Ralf Meyer develops part of the theory from scratch for locally compact groups via projective systems of Lie groups in section 5 of http://arxiv.org/abs/math/0310094
This theory speciales to the Bernstein center in the totally disconnected case and to the theory of the center of the universal envelopping algebra for complex Lie groups with trivial center. Also he relates the center to distributions.
He works on bornological vector-spaces, but states in the introduction that the theory includes smooth representation on Frechet-spaces. So the notion bornological and topological can be replaced in this case. (I assume that many of us are more comfy with topological vectorspaces than with bornological ones. At least I am.)