On Gelfand-Mazur theorem on a class of F -algebras

E. Anjidani

Topological Algebra and its Applications (2014)

  • Volume: 2, Issue: 1, page 19-23, electronic only
  • ISSN: 2299-3231

Abstract

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A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence (xn) in A, (xn) is Cauchy whenever the sequence bn(xn − xn-1) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra, then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theorem for a large class of F-algebras, containing both locally bounded algebras and locally convex algebras with bounded elements.

How to cite

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E. Anjidani. " On Gelfand-Mazur theorem on a class of F -algebras ." Topological Algebra and its Applications 2.1 (2014): 19-23, electronic only. <http://eudml.org/doc/266794>.

@article{E2014,
abstract = {A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence (xn) in A, (xn) is Cauchy whenever the sequence bn(xn − xn-1) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra, then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theorem for a large class of F-algebras, containing both locally bounded algebras and locally convex algebras with bounded elements.},
author = {E. Anjidani},
journal = {Topological Algebra and its Applications},
keywords = {Gelfand-Mazur theorem; F-algebra; fundamental topological algebra; topological algebra with bounded elements; division algebra; -algebra; topological algebra with bounded elements; fundamental algebra; dual separating the points},
language = {eng},
number = {1},
pages = {19-23, electronic only},
title = { On Gelfand-Mazur theorem on a class of F -algebras },
url = {http://eudml.org/doc/266794},
volume = {2},
year = {2014},
}

TY - JOUR
AU - E. Anjidani
TI - On Gelfand-Mazur theorem on a class of F -algebras
JO - Topological Algebra and its Applications
PY - 2014
VL - 2
IS - 1
SP - 19
EP - 23, electronic only
AB - A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence (xn) in A, (xn) is Cauchy whenever the sequence bn(xn − xn-1) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra, then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theorem for a large class of F-algebras, containing both locally bounded algebras and locally convex algebras with bounded elements.
LA - eng
KW - Gelfand-Mazur theorem; F-algebra; fundamental topological algebra; topological algebra with bounded elements; division algebra; -algebra; topological algebra with bounded elements; fundamental algebra; dual separating the points
UR - http://eudml.org/doc/266794
ER -

References

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  10. [10] W. Zelazko, A theorem on B0 division algebras, Bull. Polon. Acad. Sc. 8 (1960) 373–375. Zbl0095.31303
  11. [11] W. Zelazko, F-algebras: Some results and open problems, Functional Analysis and its Applications 197 (2004) 317–326. Zbl1096.46028
  12. [12] W. Zelazko, On the locally bounded and m-convex topological algebras, Studia Math. 19 (1960) 333–356. Zbl0096.08303
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