Summary.
It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. Let A be a complex semisimple Banach algebra and let D : A → A be a linear mapping satisfying the relation D(x 2) = 2xD(x) for all x ∈ R. In this case D = 0.
Throughout, R will represent an associative ring with center Z(R). A ring R is n-torsion free, where n > 1 is an integer, if nx = 0, x ∈ R implies x = 0. As usual the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identities [xy, z] = [x, z] y + x [y, z] and [x, yz] = [x, y] z + y [x, z] for all x, y, z ∈ R. Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies that a = 0. An additive mapping D is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R, and is called a Jordan derivation in case D(x 2) = D(x)x + xD(x) is fulfilled for all x ∈ R. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein ([8]) has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation (see also [1]). Cusack ([5]) has generalized Herstein’s result to 2-torsion free semiprime rings (see [2] for an alternative proof). An additive mapping D : R → R is called a left derivation if D(xy) = yD(x) + xD(y) holds for all pairs x, y ∈ R and is called a left Jordan derivation (or Jordan left derivation) in case D(x 2) = 2xD(x) is fulfilled for all x ∈ R. In this paper by a Banach algebra we mean a Banach algebra over the complex field.
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Manuscript received: July 12, 2006 and, in final form, September 18, 2006.
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Vukman, J. On left Jordan derivations of rings and Banach algebras. Aequ. math. 75, 260–266 (2008). https://doi.org/10.1007/s00010-007-2872-z
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DOI: https://doi.org/10.1007/s00010-007-2872-z