follows that f χ {f >v} ∈ L p (R + ) and f χ {f ≤v} ∈ L q (R + ). If C p,q = max{C p , C q }, then by subconvexity, By (13) and Corollary 3.6, Suppose now that (14) holds. If q < ∞, then by Corollary 3.6, Since p ≤ q we have On the other hand, if q = ∞, then it is clear that Moreover, for any v > 0 we have This completes the proof.
The following result shows that inequality (12) also implies Φ-moment inequalities.
Lemma 3.9. Let Φ be an Orlicz function on R + which satisfies the global ∆ 2condition. Let α > 0 and f ∈ (L Φ ) + . Suppose that either p < p Φ ≤ q Φ < q < ∞ or p < p Φ and q = ∞ and g ∈ S(R + ) satisfies (12). Then g ∈ L Φ and Proof. Suppose that q Φ < q < ∞. Let λ f denote the pull-back measure on R + associated with f and λ. By corollary 3.6 we can rewrite (12) as Integrating with respect to Φ and using Fubini's theorem yields By (7) and (8), we find Similarly, We conclude that The statement for q = ∞ is proved analogously.
Remark 3.10. From the presented proof it is clear that the result in Lemma 3.9, and hence the Φ-moment inequalities discussed below, remain valid if Φ is nonconvex, provided that it satisfies (6) and (7), and p Φ , q Φ are understood as in (8).
It should be noted that in this case L Φ is in general no longer a quasi-Banach space.
Boyd's interpolation theorem for Marcinkiewicz weak type operators, as well as a Φ-moment version, are now an immediate consequence of the previous observations.
be a subconvex operator of Marcinkiewicz weak types (p, p) and (q, q). If E is a symmetric quasi-Banach function space on R + , and either p < p E ≤ q E < q < ∞ or p < p E and q = ∞ holds, then On the other hand, if Φ is an Orlicz function on R + satisfying the global ∆ 2condition and either p < p Φ ≤ q Φ < q < ∞ or p < p Φ and q = ∞ holds, then Proof. As we have seen in the proof of Theorem 3.8, The assertions now follow from Lemmas 3.7 and 3.9, respectively.
Since k(f ) X = 1, it follows that and hence by the scalar-valued Boyd interpolation theorem, .
Taking g = f X yields

Noncommutative Boyd interpolation theorems
In this section we prove a noncommutative version of Boyd's theorem, Theorem 4.8 below. We first recall some terminology and preliminary results for noncommutative symmetric spaces. Let M be a semi-finite von Neumann algebra acting on a complex Hilbert space H, which is equipped with a normal, semi-finite, faithful trace τ . The distribution function of a closed, densely defined operator x on H, which is affiliated with M, is given by where e |x| is the spectral measure of |x|. The decreasing rearrangement of x is defined by We let S(τ ) be the linear space of all τ -measurable operators, which is a metrizable, complete topological * -algebra with respect to the measure topology. We denote by S 0 (τ ) the linear subspace of all x ∈ S(τ ) such that d(v; x) < ∞ for all v > 0. One can introduce a partial order on the linear subspace S(τ ) h of all self-adjoint operators in S(τ ) by setting, for a self-adjoint operator x, x ≥ 0 if and only if xξ, ξ H ≥ 0 for all ξ ∈ D(x), where D(x) is the domain of x in H. We write x ≤ y for x, y ∈ S(τ ) h if and only if y − x ≥ 0. Under this partial ordering S(τ ) h is a partially ordered vector space. Let S(τ ) + denote the positive cone of all x ∈ S(τ ) h satisfying x ≥ 0. It can be shown that S(τ ) + is closed with respect to the measure topology ( [11], Proposition 1.4).
Throughout our exposition, we will tacitly use many properties of distribution functions and decreasing rearrangements. For the convenience of the reader we collect these facts in the following two propositions. The first result is essentially contained in the proof of [30], Theorem 1.
The following properties of decreasing rearrangements can be found in [13]. If p is a projection in M, then we let p ⊥ := 1 − p denote its orthogonal complement.
