On the continuity of some linear maps on certain Banach algebras
محورهای موضوعی : Functional analysis
1 - Department of Mathematics, University of Kurdistan, Sanandaj, Iran
2 - Department of Mathematics, University of Kurdistan, Sanandaj, Iran
کلید واژه: derivation, Multiplier, Automatic continuity, Lau product,
چکیده مقاله :
Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras, $\theta$ be a nonzero character on $\mathcal{A}$ and let ${\mathcal{A}}\times_{\theta}{\mathcal{U}}$ be the corresponding Lau product Banach algebra. In this paper we investigate derivations and multipliers of ${\mathcal{A}}\times_{\theta}{\mathcal{U}}$ and study the automatic continuity of these maps. We also study continuity of the derivations for some special cases of $\mathcal{U}$ and the Banach $({{\mathcal{A}}\times_{\theta}\mathcal{U}})$-bimodule ${\mathcal{X}}$ and establish various results in this respect. Some of the results are devoted to find conditions under which one can represent a derivation on ${{\mathcal{A}}\times_{\theta}\mathcal{U}}$ as sum of two derivations in such a way that one of them is continuous. Some examples are also given.
[1] W. G. Bade, P. C. Curtis, The continuity of derivation of Banach algebras, J. Funct. Anal. 16 (1974), 372-378.
[2] W. G. Bade, P. C. Curtis, the structure of module derivations of Banach algebras of differentiable functions, J. Fun. 28 (1978), 226-247.
[3] F. T. Birtal, Isomorphism and isometric multipliers, Proc. Amer. Math. Soc. 13 (1962), 204-210.
[4] G. Brown, W. Moran, Point derivations on M(G), Bull. London Math. Soc. 8 (1976), 57-64.
[5] E. Christensen, Derivations of nest algebras, Math. Ann. 229 (1977), 155-161.
[6] H. G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monographs, Oxford University Press, Oxford, 2000.
[7] R. V. Garimella, Continuity of derivations on some semiprime Banach algebra, Proc. Amer. Math. Soc. 99 (1987), 289-292.
[8] G. H. Esslamzadeh, H. Ghahramani, Existence, automatic continuity and invariant submodules of generalized derivations on modules, Aequat. Math. 1 (2012) 84-185.
[9] H. Farhadi, H. Ghahramani, Automatic continuity of derivations on semidirect products of Banach algebras, Bull. Iran. Math. Soc. 47 (2021), 1925-1946.
[10] H. Farhadi, H. Ghahramani, Some notes on semidirect products of Banach algebras, Results Math. (2019), 74:107.
[11] H. Farhadi, H. Ghahramani, The first cohomology group of semidirect products of Banach algebras, Iran. J. Sci. Technol. Trans. Sci. 45 (2021), 695-706.
[12] H. Ghahramani, Additive mappings derivable at non-trivial idempotents on Banach algebras, Linear Multilinear Algebra. 60 (2012), 725-742.
[13] H. Ghahramani, Additive maps on some operator algebras behaving like (α, β)-derivations or generalized (α,β)-derivations at zero-product elements, Acta Math. Sci. Ser. B. Engl. Ed. 34 (2014), 1287-1300.
[14] N. Gronbak, Commutative Banach algebras, module derivations and semigroups, J. London Math. Soc. 40 (2) (1989), 137-157.
[15] S. Helgason, Multipliers of Banach algebras, Ann. Math. 64 (1956), 240-254.
[16] B. E. Johnson, A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073.
[17] R. Larsen, An Introduction to the Theory of Multipliers, Springer-Verlag, New York, 1971.
[18] A. T. Lau, Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), 161-175.
[19] K. B. Laursen, M. M. Neumann, An Introduction to Local Spectral Theory, Oxford University Press, New York, 2000.
[20] R. Loy, Continuity of derivations on topological algebras of power series, Bull. Aust. Math. Soc. 1 (3) (1969), 419-424.
[21] R. J. Loy, G. A. Willis, Continuity of derivations on B(E) for certain Banach spaces E, J. London. Math. Soc. s2-40 (2) (1989), 327-346.
[22] M. S. Monfared, On certain products of Banach algebras with applications to harmonic analysis, Studia Math. 178 (2007), 227-294.
[23] A. M. Peralta, B. Russo, Automatic continuity of derivations on C∗-algebras and JB∗-triples, J. Algebra. 399 (2014), 960-977.
[24] J. R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. 5 (1972), 432-438.
[25] V. Runde, Automatic continuity of derivations and epimorphisms, Pacific J. Math. 147 (2) (1991), 365-374.
[26] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460.
[27] A. R. Villena, Continuity of derivations on H∗-algebras, Proc. Amer. Math. Soc. 122 (1994), 821-826.
[28] A. R. Villena, Derivations with a hereditary domain, J. London Math. Soc. 57 (1998), 469-477.
[29] A. R. Villena, Derivations with a hereditary domain, II, Studia Math. 130 (1998), 275-291.
[30] J. K. Wang, Multipliers of commutative Banach algebras, Pacific J. Math. 11 (1961), 1131-1149.
