The Gelfand transform in commutative algebra

We consider the transformation ev which associates to any element in a K-algebra A a function on the the set of its K-points. This is the analogue of the fundamental Gelfand transform. Both ev and its dual ev* are the maps from a discrete K-module to a topological K-module and we investigate in which case the image of each map is dense. This question arises in the classical problem of the reconstruction of a function by its values at a given set of points. The answer is nontrivial for various choices of K and A already for A=K[x], the polynomial ring in one variable. Applications to the structure of algebras of cohomology operations are given.