Proposition 4.2. If x, y ∈ S(τ ), then: (a) µ t (λx) = |λ|µ t (x) for all λ ∈ C and t ≥ 0; (b) µ s+t (x + y) ≤ µ s (x) + µ t (y) for all s, t ≥ 0; (c) if |x| ≤ |y| then µ t (x) ≤ µ t (y) for all t ≥ 0; (d) µ t (uxv) ≤ u µ t (x) y for all u, v ∈ M and t ≥ 0; For a symmetric (quasi-)Banach function space E on R + , we define We usually denote E(M, τ ) by E(M) for brevity. We call E(M) the noncommutative (quasi-)Banach function space associated with E and M. In the quasi-Banach case these space were first considered by Xu in [35]. The following fundamental result is proved in [22], Theorem 8.11 (see also [11,35] for earlier proofs of this result under additional assumptions). Using the construction above, we obtain noncommutative versions of many important spaces in analysis, such as L p -spaces, weak L p -spaces, Lorentz spaces and Orlicz spaces. For more details on measurable operators we refer to [12,13,30] and for the theory of noncommutative symmetric spaces to [9,10,11,12,22]. We will now proceed to prove the noncommutative version of Boyd's theorem. We first show that the noncommutative symmetric space E(M) is intermediate for the couple (L p (M), L q (M)) if p < p E ≤ q E < q, using the following observation.  4.5. Let 0 < p < q ≤ ∞ and let E be a symmetric quasi-Banach function space R + which is r-convex for some 0 < r < ∞. If 0 < p < p E and either q E < q < ∞ or q = ∞, then To formulate our main result the following definition is convenient. The notion of subconvexity given below weakens the notion of sublinear operators on spaces of measurable operators introduced by Q. Xu (see [14], where it first appeared in published form).
Definition 4.6. Let M and N be von Neumann algebras equipped with normal, semi-finite, faithful traces τ and σ, respectively. Let D be a convex subset of S(τ ).
It is a well-known fact (see e.g. [13], Lemma 4.3) that for any x, y ∈ S(σ) there are partial isometries u, v ∈ N such that |x + y| ≤ u * |x|u + v * |y|v.
For further reference we state Chebyshev's inequality.
Theorem 4.8. Let E be a symmetric quasi-Banach function space on R + which is s-convex for some 0 < s < ∞. Let M, N be von Neumann algebras equipped with normal, semi-finite, faithful traces τ and σ, respectively. Suppose that 0 < p < q ≤ ∞ and let T : L p (M) + + L q (M) + → S(σ) be a midpoint subconvex map such that for some constants C p , C q > 0 depending only on p and q, respectively, If p < p E ≤ q E < q < ∞ or p < p E and q = ∞, then The same result holds if T : L p (M) + + L q (M) + → S(σ) h is a midpoint convex map satisfying (17). (17) is clearly the noncommutative version of Marcinkiewicz weak type (r, r). The reader should be warned, however, that in the noncommutative literature it is nowadays customary to simply refer to this property as 'weak type (r, r)'.

Remark 4.9. Property
Proof. We may assume that max{C p , C q } ≤ 1. By Lemma 4.5 T is well-defined on E(M) + . Let x ∈ E(M) + and let e v = e x [0, v]. By midpoint subconvexity, there exist partial isometries u 1 , u 2 ∈ N such that |T x| ≤ 1 Suppose first that q E < q < ∞. By (16) and (17) we have Therefore, and from Proposition 4.2 it follows that Therefore, by Corollary 3.6, The result now follows from Lemma 3.7, using that d(v; T x) = d(v; µ(T x)). Suppose now that q = ∞. Then (16) and (17) we have and therefore (18) implies that The original version of Boyd's theorem allows for the interpolation of operators of weak-type (p, p), i.e., which are bounded from L p,1 into L p,∞ . Theorem 4.8 only applies for Marcinkiewicz weak type (p, p) operators. In Theorem A.3 in the appendix we will show how to obtain a full noncommutative analogue of Boyd's theorem using a different approach.

Interpolation of noncommutative probabilistic inequalities.
