Density of C^{∞,∞}(Y_0) in (L^2(μ))^∞ under the L-topology

Ascertain whether the algebra A = C^{\infty,\infty}(Y_0) is dense in L = (L^2(\mu))^\infty with respect to the topology induced by smooth G–action (the L-topology); equivalently, determine whether every f \in L can be approximated by a sequence \alpha_i \in A in the L-topology, not merely in the L^2(\mu) norm.

Background

While Lusin’s theorem and smoothing show that elements of L can be approximated in L^2(\mu) by functions from A, the topology relevant to the cochain complex involves smooth dependence under the G–action. The derivative operators are unbounded in the L^2 norm, making it unclear whether convergence in L^2 can be upgraded to convergence in the L-topology. This affects comparisons between A_0 and L_0 and limits the transfer of vanishing results from L_0 to A_0.