Left derivable or Jordan left derivable mappings on Banach algebras

Document Type : Research Paper


Department of Mathematics‎, ‎East China University of Science and Technology‎, ‎Shanghai‎, ‎China.


‎Let $\mathcal{A}$ be a unital Banach algebra‎, ‎$\mathcal{M}$ be a left $\mathcal{A}$-module‎, ‎and $W$ in $\mathcal{Z}(\mathcal{A})$ be a left separating point of $\mathcal{M}$‎. ‎We show that if $\mathcal{M}$ is a unital left $\mathcal{A}$-module and $\delta$ is a linear mapping from $\mathcal{A}$ into $\mathcal{M}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $\delta$ is a Jordan left derivation; (ii)$\delta$ is left derivable at $W$; (iii) $\delta$ is Jordan left derivable at $W$; (iv)$A\delta(B)+B\delta(A)=\delta(W)$ for each $A,B$ in $\mathcal{A}$ with $AB=BA=W$‎.
‎Let $\mathcal{A}$ have property ($\mathbb{B}$) (see Definition \ref{Prop_B})‎, ‎$\mathcal{M}$ be a Banach left $\mathcal{A}$-module‎, ‎and $\delta$ be a continuous linear operator from $\mathcal{A}$ into $\mathcal{M}$‎. ‎Then $\delta$ is a generalized Jordan left derivation if and only if $\delta$ is Jordan left derivable at zero‎. ‎In addition‎, ‎if there exists an element $C\in\mathcal{Z}(\mathcal{A})$ which is a left separating point of $\mathcal{M}$‎, ‎and $Rann_{\mathcal{M}}(\mathcal{A})=\{0\}$‎, ‎then $\delta$ is a generalized left derivation if and only if $\delta$ is left derivable at zero.


Main Subjects

‎J‎. ‎Alaminos‎, ‎J‎. ‎Extremera‎, ‎A.R‎. ‎Villena and M‎. ‎Bresar‎, Characterizing homomorphisms and derivations on C*-algebras‎, Proc. Roy. Soc. Edinburgh Sect. A‎ ‎137 (2007) 1--7‎.
‎J‎. ‎Alaminos‎, ‎M‎. ‎Bresar‎, ‎J‎. ‎Extremera and A.R‎. ‎Villena‎, Characterizing Jordan maps on C*-algebras through zero products‎, Proc. Edinb. Math. Soc. (2) ‎53 (2010) 543--555‎.
‎J‎. ‎Alaminos‎, ‎M‎. ‎Bresar‎, ‎J‎. ‎Extremera and A.R‎. ‎Villena‎, Maps preserving zero products‎, Studia Math. ‎193 (2009) 131--159‎.
‎M‎. ‎Ashraf‎, ‎N‎. ‎Rehman and S‎. ‎Ali‎, On Jordan left derivations of Lie ideals in prime rings‎, Southeast Asian Bull. Math.‎ ‎25 (2002) 379--382‎.
‎M‎. ‎Brešar and J‎. ‎Vukman‎, On left derivations and related mappings‎, Proc. Amer. Math. Soc. ‎110 (1990) 7--16‎.
‎M.A‎. ‎Chebotar‎, ‎W‎. ‎Ke and P‎. ‎Lee‎, Maps characterrized by action on zero products‎, Pacific J. Math. ‎216 (2004) 217--228‎.
‎M.A‎. ‎Chebotar‎, ‎W‎. ‎Ke‎, ‎P‎. ‎Lee and R‎. ‎Zhang‎, On maps preserving zero Jordan products‎, Monatsh. Math. ‎149 (2006) 91--101‎.
‎A‎. ‎Ebadiana and M‎. ‎Eshaghi Gordji‎, Left Jordan derivations on Banach algebras‎, Iran. J. Math. Sci. Inform. ‎6 (2011) 1--6‎.
‎B‎. ‎Fadaee and H‎. ‎Ghahramani‎, Continuous linear maps on reflexive algebras behaving like Jordan left derivations at idempotent-product elements‎, ArXiv:1312.6953 [math.OA] (2014)‎.
‎H‎. ‎Ghahramani‎, Additive mappings derivable at nontrivial idempotents on Banach algebras‎, Linear Multilinear Algebra‎ ‎60 (2012) 725--742‎.
‎H‎. ‎Ghahramani‎, Characterizations of left derivable maps at non-trivial idempotents on nest algebras‎, ArXiv:1312.6959 [math‎. ‎OA] (2014)‎.
‎J‎. ‎Hou and L‎. ‎Zhao‎, Zero-product preserving additive maps on symmetric operator spaces and self-adjoint operator spaces‎, Linear Algebra Appl.‎ ‎399 (2005) 235--244‎.
‎J‎. ‎Li and J‎. ‎Zhou‎, Jordan left derivations and some left derivable maps‎, Oper. Matrices‎ ‎4 (2010)‎, ‎no‎. ‎1‎, ‎127--138‎.
‎J‎. ‎Vukman‎, Jordan left derivations on semiprime rings‎, Math. J. Okayama Univ. ‎39 (1997) 1--6‎.
‎J‎. ‎Vukman‎, On left Jordan derivations of rings and Banach algebras‎, Aequationes Math. ‎75 (2008)‎, ‎no‎. ‎3‎, ‎260--266‎.
‎J‎. ‎Zhu and C‎. ‎Xiong‎, Generalized derivable mappings at the point zero on nest algebras‎, Acta Math. Sinica (Chin. Ser.) ‎45 (2002)‎, ‎no‎. ‎4‎, ‎783--788‎.
‎J‎. ‎Zhu and C‎. ‎Xiong‎, Generalized derivable mappings at zero point on some reflexive operator algebras‎, Linear Algebra Appl. ‎397 (2005) 367--379‎.