On \(\mathbf{\beta}\)-regular families in bornological algebras. (Q2516942)

Let \(A\) be a bornological algebra. An element \(x\in A\) is called left (resp. right) bounding if there is an unbounded set \(D\subset A\) such that the set \(xD\) (resp. \(Dx\)) is bounded. This concept is analogous to that of a topological divisor of zero in the case of a topological algebra. A subset \(S\) of a bornological algebra \(A\) is called left (resp. right) \({\mathcal B}\)-regular if there is a bornological extension of \(A\) in which all elements of \(S\) are left (resp. right) invertible. The main result of the paper states that for an arbitrary bornological algebra there exists a bornological extension in which all non-left-bounding (resp. non-right-bounding) are left (resp. right) invertible.