Algebras of analytic germs

Abstract:

Let S be a Stein-Riemann domain with global local coordinates ${\sigma _1}, \cdots ,{\sigma _n}$. Let X be a compact subset of S. Denote by $\mathcal {O}(X)$ the algebra of germs on X of functions analytic near X. A subalgebra of $\mathcal {O}(X)$ containing the germs of ${\sigma _1}, \cdots ,{\sigma _n}$ and the constants is stable if it is closed under differen tiation with respect to the coordinates ${\sigma _1}, \cdots ,{\sigma _n}$. In this paper the relationship of a stable algebra to its spectrum is investigated. In general, there is no natural imbedding of the spectrum into a Stein manifold. We give necessary and sufficient conditions that such an imbedding exists, and show that a stable algebra whose spectrum admits such an imbedding has a simple description. More generally, we show that a stable algebra is determined by its spectrum. This leads to certain approximation theorems.