\(C^{\infty}\)-algebras from the functional analytic view point
Roughly speaking a \(C^{\infty}\)-algebra is a commutative real algebra, in which one may not only evaluate polynomials but also \(C^{\infty}\)- functions. In this paper, we put a natural locally-m-convex topology on each \(C^{\infty}\)-algebra. This topology need not be Hausdorff, but if it is we derive nice results: The \(C^{\infty}\)-homomorphism are exactly the continuous algebra homomorphisms, countably generated Hausdorff \(C^{\infty}\)-algebras are nuclear, the action of smooth functions on Hausdorff \(C^{\infty}\)- algebras is smooth and continuous, and the coproduct equals the bornological tensor product in the most important cases. We also consider \(C^{\infty}\)-modules.
scientific article; zbMATH DE number 4007001
scientific article