Title: 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 $\varepsilon$-representations $G\to\mathcal
U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.