Characterization of Uniform Fr´echet Algebras in which every Element is the Square of Another

Let X be a hemicompact k-space and A be a uniform Fr´echet algebra on X. Inthis note we first show that if each element of a dense subset of A has square rootin A then A = C(X) under certain condition. Then we show that G(C(X)), thegroup of invertible elements of C(X), is dense in C(X) if and only if dimX, thecovering dimension of X, does not exceed 1. Using this result we give a necessaryand sufficient condition under which each continuous function on X is the squareof another.