Dales, H.G. and Patel, Shital R. and Read, Charles J.
(2010)
*Fréchet algebras of power series.*
In:
Banach Algebras 2009.
Banach Center Publications
.
Polish Academy of Sciences, Warsaw, pp. 123-158.
ISBN 9788386806102

## Abstract

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.

Item Type: | Contribution in Book/Report/Proceedings |
---|---|

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 67563 |

Deposited By: | ep_importer_pure |

Deposited On: | 15 Nov 2013 11:48 |

Refereed?: | No |

Published?: | Published |

Last Modified: | 03 Dec 2019 00:36 |

URI: | https://eprints.lancs.ac.uk/id/eprint/67563 |

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