×

zbMATH — the first resource for mathematics

Multiplicative derivations on \(C(X)\). (English) Zbl 0843.46018
Summary: Let \(X\) be a completely regular topological space satisfying the first axiom of countability with no isolated points, and let \(C(X)\) be the algebra of all continuous functions on \(X\). A mapping \(d: C(X)\to C(X)\) is called a multiplicative derivation if \(d(fg)= fd(g)+ gd(f)\) for every pair of functions \(f,g\in C(X)\) (no linearity or continuity of \(d\) is assumed). We obtain a complete description of such mappings and give examples to show that the above assumptions on the space \(X\) are essential.

MSC:
46E25 Rings and algebras of continuous, differentiable or analytic functions
47B47 Commutators, derivations, elementary operators, etc.
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Aczél, J., Dhombres, J.: Functional equations in several variables. In: Encyclopedia of Mathematics and Its Applications, Vol. 31. London New York: Cambridge Univ. Press. 1989. · Zbl 0685.39006
[2] Chernoff, P. R.: Representations, automorphisms and derivations of some operator algebras. J. Funct. Anal.12, 275–289 (1973). · Zbl 0252.46086
[3] Johnson, B. E., Sinclair, A. M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math.90, 1067–1073 (1968). · Zbl 0179.18103
[4] Kaplansky, I.: Ring isomorphisms of Banach algebras. Canad. J. Math.6, 374–381 (1954). · Zbl 0058.10505
[5] Kaplansky, I.: Algebraic and analytic aspects of operator algebras. Regional Conference Series in Mathematics. Providence, R. I.: Amer. Math. Soc. 1970. · Zbl 0217.44902
[6] Samue, P., Zariski, O.: Commutative Algebra, New York: Van Nostrand. 1958.
[7] Šemrl, P.: Ring derivations on standard operator algebras. J. Funct. Anal.112, 318–324 (1993). · Zbl 0801.47024
[8] Šemrl, P.: The functional equation of multiplicative derivation is superstable on standard operator algebras. Integral Equations Operator Theory18, 118–122 (1994). · Zbl 0810.47029
[9] Singer, I. M., Wermer, J.: Derivations on commutative normed algebras. Math. Ann.129, 260–264 (1955). · Zbl 0067.35101
[10] Thomas, M. P.: The image of a derivation is contained in the radical, Ann. of Math.128, 435–460 (1988). · Zbl 0681.47016
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.