Zinedine, Ahmed; Tajmouati, Abdelaziz On \(\mathbf{\beta}\)-regular families in bornological algebras. (English) Zbl 1329.46005 Afr. Mat. 26, No. 3-4, 503-507 (2015). 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. Reviewer: Wiesław Tadeusz Żelazko (Warszawa) MSC: 46A17 Bornologies and related structures; Mackey convergence, etc. 46H05 General theory of topological algebras Keywords:bornological algebras; topological algebras; permanently singular elements; bounding elements × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Akkar, M., Tajmouati, A., Zinedine, A.: Idéaux non removables dans la classe des algèbres bornologiques. Functiones et Approximatio 29, 7-16 (2001) [2] Arens, R.: Inverse-producing extensions of normed algebras. Trans. Am. Math. Soc. 88, 536-548 (1958) · Zbl 0082.11105 · doi:10.1090/S0002-9947-1958-0095419-5 [3] Bollobàs, B.: Adjoining inverses to commutative Banach algebras. Trans. Am. Math. Soc. 181, 165-179 (1973) · Zbl 0274.46039 · doi:10.1090/S0002-9947-1973-0324418-9 [4] Hogbe-Nlend, H.: Théorie des bornologies et applications. Lecture Notes in Mathematics, Vol. 213. Springer (1971) · Zbl 0225.46005 [5] Muller, V.: Adjoining inverses to noncommutative Banach algebras and extensions of operators. Studia Math. 91, 73-77 (1988) · Zbl 0677.46031 [6] Müller, V.: Adjoining the one-side inverses to non-commutative Banach algebras. Bull. Polish Acad. Sci. 37, 415-419 (1989) · Zbl 0758.46036 [7] Tajmouati, A., Zinedine, A.: Eléments singuliers permanents dans la classe des algèbres bornologiques. Functiones et Approximatio XXVII, 31-38 (1999) · Zbl 0988.46038 [8] Tajmouati, A., Zinedine, A.: Eléments singuliers permanents dans la classe des algèbres bornologiques multiplicativement convexes. Functiones et Approximatio XXX, 117-126 (2002) · Zbl 1082.46002 [9] Tajmouati, A., Zinedine, A.: Sur les J-diviseurs topologiques de zéro dans les algèbres de Jordan métrisables. Functiones et Approximatio XXXII, 1-7 (2004) · Zbl 1072.46512 [10] Mouanis, H., Zinedine, A.: Sur le radical permanent dans les algèbres topologiques. Rend. Sem. Mat. Univ. Padova 114, 103-107 (2005) · Zbl 1166.46310 [11] Żelazko, W.: Concerning a characterization of permanently singular elements in commutative locally convex algebras. Math. Structures-Computational Mathematics-Mathematical Modeling, 2, 326-333 (1984) · Zbl 0607.46028 [12] Żelazko, W.: On permanently singular elements in commutative m-convex locally convex algebras. Studia Math. 37, 181-190 (1971) · Zbl 0214.13704 [13] Żelazko, W.: Selected topics in topological algebras, Lecture Notes Serie 31. Mathematisk Institut. Aarhus universitet, Aarhus (1971) · Zbl 0221.46041 [14] Zinedine, A.: Eléments singuliers permanents et idéaux non relevables dans certaines classes d’algèbres topologiques ou bornologiques. Thèse doctorale, Université de Fès (2001) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.