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Fréchet algebras of power series

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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.
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  • Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK
  • Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208 016, Uttar Pradesh, India
  • Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK
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bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-7
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