Research output: Contribution in Book/Report/Proceedings › Chapter

Published

Publication date | 2010 |
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Host publication | Banach Algebras 2009 |

Editors | Richard J. Loy, Volker Runde, Andrzej Sołtysiak |

Place of Publication | Warsaw |

Publisher | Polish Academy of Sciences |

Pages | 123-158 |

Number of pages | 38 |

ISBN (Print) | 9788386806102 |

<mark>Original language</mark> | English |

Name | Banach Center Publications |
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Volume | 91 |

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.