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 $\mathrm{ev}^*$ 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.