Operator algebraic approach to inverse and stability theorems for amenable groups

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 $\epsilon$-representations $G\to \mathcal{U}(\mathcal{M})$ with respect to $\Vert \cdot \Vert$.

21 pages, no figures; v2 minor changes

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