I am sure that this is well known in the right places, but: Is the C* completion of a star nuclear Fréchet algebra a nuclear C* algebra? (Suppose that the C* norm is continuous with respect to the Fréchet topology.)
[Basic, but commutative, example to explain topology - smooth functions on a compact manifold completed to continuous functions.]
To explain more: I expect that if the result was true, all that would be needed would be that the product and star were continuous. If we take continuous seminorms $\|.\|_1 \le \|.\|_2 \le \dots$ then likely we should have $\|a\,b\|_n\le C\, \|a\|_m\,\|b\|_m$ where $m,C$ just depend on $n$. By C* completion, this would be via some star representation of the algebra on a Hilbert space $H$, so that the map from the algebra to $B(H)$ was continuous.