On the denseness of the invertible group in uniform Frechet algebras

We first extend the Arens-Royden theorem to unital, commutative Fr´echet algebras under certain conditions. Then, we show that if A is a uniform Fr´echet algebra on X = MA, where MA is the continuous character space of A, then A does not have dense invertible group, if we impose some conditions on X. On the other hand, if A has dense invertible group, then it is shown that A = C(X), with certain conditions on X. Thus, the results of Dawson and Feinstein on denseness of the invertible group in Banach algebras are extended to uniform Fr´echet algebras.