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On Gelfand-Mazur theorem on a class ofF-algebras

100%
Anjidani E.
Topological Algebra and its Applications
|
2014
|
tom 2
|
nr 1
EN
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.
2
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Topological algebras with maximal regular ideals closed

76%
Abel M.
Open Mathematics
|
2012
|
tom 10
|
nr 3
1054-1059
EN
It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.
3
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Description of quotient algebras in function algebras containing continuous unbounded functions

76%
Abel M., Arhippainen J., Kauppi J.
Open Mathematics
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2012
|
tom 10
|
nr 3
1060-1066
EN
Let X be a completely regular Hausdorff space, $$\mathfrak{S}$$ a cover of X, and $$C_b (X,\mathbb{K};\mathfrak{S})$$ the algebra of all $$\mathbb{K}$$ -valued continuous functions on X which are bounded on every $$S \in \mathfrak{S}$$ . A description of quotient algebras of $$C_b (X,\mathbb{K};\mathfrak{S})$$ is given with respect to the topologies of uniform and strict convergence on the elements of $$\mathfrak{S}$$ .
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