Topological localization in Fréchet algebras (Q1842046)

Let \(A\) be a symmetric Fréchet algebra. The topological spectrum \(X\) of \(A\) is the set of all continuous algebra morphisms of \(A\) into the complex numbers (endowed with the initial topology defined by the maps \(a: X\to \mathbb{C}\), \(a(x)= x(a)\), where \(a\in A\)). If \(U\) is an open set in \(X\), then \(A_U\) denotes the localization of \(A\) with respect to the multiplicatively closed subset \(S_U= \{a\in A\): \(0\not\in a(U)\}\). Thus we have a presheaf of algebras \(U\to A_U\) on \(X\), and the associated sheaf \(\widetilde {A}\) of algebras, which is the structural sheaf of the topological spectrum \(X\) of \(A\), considered as a sheaf of functions on \(X\). Hence, for non-semisimple algebras, the classical concept of regularity must be replaced by a stronger condition, namely: given \(x\in X\) and a closed subset \(Y\) not containing \(x\), there exists \(f\in A\) such that \(f(x) \neq 0\) and the germ of \(f\) (considered as a global section of \(\widetilde {A}\)) at any point of \(Y\) is zero. Algebras with this property are called strictly regular. For semisimple algebras, strict regularity coincides with regularity. The main result of the paper is: Let \(A\) be a strictly regular symmetric Fréchet algebra and let \(U\) be an open subset of the topological spectrum \(X\) of \(A\). If there exists \(a\in A\) such that \(U= \{x\in X\): \(a(x)\neq 0\}\), then the natural algebra morphism \(A_U\to \widetilde {A} (U)\) is an isomorphism. Moreover, \(A_U\) is a strictly regular symmetric Fréchet algebra with spectrum \(U\).