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.
@article{bwmeta1.element.doi-10_2478_taa-2014-0004,
author = {E. Anjidani},
title = {
On Gelfand-Mazur theorem on a class of
F
-algebras
},
journal = {Topological Algebra and its Applications},
volume = {2},
year = {2014},
zbl = {1316.46040},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_taa-2014-0004}
}
E. Anjidani.
On Gelfand-Mazur theorem on a class of
F
-algebras
. Topological Algebra and its Applications, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_taa-2014-0004/
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