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**Fréchet algebras of power series.** / Dales, H.G.; Patel, Shital R.; Read, Charles J.

Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Chapter

Dales, HG, Patel, SR & Read, CJ 2010, Fréchet algebras of power series. in RJ Loy, V Runde & A Sołtysiak (eds), *Banach Algebras 2009.* Banach Center Publications, vol. 91, Polish Academy of Sciences, Warsaw, pp. 123-158. https://doi.org/10.4064/bc91-0-7

Dales, H. G., Patel, S. R., & Read, C. J. (2010). Fréchet algebras of power series. In R. J. Loy, V. Runde, & A. Sołtysiak (Eds.), *Banach Algebras 2009 *(pp. 123-158). (Banach Center Publications; Vol. 91). Warsaw: Polish Academy of Sciences. https://doi.org/10.4064/bc91-0-7

Dales HG, Patel SR, Read CJ. Fréchet algebras of power series. In Loy RJ, Runde V, Sołtysiak A, editors, Banach Algebras 2009. Warsaw: Polish Academy of Sciences. 2010. p. 123-158. (Banach Center Publications). https://doi.org/10.4064/bc91-0-7

@inbook{100f0cf7f44b42568c76d7cf23809aa1,

title = "Fr{\'e}chet algebras of power series",

abstract = "We consider Fr{\'e}chet 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 Fr{\'e}chet 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 Fr{\'e}chet 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 Fr{\'e}chet algebra of power series which is a test case for Michael's problem. We also discuss homomorphisms from Fr{\'e}chet 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.",

author = "H.G. Dales and Patel, {Shital R.} and Read, {Charles J.}",

year = "2010",

doi = "10.4064/bc91-0-7",

language = "English",

isbn = "9788386806102",

series = "Banach Center Publications",

publisher = "Polish Academy of Sciences",

pages = "123--158",

editor = "Loy, {Richard J.} and Runde, {Volker } and {Sołtysiak }, Andrzej",

booktitle = "Banach Algebras 2009",

}

TY - CHAP

T1 - Fréchet algebras of power series

AU - Dales, H.G.

AU - Patel, Shital R.

AU - Read, Charles J.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

U2 - 10.4064/bc91-0-7

DO - 10.4064/bc91-0-7

M3 - Chapter

SN - 9788386806102

T3 - Banach Center Publications

SP - 123

EP - 158

BT - Banach Algebras 2009

A2 - Loy, Richard J.

A2 - Runde, Volker

A2 - Sołtysiak , Andrzej

PB - Polish Academy of Sciences

CY - Warsaw

ER -