Duke Mathematical Journal

Free Hilbert transforms

Tao Mei and Éric Ricard

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We study Fourier multipliers of Hilbert transform type on free groups. We prove that they are completely bounded on noncommutative Lp-spaces associated with the free group von Neumann algebras for all 1<p<. This implies that the decomposition of the free group F into reduced words starting with distinct free generators is completely unconditional in Lp. We study the case of Voiculescu’s amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness problem posed by Ozawa, a length-independent estimate for Junge–Parcet–Xu’s free Rosenthal’s inequality, a Littlewood–Paley–Stein-type inequality for geodesic paths of free groups, and a length reduction formula for Lp-norms of free group von Neumann algebras.

Article information

Duke Math. J., Volume 166, Number 11 (2017), 2153-2182.

Received: 5 July 2016
Revised: 10 February 2017
First available in Project Euclid: 28 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25]
Secondary: 46L54: Free probability and free operator algebras 46L52: Noncommutative function spaces

Hilbert transforms free group von Neumann algebra noncommutative $L^{p}$-spaces


Mei, Tao; Ricard, Éric. Free Hilbert transforms. Duke Math. J. 166 (2017), no. 11, 2153--2182. doi:10.1215/00127094-2017-0007. https://projecteuclid.org/euclid.dmj/1493344841

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