### Representations of topological algebras by projective limits.

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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.

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 ${r}_{A}\left(a\right)\le {\beta}_{A}\left(a\right)$ or ${r}_{A}\left(a\right)={\beta}_{A}\left(a\right)$ for each a ∈ A are described.

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.

We investigate stability of various classes of topological algebras and individual algebras under small deformations of multiplication.

We show that a real Banach algebra A such that ||a²|| = ||a||² for a ∈ A is a subalgebra of the algebra ${C}_{\mathbb{H}}\left(X\right)$ of continuous quaternion-valued functions on a compact set X.

We characterize unital topological algebras in which all maximal two-sided ideals are closed.

Properties of topologically invertible elements and the topological spectrum of elements in unital semitopological algebras are studied. It is shown that the inversion $x\mapsto {x}^{-1}$ 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.

Let X be a completely regular Hausdorff space, $$\U0001d516$$ a cover of X, and $${C}_{b}(X,\mathbb{K};\U0001d516)$$ the algebra of all $$\mathbb{K}$$ -valued continuous functions on X which are bounded on every $$S\in \U0001d516$$ . A description of quotient algebras of $${C}_{b}(X,\mathbb{K};\U0001d516)$$ is given with respect to the topologies of uniform and strict convergence on the elements of $$\U0001d516$$ .

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