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Quasimultipliers on \(F\)-algebras. (English) Zbl 1230.46042

Summary: We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of \(F\)-algebras, in particular on complete \(k\)-normed algebras, \(0 < k \leq 1\), not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general \(F\)-algebras. The bilinearity and joint continuity of quasimultipliers on an \(F\)-algebra \(A\) are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra \(QM(A)\) of quasimultipliers of a complete \(k\)-normed algebra \(A\) having a minimal ultra-approximate identity.

MSC:

46H05 General theory of topological algebras
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[1] C. A. Akemann and G. K. Pedersen, “Complications of semicontinuity in C\ast -algebra theory,” Duke Mathematical Journal, vol. 40, pp. 785-795, 1973. · Zbl 0287.46073 · doi:10.1215/S0012-7094-73-04070-2
[2] K. McKennon, “Quasi-multipliers,” Transactions of the American Mathematical Society, vol. 233, pp. 105-123, 1977. · Zbl 0377.46038 · doi:10.2307/1997825
[3] R. Vasudevan and S. Goel, “Embedding of quasimultipliers of a Banach algebra into its second dual,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95, no. 3, pp. 457-466, 1984. · Zbl 0569.46024 · doi:10.1017/S0305004100061788
[4] M. S. Kassem and K. Rowlands, “The quasistrict topology on the space of quasimultipliers of a B\ast -algebra,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 101, no. 3, pp. 555-566, 1987. · Zbl 0631.46046 · doi:10.1017/S0305004100066913
[5] H. X. Lin, “Fundamental approximate identities and quasi-multipliers of simple AFC\ast -algebras,” Journal of Functional Analysis, vol. 79, no. 1, pp. 32-43, 1988. · Zbl 0678.46048 · doi:10.1016/0022-1236(88)90028-6
[6] H. X. Lin, “Support algebras of \sigma -unital C\ast -algebras and their quasi-multipliers,” Transactions of the American Mathematical Society, vol. 325, no. 2, pp. 829-854, 1991. · Zbl 0741.46025 · doi:10.2307/2001650
[7] B. Dearden, “Quasi-multipliers of Pedersen’s ideal,” The Rocky Mountain Journal of Mathematics, vol. 22, no. 1, pp. 157-163, 1992. · Zbl 0810.46061 · doi:10.1216/rmjm/1181072800
[8] Z. Argün and K. Rowlands, “On quasi-multipliers,” Studia Mathematica, vol. 108, no. 3, pp. 217-245, 1994. · Zbl 0824.46053
[9] M. Grosser, “Quasi-multipliers of the algebra of approximable operators and its duals,” Studia Mathematica, vol. 124, no. 3, pp. 291-300, 1997. · Zbl 0893.46041
[10] R. Yılmaz and K. Rowlands, “On orthomorphisms, quasi-orthomorphisms and quasi-multipliers,” Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 120-131, 2006. · Zbl 1110.46014 · doi:10.1016/j.jmaa.2005.05.074
[11] M. Kaneda, “Quasi-multipliers and algebrizations of an operator space,” Journal of Functional Analysis, vol. 251, no. 1, pp. 346-359, 2007. · Zbl 1157.46030 · doi:10.1016/j.jfa.2006.12.014
[12] M. Kaneda and V. I. Paulsen, “Quasi-multipliers of operator spaces,” Journal of Functional Analysis, vol. 217, no. 2, pp. 347-365, 2004. · Zbl 1067.46050 · doi:10.1016/j.jfa.2004.06.002
[13] G. Köthe, Topological Vector Spaces I, Springer, Berlin, Germany, 1969. · Zbl 0179.17001
[14] S. Rolewicz, Metric Linear Spaces, vol. 20 of Mathematics and Its Applications (East European Series), D. Reidel, Dordrecht, The Netherlands, 2nd edition, 1985. · Zbl 0573.46001
[15] J. L. Kelley and I. Namioka, Linear Topological Spaces, D. Van Nostrand, Princeton, NJ, USA; Springer, New York, NY, USA, 1976.
[16] L. Waelbroeck, Topological Vector Spaces and Algebras, Lecture Notes in Mathematics, Vol. 230, Springer, Berlin, Germany, 1971. · Zbl 0225.46001 · doi:10.1007/BFb0061234
[17] H. H. Schaefer, Topological Vector Spaces, Springer, New York, NY, USA, 1971. · Zbl 0217.16002
[18] A. Mallios, Topological Algebras. Selected Topics, vol. 124 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1986. · Zbl 0597.46046
[19] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, NY, USA, 1965. · Zbl 0182.16101
[20] N. J. Kalton, N. T. Peck, and J. W. Roberts, An F-Space Sampler, vol. 89 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1984. · Zbl 0556.46002
[21] M. Fragoulopoulou, Topological Algebras with Involution, vol. 200 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2005. · Zbl 1197.46001
