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Topological algebras with maximal regular ideals closed

Mati Abel — 2012

Open Mathematics

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.

Topological algebras with pseudoconvexly bounded elements

Mati Abel — 2005

Banach Center Publications

It is shown that every commutative sequentially bornologically complete Hausdorff algebra A with bounded elements is representable in the form of an (algebraic) inductive limit of an inductive system of locally bounded Fréchet algebras with continuous monomorphisms if the von Neumann bornology of A is pseudoconvex. Several classes of topological algebras A for which r A ( a ) β A ( a ) or r A ( a ) = β A ( a ) for each a ∈ A are described.

On a problem of Bertram Yood

Mart AbelMati Abel — 2014

Topological Algebra and its Applications

In 1964, Bertram Yood posed the following problem: whether the intersection of all closed maximal regular left ideals of a topological ring coincides with the intersection of all closed maximal regular right ideals of this ring. It is proved that these two intersections coincide for advertive and simplicial topological rings and, using this result, it is shown that the topological left radical and the topological right radical for every advertive and simplicial topological algebra coincide.

Topologically Invertible Elements and Topological Spectrum

Mati AbelWiesław Żelazko — 2006

Bulletin of the Polish Academy of Sciences. Mathematics

Properties of topologically invertible elements and the topological spectrum of elements in unital semitopological algebras are studied. It is shown that the inversion x x - 1 is continuous in every invertive Fréchet algebra, and singly generated unital semitopological algebras have continuous characters if and only if the topological spectrum of the generator is non-empty. Several open problems are presented.

Description of quotient algebras in function algebras containing continuous unbounded functions

Mati AbelJorma ArhippainenJukka Kauppi — 2012

Open Mathematics

Let X be a completely regular Hausdorff space, 𝔖 a cover of X, and C b ( X , 𝕂 ; 𝔖 ) the algebra of all 𝕂 -valued continuous functions on X which are bounded on every S 𝔖 . A description of quotient algebras of C b ( X , 𝕂 ; 𝔖 ) is given with respect to the topologies of uniform and strict convergence on the elements of 𝔖 .

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