Spectral gap and stability for groups and non-local games

TL;DR

This paper explores how spectral gap estimates can ensure stability in group representations and non-local games, providing new results on stability preservation and simplifying aspects of the Connes' embedding problem resolution.

Contribution

It introduces spectral gap-based methods for stability in operator algebras and non-local games, including stability of product groups and a simplified non-local game analysis.

Findings

Product of Hilbert-Schmidt stable groups with property (T) remains stable

A simple non-local game with strategies close to perfect ones involving large Pauli matrices

Simplification of the question reduction step in the Connes' embedding problem resolution

Abstract

The word stable is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and quantum synchronous strategies for non-local games. We observe in particular that simple spectral gap estimates can lead to strong quantitative forms of stability. For example, we prove that the direct product of two (flexibly) Hilbert-Schmidt stable groups is again (flexibly) Hilbert-Schmidt stable, provided that one of them has Kazhdan's property (T). We also provide a simple form and simple analysis of a non-local game with few questions, with the property that synchronous strategies with large value are close to perfect strategies involving large Pauli matrices. This simplifies one of the steps (the question reduction) in the recent announced resolution of Connes' embedding problem by Ji, Natarajan, Vidick, Wright and Yuen.