Reduction of discontinuity for derivations on Fréchet algebras

Abstract

AbstractWe consider strictly irreducible representations with which the discontinuity of a derivation on a (locally multiplicatively convex) Frechet algebra must be associated. Only those strictly irreducible representations which are compatible with the topology of the algebra are considered. The main results show that when consideration is fixed upon each seminorm, the exceptional set of primitive ideals supporting the discontinuity must be a finite set, with each ideal being the kernel of some finite-dimensional irreducible representation. This result is the best possible, as can be seen by considering the radical Frechet algebra constructed by Charles Read with identity adjoined which has a derivation with separating ideal that is the entire algebra, and one could take (countable) Frechet products of his counterexample. It is also proved that derivations on commutative Frechet algebras, the structure spaces of which are compact metric in the weak* topology, have only finitely many such exceptional points overall.