Fréchet Algebras of Power Series

We consider Fréchet algebras which are subalgebras of the algebra F = C [[X]] of formal power series in one variable and of Fn = C [[X1, . . . , Xn]] of formal power series in n variables where n ∈ 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’ U for Michael’s problem for commutative Fréchet algebras has been described by Dixon and Esterle. We prove that there is an embedding of U into 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 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 F containing C[X] that is a Banach algebra is already a Banach algebra of power series, in the sense that the embedding of A into 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 F by F2. 2000 Mathematics Subject Classification: Primary 46H10, secondary 13F25, 46H40