On a class of vector lattices of continuous function spaces and related approximation-density problems

We review some recent results concerning a class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space, named regular vector lattices. This class of vector lattices seems to be well adapted to develop both Stone-Weierstrass-type theorems and Korovkin-type approximation theorems. As a chief tool to get our main results on these topics and on their deep connection, we make extensive use of the notion of Choquet boundary and the space of the so-called generalized affine functions. The survey also contains some examples as well as applications and open problems.