Contractible Fréchet algebras

A unital Fréchet algebra is called contractible if there exists an element such that and for all where is the canonical Fréchet -bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fréchet algebra to be a direct sum of a finite-dimensional semisimple algebra and a contractible Fréchet algebra without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative lmc Fréchet -algebra is contractible if, and only if, it is algebraically and topologically isomorphic to for some . On the other hand, we show that a Fréchet algebra, that is, a locally -algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.