Fréchet algebras of power series

H. Garth Dales, Shital R. Patel, and Charles J. Read. "Fréchet algebras of power series." Banach Center Publications 91.1 (2010): 123-158. <http://eudml.org/doc/281940>.

@article{H2010,
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.},
author = {H. Garth Dales, Shital R. Patel, Charles J. Read},
journal = {Banach Center Publications},
keywords = {(F)-algebras; Fréchet algebras of power series; uniqueness of topology; discontinuous homomorphisms; separating space},
language = {eng},
number = {1},
pages = {123-158},
title = {Fréchet algebras of power series},
url = {http://eudml.org/doc/281940},
volume = {91},
year = {2010},
}

TY - JOUR
AU - H. Garth Dales
AU - Shital R. Patel
AU - Charles J. Read
TI - Fréchet algebras of power series
JO - Banach Center Publications
PY - 2010
VL - 91
IS - 1
SP - 123
EP - 158
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.
LA - eng
KW - (F)-algebras; Fréchet algebras of power series; uniqueness of topology; discontinuous homomorphisms; separating space
UR - http://eudml.org/doc/281940
ER -