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