2-Universally complete Riesz spaces and inverse-closed Riesz spaces ⋆

In this paper we show mainly two results about uniformly closed Riesz subspaces of ℝX containing the constant functions. First, for such a Riesz subspace E, we solve the problem of determining the properties that a real continuous functiondefined on a proper open interval of ℝshould have in order that the conditions “E is closed under composition with ” and “E is closed under inversion in X” become equivalent. The second result, reformulated in the more general frame of the Archimedean Riesz spaces with weak order unit e, establishes that E (e-uniformly complete and e-semisimple) is closed under inversion in C(Spec E) if and only if E is 2-universally e-complete.