Main properties of the topology defined by a bornology on a topological linear space and main properties of Mackey Q-algebras are presented. Relationships of Mackey Q-algebras with other classes of topological algebras are described. It is shown that every Mackey Q-algebra is an advertibly Mackey complete algebra, every strongly sequential Mackey Q-algebra is a Q-algebra, every infrasequential Mackey Q-algebra is an advertibly complete algebra, and every infrasequential advertive Hausdorff algebra is a Mackey Q-algebra.
1. Abel, M. Advertive topological algebras. In General Topological Algebras. Tartu, 1999, 14–24; Math. Stud. (Tartu), 1. Est. Math. Soc., Tartu, 2001.
2. Abel, M. Topological algebras with pseudoconvexly bounded elements. In Topological Algebras, Their Applications, and Related Topics. Banach Center Publications, 67, 21–33. Polish Academy of Sciences, Warsaw, 2005.
https://doi.org/10.4064/bc67-0-2
3. Abel, M. Topological algebras with idempotently pseudoconvex von Neumann bornology. Contemp. Math., 2007, 427, 15–29. M. Abel: Mackey Q-algebras 53
4. Abel, M. Structure of locally idempotent algebras. Banach J. Math. Anal., 2007, 1(2), 195–207.
https://doi.org/10.15352/bjma/1240336216
5. Abel, M. Topological algebras with all elements bounded. In Proceedings of the 8th International Conference of Topological Algebras and Their Applications, 2015. De Gruyter Proc. Math. (accepted).
6. Akkar, M. Sur le groupe des ´el´ements inversibles d’une alg`ebre bornologique convexe, Q-alg`ebres bornologiques convexes. C. R. Acad. Sc. Paris, Ser. I, 1985, 300(2), 35–38.
7. Akkar, M. Caract´erisation des alg`ebres localement m-convexes dont l’ensemble des caract`eres est ´equiborn´e. Colloq. Math., 1995, 68(1), 59–65.
8. Arnold, B. H. Topologies defined by bounded sets. Duke Math. J., 1951, 18, 631–642.
https://doi.org/10.1215/S0012-7094-51-01855-8
9. Bedda, A. and Cheikh, O. H. Boundedness of characters in locally A-convex algebras. Rend. Circ. Mat. Palermo Ser. II, 1998, 47, 91–98.
https://doi.org/10.1007/BF02844724
10. El Adlouni, H. and Oubbi, L. Groupe des inversibles et rayon spectral dans les alg`ebres topologiques. Rend. Circ. Mat. Palermo, 2000, 49, 527–539.
https://doi.org/10.1007/BF02904263
11. El Kinani, A. Advertible compl´etute et structure de Q-alg`ebre. Rend. Circ. Mat. Palermo, 2001, 50, 427–442.
https://doi.org/10.1007/BF02844423
12. Hogbe-Nlend, H. Th´eories des bornologies et applications. Lecture Notes Math., 213. Springer-Verlag, Berlin, 1971.
13. Hogbe-Nlend, H. Les fondements de la th´eorie spectrale des alg`ebres bornologiques. Bol. Soc. Brasil. Mat., 1972, 3(1), 19–56.
14. Hogbe-Nlend, H. Bornologies and Functional Analysis. Introductory Course on the Theory of Duality Topology-Bornology and Its Use in Functional Analysis. North-Holland Math. Studies, 62. North-Holland Publ. Co., Amsterdam, 1977.
15. Husain, T. Multiplicative Functionals on Topological Algebras. Res. Notes in Math., 85. Bitman, Boston, 1983.
16. Mallios, A. Topological Algebras. Selected Topics. North-Holland Math. Studies, 124. North-Holland Publ. Co., Amsterdam, 1986.
17. Oudadess, M. A note on m-convex and pseudo-Banach structures. Rend. Circ. Mat. Palermo, 1992, 41(1), 105–110.
https://doi.org/10.1007/BF02844467
18. Rickart, C. E. General Theory of Banach Algebras. D. van Nostrand Co., Inc., Princeton, NJ, 1960.
19. Schaefer, H. H. Topological Vector Spaces. Second edition. Graduate Texts in Math., 3. Springer-Verlag, New York, 1999.
https://doi.org/10.1007/978-1-4612-1468-7
https://doi.org/10.1215/S0012-7094-56-02301-8
