Kakeya conjecture and High-Rank Lattice von Neumann algebras

TL;DR

This paper explores the connection between the approximation properties of non-commutative Lp spaces of SLn(Z) and the Kakeya conjecture, introducing harmonic analysis questions and proving related smoothness and boundedness conditions.

Contribution

It establishes a novel link between operator algebra properties and geometric measure theory, proposing new harmonic analysis questions and partial results.

Findings

A spherical analogue of a harmonic analysis question is formulated and investigated.

Necessary conditions for Lp-boundedness of Fourier multipliers are derived in terms of Besov spaces.

A connection between the approximation property of non-commutative Lp spaces and the Kakeya conjecture is demonstrated.

Abstract

If the non-commutative L p space of SLn(Z) has the completely bounded approximation property for some non-trivial value of p, then some form of the Kakeya conjecture holds in dimension d, for all d $\le$ n+1 2 . The proof relies on a spherical analogue of the following question in Euclidean harmonic analysis, that we raise and investigate: does a radially symmetric Fourier multiplier that is bounded on Lp(R d ) for some p __ = 2 necessarily have a continuous symbol? We leave the question open, but we prove that the primitive of such function is smooth in the sense of Zygmund, give some necessary conditions for Lp-boundedness in terms of Besov spaces and Littlewood-Paley decomposition for the symbol, and observe that a negative answer implies some form of the Kakeya conjecture in dimension d. We then provide spherical forms of these results, which, when combined with a refinement of Lafforgue's rank 0 reduction, leads to the claimed result.