Contractible Fréchet algebras

Abstract:

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 ${\mathbb {C}}^ n$ for some $n \in \mathbb {N}$. On the other hand, we show that a Fréchet 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.