Noncommutative Boyd interpolation theorems

Dirksen, Sjoerd

doi:10.1090/S0002-9947-2014-06185-0

Noncommutative Boyd interpolation theorems
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Trans. Amer. Math. Soc. 367 (2015), 4079-4110 Request permission

Abstract:

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 $\ell ^1$-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.

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