To illustrate the flexibility of the method used to prove Theorem 4.8, we modify it to interpolate several noncommutative probabilistic inequalities. In particular we prove the dual version of Doob's maximal inequality in noncommutative symmetric spaces, see Corollary 4.13 below. The latter result is a consequence of Theorem 4.11, which we will interpret in the following section as an interpolation result for operators on noncommutative l 1 -valued symmetric spaces. For its proof, we shall need the following observation. x ≤ 2(exe + e ⊥ xe ⊥ ).
Proof. By writing x = exe + e ⊥ xe + exe ⊥ + e ⊥ xe ⊥ , we see that the asserted inequality is equivalent to  . Let E be a symmetric quasi-Banach function space on R + which is s-convex for some 0 < s < ∞. Let M, N be von Neumann algebras equipped with normal, semi-finite, faithful traces τ and σ, respectively. Suppose that 0 < p < q ≤ ∞ and for every k ≥ 1 let T k : L p (M) + + L q (M) + → S(σ) + be positive midpoint convex maps such that for some constants C p , C q > 0 depending only on p and q, respectively, If p < p E ≤ q E < q < ∞ or p < p E and q = ∞, then for any sequence (x k ) k≥1 in E(M) + , where the sums converge in norm.
Proof. We may assume C p , C q ≤ 1. Suppose first that q E < q < ∞. By completeness it suffices to prove (20) for a finite sequence ( . By Lemma 4.10 and positivity and convexity of the T k , . By (19), where the final equality follows from Corollary 3.6. The result is now immediate from Lemma 3.7. The case q = ∞ follows analogously as in the proof of Theorem 4.8.
As an application of Theorem 4.11, we can interpolate the following dual Doob inequality in noncommutative L p -spaces, due to M. Junge.
. Theorems 4.12 and 4.11 together yield the following extension.
Corollary 4.13. Let E be a symmetric quasi-Banach function space on R + which is s-convex for some 0 < s < ∞ and let M be a semi-finite von Neumann algebra.
Let (E k ) k≥1 be an increasing sequence of conditional expectations in M.
where the sums converge in norm.
In [18], Theorem 7.1, it was shown that any conditional expectation E is 'antibounded' for the L p -norm if 0 < p < 1, i.e., (22) x L p (M) ≤ 2 Even though (22) does not correspond to the boundedness of an operator, we can still 'interpolate' this estimate.
Proposition 4.14. Let M be a finite von Neumann algebra and let E be a conditional expectation on M. If E is a symmetric quasi-Banach function space on R + with q E < 1, then Proof. Let y = E(x) and for v > 0 set e v = e y [0, v]. As was remarked after Lemma 2.2, we have p E > 0 and hence we can fix 0 < p < p E and q E < q < 1. By Chebyshev's inequality and (22), where the final equality follows from Corollary 3.6. The conclusion now follows from Lemma 3.7.
The following result facilitates interpolation of noncommutative square function estimates.
Theorem 4.15. Let E be a symmetric quasi-Banach function space on R + which is s-convex for some 0 < s < ∞. Let M, N be von Neumann algebras equipped with normal, semi-finite, faithful traces τ and σ, respectively. Suppose that 0 < p < q ≤ ∞ and for k ≥ 1 let T k : L p (M) + L q (M) → S(σ) be linear maps such that for some constants C p , C q > 0 depending only on p and q, respectively, If p < p E ≤ q E < q < ∞ or p < p E and q = ∞, then for any finite sequence ( By (4.15), The result now follows from Lemma 3.7. The case q = ∞ is similar.
As a corollary, we find the following version of Stein's inequality for noncommutative symmetric spaces, which will be needed in the proof of Theorem 6.2 below. A different proof of this result was found in [16].
Corollary 4.16. Let E be a symmetric quasi-Banach function space on R + which is s-convex for some 0 < s < ∞ and let M be a semi-finite von Neumann algebra. Let (E k ) k≥1 be an increasing sequence of conditional expectations in M. If 1 < p E ≤ q E < ∞, then for any finite sequence (x k ) in E(M), Proof. Let e kl be the standard matrix units, let y k = x k ⊗ e k1 and let T k = E k ⊗ 1 B(l 2 ) . Then (23) is equivalent to .
