Fréchet algebras of power series
Tom 91 / 2010
Streszczenie
We consider Fréchet algebras which are subalgebras of the algebra ${\frak F} = \mathbb C\,[[X]]$ of formal power series in one variable and of ${\frak F}_n = \mathbb C\,[[X_1,\dots, X_n]]$ of formal power series in $n $ variables, where $n\in{\mathbb N}$. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, $(F)$-algebras, and other topological algebras, and recall some of their properties; we discuss Michael's problem from 1952 on the continuity of characters on these algebras and some results on uniqueness of topology. A `test algebra' $\mathcal U$ for Michael's problem for commutative Fréchet algebras has been described by Clayton and by Dixon and Esterle. We prove that there is an embedding of $\mathcal U$ into ${\frak F}$, and so there is a Fréchet algebra of power series which is a test case for Michael's problem. We also discuss homomorphisms from Fréchet algebras into ${\frak F}$. We prove that such a homomorphism is either continuous or a surjection, so answering a question of Dales and McClure from 1977. As corollaries, we note that a subalgebra $A$ of ${\frak F}$ containing $\mathbb C[X]$ that is a Banach algebra is already a Banach algebra of power series, in the sense that the embedding of $A$ into ${\frak F}$ is automatically continuous, and that each $(F)$-algebra of power series has a unique $(F)$-algebra topology. We also prove that it is not true that results analogous to the above hold when we replace ${\frak F}$ by ${\frak F}_2$.