[22] W. \DZelazko, Banach Algebras, Elsevier, Amsterdam, The Netherlands, 1973.
[23] H. G. Heuser, Functional Analysis, John Wiley & Sons, Chichester, UK, 1982. · Zbl 0465.47001
[24] C. Swartz, “Continuity and hypocontinuity for bilinear maps,” Mathematische Zeitschrift, vol. 186, no. 3, pp. 321-329, 1984. · Zbl 0545.46004 · doi:10.1007/BF01174886
[25] T. Husain, “Multipliers of topological algebras,” Dissertationes Mathematicae. Rozprawy Matematyczne, vol. 285, p. 40, 1989. · Zbl 0676.46038
[26] R. S. Doran and J. Wichmann, Approximate Identities and Factorization in Banach Modules, vol. 768 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1979. · Zbl 0418.46039
[27] E. Ansari-Piri, “A class of factorable topological algebras,” Proceedings of the Edinburgh Mathematical Society, vol. 33, no. 1, pp. 53-59, 1990. · Zbl 0699.46027 · doi:10.1017/S001309150002887X
[28] A. Bayoumi, Foundations of Complex Analysis in Non-locally Convex Spaces, Functions Theory without Convexity Conditions, vol. 193 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2003. · Zbl 1082.46001
[29] N. C. Phillips, “Inverse limits of C\ast -algebras,” Journal of Operator Theory, vol. 19, no. 1, pp. 159-195, 1988. · Zbl 0662.46063
[30] R. C. Busby, “Double centralizers and extensions of C\ast -algebras,” Transactions of the American Mathematical Society, vol. 132, pp. 79-99, 1968. · Zbl 0165.15501 · doi:10.2307/1994883
[31] B. E. Johnson, “An introduction to the theory of centralizers,” Proceedings of the London Mathematical Society, vol. 14, pp. 299-320, 1964. · Zbl 0143.36102 · doi:10.1112/plms/s3-14.2.299
[32] B. E. Johnson, “Continuity of centralisers on Banach algebras,” Journal of the London Mathematical Society, vol. 41, pp. 639-640, 1966. · Zbl 0143.36103 · doi:10.1112/jlms/s1-41.1.639
[33] L. A. Khan, N. Mohammad, and A. B. Thaheem, “Double multipliers on topological algebras,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 3, pp. 629-636, 1999. · Zbl 1007.46042 · doi:10.1155/S0161171299226294
[34] C. A. Akemann, G. K. Pedersen, and J. Tomiyama, “Multipliers of C\ast -algebras,” Journal of Functional Analysis, vol. 13, pp. 277-301, 1973. · Zbl 0258.46052 · doi:10.1016/0022-1236(73)90036-0
[35] R. A. Fontenot, “The double centralizer algebra as a linear space,” Proceedings of the American Mathematical Society, vol. 53, no. 1, pp. 99-103, 1975. · Zbl 0315.46049 · doi:10.2307/2040376
[36] R. Larsen, An Introduction to the Theory of Multipliers, Springer, New York, NY, USA, 1971. · Zbl 0213.13301
[37] F. D. Sentilles and D. C. Taylor, “Factorization in Banach algebras and the general strict topology,” Transactions of the American Mathematical Society, vol. 142, pp. 141-152, 1969. · Zbl 0185.21103 · doi:10.2307/1995349
[38] D. C. Taylor, “The strict topology for double centralizer algebras,” Transactions of the American Mathematical Society, vol. 150, pp. 633-643, 1970. · Zbl 0204.14701 · doi:10.2307/1995543
[39] B. J. Tomiuk, “Multipliers on Banach algebras,” Studia Mathematica, vol. 54, no. 3, pp. 267-283, 1976. · Zbl 0319.46033
[40] J. Wang, “Multipliers of commutative Banach algebras,” Pacific Journal of Mathematics, vol. 11, pp. 1131-1149, 1961. · Zbl 0127.33302 · doi:10.2140/pjm.1961.11.1131
[41] S. K. Jain and A. I. Singh, “Quotient rings of algebras of functions and operators,” Mathematische Zeitschrift, vol. 234, no. 4, pp. 721-737, 2000. · Zbl 0982.46027 · doi:10.1007/s002090000097
[42] L. A. Khan, “The general strict topology on topological modules,” in Function Spaces, vol. 435 of Contemporary Mathematics, pp. 253-263, American Mathematical Society, Providence, RI, USA, 2007. · Zbl 1148.46306
[43] L. A. Khan, “Topological modules of continuous homomorphisms,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 141-150, 2008. · Zbl 1152.46039 · doi:10.1016/j.jmaa.2008.01.025
[44] L. A. Khan, N. Mohammad, and A. B. Thaheem, “The strict topology on topological algebras,” Demonstratio Mathematica, vol. 38, no. 4, pp. 883-894, 2005. · Zbl 1101.46314
[45] W. Ruess, “On the locally convex structure of strict topologies,” Mathematische Zeitschrift, vol. 153, no. 2, pp. 179-192, 1977. · Zbl 0361.46002 · doi:10.1007/BF01179791
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