By [32], Theorem 2.3, this inequality holds if E = L p with 1 < p < ∞. Hence, the result follows immediately from Theorem 4.15.
Further examples of probabilistic inequalities which can be interpolated using the presented method are given by the 'upper' noncommutative Khintchine inequalities, see [7], Theorem 4.1, and [8], Corollary 2.2.  16, and Proposition 4.14 all have an appropriate 'Φ-moment version'. Indeed, these versions follow immediately by using Lemma 3.9 instead of Lemma 3.7 in the proofs of the latter results. In particular, by following the proof of Theorem 4.8 and taking Remark 3.10 into account, we find an extension of [1], Theorem 2.1 for non-convex Orlicz functions.

Interpolation of noncommutative maximal inequalities
In this section we present a Boyd-type interpolation theorem for noncommutative maximal inequalities. To formulate maximal inequalities in noncommutative symmetric spaces and their dual versions, we first introduce the two 'noncommutative vector-valued symmetric spaces' E(M; l ∞ ) and E(M; l 1 ). We define these in analogy with the noncommutative vector-valued L p -spaces L p (M; l ∞ ) and L p (M; l 1 ), which were introduced in [31] for hyperfinite von Neumann algebras, and considered in general in [17]. From now on, we let E be a symmetric Banach function space on R + .
We define E(M; l ∞ ) to be the space of all sequences x = (x k ) k≥1 in E(M) for which there exist a, b ∈ E (2) (M) and a bounded sequence y = (y k ) k≥1 such that where the infimum is taken over all possible factorizations of x as above. We can think of the quantity (24) as ' sup k≥1 x k E(M) ', even though sup k≥1 x k need not be defined at all. We define E(M; l 1 ) to be the space of all sequences x = (x k ) k≥1 in E(M) which can be decomposed as for two families (u jk ) j,k≥1 and (v jk ) j,k≥1 in E (2) (M) satisfying j,k u * jk u jk ∈ E(M) and where the series converge in norm. For x ∈ E(M; l 1 ) we define where the infimum is taken over all decompositions of x as above. In what follows, we will mostly consider elements x = (x k ) k≥1 ∈ E(M; l 1 ) for which x k ≥ 0 for all k. In this case, .
The theory for the spaces E(M; l ∞ ) and E(M; l 1 ) can be developed in full analogy with the special case E = L p considered in [17,19,34]. In fact, most of the basic results follow verbatim as soon as we replace L p by E, L p ′ by E × , where 1 p + 1 p ′ = 1, and L 2p by E (2) in the proofs of these results. For example, the following observation is immediate. Our strategy to prove a Boyd-type interpolation theorem for maximal inequalities is to dualize Theorem 4.11, which can be viewed as a Boyd-type interpolation theorem for l 1 -valued noncommutative symmetric spaces. We shall need the duality stated in Theorem 5.3 below. The proof is essentially an adaptation of the Hahn-Banach separation argument in [17], Proposition 3.6 (see also [34], Theorem 4.11) to our context. We need the following observation, proved in [11], Theorem 5.6 and p. 745.
