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Fréchet algebras of power series

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Publication date2010
Host publicationBanach Algebras 2009
EditorsRichard J. Loy, Volker Runde, Andrzej Sołtysiak
Place of PublicationWarsaw
PublisherPolish Academy of Sciences
Number of pages38
ISBN (Print)9788386806102
Original languageEnglish

Publication series

NameBanach Center Publications
ISSN (Print)0137-6934
ISSN (Electronic)1730-6299


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 Clayton and 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.