Fréchet algebras of power series

• Autorzy: H. Garth Dales , Shital R. Patel , Charles J. Read
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 13J05: Power series rings
• 46H40: Automatic continuity
• 46J99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 91
• Numer: 1
• Strony: 123-158

Daty

wydano

2010

Twórcy

autor

H. Garth Dales

• Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

autor

Shital R. Patel

• Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208 016, Uttar Pradesh, India

autor

Charles J. Read

• Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

Identyfikatory

DOI

10.4064/bc91-0-7