Localness of A (Ψ) Algebras

Let d and r be positive integers. Given Ψ = (v/₁,…,v/ᵣ) ∈ C∞ (Rᵈ,Rr), we consider the unital algebra R[Ψ] = R[v/₁,…,v/ᵣ] generated by {v/₁,…,v/ᵣ}, and its closure A(Ψ) in C∞ topology. We identify the space of closed maximal ideals of A(Ψ), we establish that it is a regular algebra, and we show that the approximation problem, to provide an explicit description of Ψ, we establish that it is a regular algebra, and we show that the approximation problem, to provide an explicit description of Ψ, is local to the level sets of Ψ