Theorem 5.2. If E is a separable symmetric Banach function space on R + , then E(M) * = E × (M) isometrically, with associated duality bracket given by Below we will implicitly use the trace property a number of times, i.e., we will use that if x, y ∈ S(τ ) are such that xy, yx ∈ L 1 (M), then τ (xy) = τ (yx). In particular this holds if x ∈ E(M) and y ∈ E × (M). Theorem 5.3. Let M be a semi-finite von Neumann algebra and let E be a separable symmetric Banach function space on R + . If y = (y k ) ∈ E × (M; l ∞ ) satisfies y k ≥ 0 for all k, then Proof. We let S denote the supremum on the right hand side of (25). Let y k = az k b with a, b ∈ (E × ) (2) (M) and z k ∈ M with z k ∞ ≤ 1 and let (x k ) be a sequence in E(M) + . By Hölder's inequality, We conclude that S ≤ y E × (M;l ∞ ) . Suppose now that S = 1, we will show that y E × (M;l ∞ ) ≤ 1. Under this assumption, we have for any finite sequence x = (x k ) in E(M) + , For any finite sequence x as above we define Clearly f x is a real-valued continuous function on K and from (26) it follows that sup s∈K f x (s) ≥ 0. Let A be the subset of C(K) consisting of all f x , where x = (x k ) is a finite sequence in E(M) + . Then A is a cone in C(K). Indeed, if λ ≥ 0 then λf x = f λx . Moreover, if x,x are finite families in E(M), then f x + fx can be realized as f x+x , since without loss of generality we may assume thatx is to the right of the finite family x. Observe that A is disjoint from the cone A − = {g ∈ C(K) : sup g < 0}. By the Hahn-Banach separation theorem, there exists a real Borel measure µ on K and α ∈ R such that for all f ∈ A and g ∈ A − , Note that α = 0, as both A and A − are cones. If B is a Borel subset, then we can find a sequence (g i ) in A − such that g n ↑ −χ B . This shows that µ must be positive, and by normalization we may assume that µ is a probability measure. Hence, for all k ≥ 1, we have Clearly a ∈ K, so a ∈ E × (M) + and a E × (M) ≤ 1. By (27) and normality of τ , This implies that y k ≤ a and therefore we find a contraction u k ∈ M such that y This completes the proof. isometrically, with respect to the duality bracket where x ∈ E(M; l 1 ) and y ∈ E × (M; l ∞ ).
The following result facilitates the interpolation of noncommutative maximal inequalities.
Theorem 5.5. Let E be the Köthe dual of a separable symmetric Banach function space on R + . Suppose that 1 ≤ p < q < ∞ and let S k : L p,1 (M) + + L q,1 (M) + → S(τ ) + be positive linear operators satisfying If q E < q and either p = 1 or p E > p, then Proof. Let F be the symmetric space on R + such that F × = E. By (5) we have p F > q ′ and, if p E > 1 also q F < p ′ . Since S k is positive, so is its adjoint S * k . If r ∈ {p, q} and y = (y k ) ∈ L r ′ (M; l 1 ) with y k ≥ 0, then for any x ∈ L r,1 (M) + , It follows that . Therefore, if (y k ) ∈ F (M; l 1 ) satisfies y k ≥ 0, then by Theorem 4.11, .
Examples of sequences of operators satisfying the conditions of Theorem 5.5 are established in [19]. We give two examples which yield maximal ergodic inequalities in noncommutative symmetric spaces. Let T : M → M be a linear map such that (a) T is a contraction on M; In [19], Theorem 4.1, it is shown that for any 1 < p ≤ ∞ the ergodic averages satisfy the maximal inequality If T moreover satisfies (d) τ (T (y) * x) = τ (y * T (x)) for all x, y ∈ L 2 (M) ∩ M, then, as was observed in [19], Theorem 5.1, for every 1 < p ≤ ∞ one has Using Theorem 5.5 we can interpolate these inequalities to obtain the following result.
Theorem 5.6. Let E be the Köthe dual of a separable symmetric Banach function space on R + and suppose that 1 < p E ≤ q E < ∞. If T : M → M is a linear operator satisfying conditions (a)-(c) above, then If T moreover satisfies condition (d), then To conclude this section, we prove a version of Doob's maximal inequality for noncommutative symmetric spaces. First recall the following definitions. Let E be a symmetric Banach function space on R + and let M be a semi-finite von Neumann algebra. Suppose that (M k ) k≥1 is a filtration, i.e., an increasing sequence of von Neumann subalgebras such that τ | M k is semi-finite, and let E k be the conditional expectation with respect to M k . A sequence (y k ) in E(M) is called a martingale with respect to (M k ) if E k (y k+1 ) = y k for all k ≥ 1. We say that (y k ) is finite if there is an n ≥ 1 such that y k = y n for all k ≥ n. A sequence (x k ) in E(M) is called a martingale difference sequence if x k = y k − y k−1 for some martingale (y k ), with the convention y 0 = 0 and M 0 = C1.
It was shown by M. Junge in [17] that for every 1 < p ≤ ∞, This result implies the following version for noncommutative symmetric spaces.
Theorem 5.7. Let M be a semi-finite von Neumann algebra and let E be the Köthe dual of a separable symmetric Banach function space on R + with 1 < p E ≤ q E < ∞.
