Fourier-type convergence in groupoid C*-algebras

In the group setting, Bedos and Conti have proved the following. Let G be a discrete group with approximation property acting on compact Hausdorff topological space X and C(X)xG be its reduced crossed product. If an element a of C(X)xG has Fourier coefficients a_g, then a is in the closed span of the products a_g u_g, where g runs through G. 

Later, Brown, Exel, Fuller, Pitts, and Reznikoff have generalized it from groups to groupoids. They have shown that when an etale groupoid is amenable, any arbitrary element f in the reduced C*-algebra of G can be approximated in the reduced norm by a net of compactly supported functions within the support of f. Recently, we have generalized it further. We have shown that if an etale groupoid has rapid decay property with respect to a conditionally negative-definite length function L, then any element f in the reduced C*-algebra of G can be approximated in the reduced norm by a net of compactly supported functions within the support of f. This result is based on ongoing joint work with Dr. Adam Fuller.