# Operator algebraic approach to inverse and stability theorems for amenable groups

## TL;DR

This paper develops an operator algebraic framework to establish inverse and stability theorems for functions from amenable groups into von Neumann algebras, extending harmonic analysis tools to non-commutative settings.

## Contribution

It introduces an inverse theorem for the Gowers U^2-norm in the context of von Neumann algebras and proves stability of approximate unitary representations.

## Key findings

- Proved an inverse theorem for Gowers U^2-norm in von Neumann algebras.
- Established stability results for approximate unitary representations.
- Extended harmonic analysis techniques to operator algebraic settings.

## Abstract

We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.