For any y ∈ E(M) and any increasing sequence of conditional expectations (E k ) k≥1 , If Proof. The first statement follows immediately from Theorem 5.5 and (30). To prove the second statement, let y k = az k b with (z k ) k≥1 a bounded sequence in M and a, b ∈ E (2) (M). Since E is the Köthe dual of a symmetric space, it has the Fatou property and is hence fully symmetric. Therefore, µ(y k ) ≺≺ µ(az k )µ(b) implies that Taking the infimum over all decompositions as above gives sup k≥1 y k E(M) ≤ (y k ) k≥1 E(M;l ∞ ) .
For the reverse inequality, observe that E(M) ⊂ L p (M) + L q (M) for some 1 < p < p E ≤ q E < q < ∞. Let (y k ) k≥1 be a martingale in E(M) with sup k≥1 y k E(M) = 1. Then (y k ) is a bounded martingale in L p (M) + L q (M) and hence there exists y ∞ ∈ L p (M) + L q (M) such that y k → y ∞ in L p (M) + L q (M) and E k (y ∞ ) = y k for all k ≥ 1. Since E(M) has the Fatou property, its unit ball is closed in S(τ ) (cf. [11], Proposition 5.14). As y k → y ∞ in measure, we conclude that y ∞ ∈ E(M) and y ∞ E(M) ≤ 1. Applying (31) for y = y ∞ yields the result.
Remark 5.8. While the author was completing this paper, a preprint [2] appeared in which Theorems 5.6 and 5.7 are proved (under slightly different assumptions) by an alternative method.

Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities
As applications of the noncommutative version of Doob's maximal inequality and its dual version, we derive versions of the Burkholder-Davis-Gundy inequalities and Burkholder-Rosenthal inequalities in noncommutative symmetric spaces. Let E be a symmetric Banach function space on R + . For any finite martingale difference sequence (x k ) in E(M) we set .
These expressions define two norms on the linear space of all finite martingale difference sequences in E(M). The following Burkholder-Davis-Gundy inequalities extend the Burkholder-Gundy inequalities established in [7]. Theorem 6.1. Let E be the Köthe dual of a separable symmetric Banach function space on R + and suppose that 1 < p E ≤ q E < ∞. Let M be a semi-finite von Neumann algebra and (M k ) k≥1 a filtration in M. Then, for any martingale difference sequence (x k ) of a finite martingale (y k ) in E(M) we have r . Suppose that, moreover, E is separable. If p E > 1 and either q E < 2 or E is 2-concave, then On the other hand, if either E is 2-convex and q E < ∞ or 2 < p E ≤ q E < ∞ then Proof. If F is a symmetric Banach function space with F × = E, then by (5) 1 < p F ≤ q F < ∞. By Theorem 5.7, .
The result now follows directly from [7], Proposition 4.18.
The following result generalizes the noncommutative Rosenthal inequalities presented in [7], as well as the Burkholder-Rosenthal inequalities for noncommutative L p -spaces and Lorentz spaces obtained in [18], Theorem 5.1, and [15], Theorem 3.1, respectively. In fact, Theorem 6.2 positively answers an open question posed in [16], Problem 3.5 (2). The proof follows the general strategy of the proof of [7], Theorem 6.3. Let M n (M) denote the von Neumann algebra of n × n matrices with entries in M and for any sequence (x k ) n k=1 in E(M) we let diag(x k ) and col(x k ) be the matrices with the x k on the diagonal and first column, respectively, and zeroes elsewhere.
Theorem 6.2. (Noncommutative Burkholder-Rosenthal inequalities) Let M be a semi-finite von Neumann algebra. Suppose that E is a symmetric Banach function space on R + satisfying 2 < p E ≤ q E < ∞. Let (M k ) be a filtration in M and, for every k ≥ 1, let E k denote the conditional expectation with respect to M k . Let (x k ) be a martingale difference sequence in E(M) with respect to (M k ). Then, for any n ≥ 1, Proof. We first prove that the maximum on the right hand side is dominated by .
By interpolating this estimate for q = 2 and q > q E we obtain diag(x k ) n .