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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.