Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces

Abstract

Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the non-commutative Lp-space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the Lp-Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. For a discrete group G and in the special case when p > 2 is an even integer, we show the following. If there exists a map between Lp(LG) and an ultraproduct of Lp(M) \otimes S_p(L^2 G) that intertwines the Fourier multiplier with the Schur multiplier, then G must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.

Trans. Amer. Math. Soc., to appear

References in corpus (2)

• Harmonic analysis on quantum tori
• Approximation properties for noncommutative Lp-spaces associated with lattices in Lie groups

Cited by in corpus (6)

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• Absolute dilations of ucp self-adjoint Fourier multipliers: the non unimodular case
• Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces
• The local geometry of idempotent Schur multipliers
• A Marcinkiewicz Testing Criterion for Schur Multipliers
• Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability