Bounded cohomology with coefficients in uniformly convex Banach spaces

We show that for acylindrically hyperbolic groups $\Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $\rho$ of $\Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(\Gamma;\rho)$ is infinite dimensional. The result was known for the regular representations on $\ell^p(\Gamma)$ with $1<p<\infty$ by a different argument. But our result is new even for a non-abelian free group in this great generality for representations, and also the case for acylindrically hyperbolic groups follows as an application.