Summary
In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = − aD(a −1)a holds for all invertible elementsa ∈A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.
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References
Chernoff, P. R.,Representations, automorphisms and derivations of some operator algebras. J. Funct. Anal.12 (1973), 275–289.
Dales, H. G.,Automatic continuity. Bull. London Math. Soc.10 (1978), 129–183.
Davison, T. M. K.,Jordan derivations and quasi-bilinear forms. Comm. Algebra12 (1 (1984), 23–32.
Gleason, A. M.,The definition of a quadratic form. Amer. Math. Monthly73 (1966), 1049–1056.
Herstein, I. N.,Jordan derivations of prime rings. Proc. Amer. Math. Soc.8 (1957), 1104–1110.
Johnson, B. E.,Continuity of derivations on commutative algebras. Amer. J. Math.91 (1969), 1–10.
Kurepa, S.,The Cauchy functional equation and scalar product in vector spaces. Glasnik Mat.-Fiz. Astr.19 (1964), 23–36.
Kurepa, S.,Quadratic and sesquilinear functionals. Glasnik Mat.-Fiz. Astr.20 (1965), 79–92.
Kurepa, S.,On bimorphisms and quadratic forms on groups. Aequationes Math.9 (1973), 30–45.
Röhmel, J.,Eine Charakterisierung quadratischer Formen durch eine Funktional geichung. Aequationes Math.15 (1977), 163–168.
Sinclair, A. M.,Jordan homomorphisms and derivations on simisimple Banach algebras. Proc. Amer. Math. Soc.24 (1970), 209–214.
Singer, I. M. andWermer, J.,Derivations on commutative normed algebras. Math. Ann.129 (1955), 260–264.
Vrbová, P.,Quadratic functionals and bilinear forms. Časopis Pést. Mat.98 (1973), 159–161.
Vukman, J.,A result concerning additive functions in hermitian Banach *-algebras and an application. Proc. Amer. Math. Soc.91 (1984), 367–372.
Vukman, J.,Some results concerning the Cauchy functional equation in certain Banach algebras. Bull. Austral. Math. Soc.31 (1985), 137–144.
Vukman, J.,Some functional equations in Banach algebras and an application. Proc. Amer. Math. Soc.100 (1987), 133–136.
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Vukman, J. Some remarks on derivations in Banach algebras and related results. Aeq. Math. 36, 165–175 (1988). https://doi.org/10.1007/BF01836088
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DOI: https://doi.org/10.1007/BF01836088