# Non-Hausdorff part of measure zero?

Let $$G$$ be a locally compact group of type I.
By Theorem 4.4.5 of Dixmier’s book on C*-algebras, there exists a dense open subset $$U$$ of the unitary dual $$widehat G$$, which is a Hausdorff space.

• Can this set $$U$$ be chosen in a way that it’s complement has Plancherel-measure zero?

• Can this set $$U$$ be chosen such that for every $$piin U$$ and every $$fin C_c^infty(G)$$ the operator $$pi(f)$$ is trace-class and the map $$pimapstomathrm{tr}(pi(f))$$ is continuous?