Contractible Fréchet algebras

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