Characterization of jordan*-derivations by local action on rings with involution

Document Type : Research Paper

Authors

Department of Mathematics, Shanxi University, Taiyuan, China

Abstract

Let R be a ring with an involution ∗ and a symmetric idempotent e. It is shown that, under some mild conditions on R, an additive map δ : R → R satisfies δ(ab + ba) = δ(a)b ∗ + aδ(b) + δ(b)a ∗ + bδ(a) whenever ab = e for a, b ∈ R if and only if δ is a Jordan *-derivation.

Keywords


[1] M. Breˇsar, M. A. Chebotar and W. S. Martindale III, Functional identities,Birkh¨auser Basel: (2006).
[2] M. Bresar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 9-21.
[3] M. Breˇsar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math., 38 (1989), 178-185.
[4] C.-L. Chuang, A. Foˇsner and T.-K. Lee, Jordan τ -derivations of locally matrix rings, Algebra Represent. Theory, 16 (2013), 755-763.
[5] A. Foˇsner and T.-K.Lee, Jordan *-derivations of finite-dimensional semiprime algebras, Canad.Math.Bull.,57(2014),51-60.
[6] J.-C. Hou and X.-F. Qi, Additive maps derivable at some points on J -subspace lattice algebras, Linear Algebra Appl., 429 (2008), 1851-1863.
[7] S. Kurepa, Quadratic and sesquilinear functionals, Glasg. Math. J., 20 (1965),79-92.
[8] T.-K. Lee and Y.-Q. Zhou, Jordan *-derivations of prime rings, J. Algebra Appl.,13 (2014), 1350126.
[9] T.-K. Lee, T.-L. Wong and Y. Zhou, The structure of Jordan *-derivations of prime rings, Linear Multilinear Algebra, 63 (2015), 411-422.
[10] X.-F. Qi and F.-F. Zhang, Multiplicative Jordan *-derivations on rings with in-volution, Linear Multilinear Algebra, 64 (2016), 1145-1162.
[11] P. Semrl, ˇ Quadratic functionals and Jordan *-derivations, Studia Math., 97 (1991), 157-165.
[12] P. Semrl, ˇ Quadratic and quasi-quadratic functionals, Proc. Amer. Math. Soc.,119 (1993), 1105-1113.
[13] P. Semrl, ˇ Jordan *-derivations of standard operator algebras, Proc. Amer. Math.Soc., 120 (1994), 515-518.
[14] J. Vukman, Some functional equations in Banach algebras and an application,Proc. Amer.Math. Soc.,100(1987),133-136.
[15] F.-F. Zhang and X.-F. Qi, Characterizing local Jordan *-derivations on rings with involution, submitted.