The structure of non-commutative multiple correlation sequences and applications

Abstract

Let $\Gamma$ be a countable abelian group with Pontryagin dual $\Sigma :=\hat{\Gamma }$. A non-commutative $2$-fold multiple correlation sequence is a double-indexed sequence $$b(\gamma ,{\gamma }^{\prime })={\int }_X{T}_{\gamma }{S}_{{\gamma }^{\prime }}f\cdot T_{\gamma }g\cdot hd\mu ,$$ where $f,h\in L^2(\mu )$ and $g\in L^{\infty }(\mu )$ are functions on a probability space $\XX=(X,\mathcal{B},μ)$ and $T,S$ are two, not necessarily commuting, measure preserving $\Gamma$-actions. We prove that $b$ is of this form if and only if there are two finite Borel measures ${\sigma }_1,{\sigma }_2$ on $\Sigma$ and a bounded operator $G:L^2(\Sigma ,{\sigma }_1)\to L^2(\Sigma ,{\sigma }_2)$ with $$b(\gamma ,{\gamma }^{\prime })={\int }_{\Sigma }G({\xi }_{\gamma })\cdot {\xi }_{{\gamma }^{\prime }}d{\sigma }_2,$$ where ${\xi }_{\gamma }(\chi )=\chi (\gamma )$ is the evaluation map. We then study the extent to which the triple $(G,{\sigma }_1,{\sigma }_2)$ is positive in the case $f=g=h=1_A$. Positivity fails in the sense available for the spectral measure of a single correlation sequence, but survives on positive functions with non-negative Fourier coefficients, where we also obtain a quantitative form by generalizing an inequality of Chu. As an application we prove a multiple recurrence theorem for products of linear forms, and deduce a simultaneous partition regularity result for a family of quadratic equations, extending, under a non-degeneracy hypothesis, a theorem of Frantzikinakis and Host.

arXiv comment31 pages. Substantial revision. Corrects an error in v1. The main theorem is simplified and strengthened, and its proof no longer uses Grothendieck's inequality. We also prove the converse direction of the main result, and establish a generalization of an inequality of Chu