Contractible Fréchet algebras

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