Stable elements of Banach and Fréchet algebras

Graham Allan

Studia Mathematica (1998)

  • Volume: 129, Issue: 1, page 67-96
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Allan, Graham. "Stable elements of Banach and Fréchet algebras." Studia Mathematica 129.1 (1998): 67-96. <http://eudml.org/doc/216493>.

@article{Allan1998,
abstract = {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.},
author = {Allan, Graham},
journal = {Studia Mathematica},
keywords = {power-series embeddings; finite closed descent; inverse-limit sequences; Arens-Michael representation; algebraic notion of stability for elements of a commutative ring; commutative Fréchet algebras},
language = {eng},
number = {1},
pages = {67-96},
title = {Stable elements of Banach and Fréchet algebras},
url = {http://eudml.org/doc/216493},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Allan, Graham
TI - Stable elements of Banach and Fréchet algebras
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 1
SP - 67
EP - 96
AB - 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.
LA - eng
KW - power-series embeddings; finite closed descent; inverse-limit sequences; Arens-Michael representation; algebraic notion of stability for elements of a commutative ring; commutative Fréchet algebras
UR - http://eudml.org/doc/216493
ER -

References

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  1. [1] G. R. Allan, Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. (3) 25 (1972), 329-340. Zbl0243.46059
  2. [2] G. R. Allan, Elements of finite closed descent in a Banach algebra, J. London Math. Soc. (2) 7 (1973), 462-466. Zbl0274.46040
  3. [3] G. R. Allan, Ideals of rapidly growing functions, in: Proc. Internat. Sympos. on Functional Analysis and its Applications, Univ. of Ibadan, 1977, 85-109. Zbl0448.46027
  4. [4] G. R. Allan, Fréchet algebras and formal power series, Studia Math. 119 (1996), 271-288. Zbl0858.46041
  5. [5] G. R. Allan, Stable inverse-limit sequences, with application to Fréchet algebras, ibid. 121 (1996), 277-308. Zbl0874.46048
  6. [6] R. F. Arens, Linear topological division algebras, Bull. Amer. Math. Soc. 53 (1947), 632-630. Zbl0031.25103
  7. [7] R. F. Arens, A generalization of normed rings, Pacific J. Math. 2 (1952), 455-471. Zbl0047.35802
  8. [8] H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), 129-183. Zbl0391.46037
  9. [9] S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, Princeton, N.J., 1952. Zbl0047.41402
  10. [10] E. A. Michael, Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc. 11 (1953; third printing 1971). 
  11. [11] O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Van Nostrand, Princeton, 1960. Zbl0121.27801

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