Jordan Derivable Maps On Operator Algebras

Derivations, Jordan derivations, which are important both in theory and applications, are active topics in operator algebra and operator theory. In recent years, more and more mathematicians are interested in discussing the conditions for maps to be derivations and Jordan derivations. For example, the study of the maps derivable, Jordan derivable at some point. In this paper, we study Jordan derivable maps on operator algebras, we give a necessary and sufficient condition for every additive map on operator algebras which is Jordan derivable at some fixed point, and new characterization of derivations was obtained.The structure of this paper is as follows:In the first chapter, we introduce the background of Jordan derivable maps, and we list the main results of this thesis.In the second chapter, we give a necessary and sufficient condition for a family of linear maps δ={δn:Alg(?)â†'Alg(?),n∈N}(no continuous assumption)which is Jordan higher derivable at Ω, where Ω=0,P, I respectively. As its applications, we show that a family of linear maps δ=(δn)n∈N from an irreducible CDCSL algebra or nest subalgebras of factor von Nuemann algebras(particularly, nest algebra of Hilbert spaces)into itself, which is Jordan higher derivable at such Ω=P, I is a higher derivation.In the third chapter, we give a necessary and sufficient condition for a linear map5on an arbitrary ring with a nontrivial idempotent. which is Jordan derivable at Ω=ΩP=PΩ. As its applications, we show that a linear map δ from triangular rings, irreducible CDCSL algebras or nest algebras, which is Jordan derivable at Ω=Ω,P=PΩ is a derivation.