Vanishing of reduced cohomology with A_0 coefficients

Determine whether, for each degree d, every cocycle \beta \in C^d(\mathfrak{g}; A_0), where A_0 = {f \in C^{\infty,\infty}(Y_0) : \int_{Y_0} f\, d\mu = 0}, is a reduced coboundary in C^d(\mathfrak{g}; A_0). Equivalently, ascertain whether the reduced cohomology \overline{H}^d(\mathfrak{g}; A_0) vanishes.

Background

The paper introduces two coefficient modules: A = C^{\infty,\infty}(Y_0), an algebra of smooth G–equivariant functions on Y_0, and L = (L^2(\mu))^\infty, the smooth part of L^2(\mu) under the G–action. For mean-zero submodules A_0 and L_0, Lemma 5.4 shows \overline{H}^n(\mathfrak{g}; L_0) = 0, i.e., mean-zero cocycles with L-coefficients are reduced coboundaries. It remains unknown if the same vanishing holds with A_0-coefficients, reflecting subtle differences between A and L and the topologies involved.