Contractible Fréchet algebras

Abstract

A unital Frechet algebra A is called contractible if there exists an element d such that π A (d) = 1 and ad = da for all a ∈ A where π A : A⊗A → A is the canonical Frechet A-bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Frechet algebra A to be a direct sum of a finite-dimensional semisimple algebra M and a contractible Frechet algebra N without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative Imc Frechet Q-algebra is contractible if, and only if, it is algebraically and topologically isomorphic to C n for some n ∈ N. On the other hand, we show that a Frechet algebra, that is, a locally C*-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.