The class of elements of locally finite closed descent in a commutative Fréchet algebra is introduced. Using this notion, those commutative Fréchet algebras in which the algebra ℂ[[X]] may be embedded are completely characterized, and some applications to the theory of automatic continuity are given.
Firstly, we give extensions of results of Gelfand, Esterle and Katznelson--Tzafriri on power-bounded operators. Secondly, some results and questions relating to power-bounded elements in the unitization of a commutative radical Banach algebra are discussed.
A lemma of Gelfand-Hille type is proved. It is used to give an improved version of a result of Kalton on sums of idempotents.
The elementary theory of stable inverse-limit sequences, introduced in stable inverse-limit sequences, is used to extend the 'stability lemma' of automatic continuity theory.
We introduce an algebraic notion-stability-for an element of a commutative ring. It is shown that the stable elements of Banach algebras, and of Fréchet algebras, may be simply described. Part of the theory of power-series embeddings, given in [1] and [4], is seen to be of a purely algebraic nature. This approach leads to other natural questions.
The notion of a stable inverse-limit sequence is introduced. It provides a sufficient (and, for sequences of abelian groups, necessary) condition for the preservation of exactness by the inverse-limit functor. Examples of stable sequences are provided through the abstract Mittag-Leffler theorem; the results are applied in the theory of Fréchet algebras.
We give a simple complex-variable proof of an old result of Zemánek and Le Page on the radical of a Banach algebra. Incidentally, the argument also proves a recent result of Harris and Kadison.
This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.
This paper will give a brief survey of ideas related to 'elements of finite closed descent' in certain kinds of topological algebra.
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