Characterization of derivations on reflexive algebras |
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Authors: | Runling An Jinchuan Hou |
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Institution: | 1. Department of Mathematics , Taiyuan University of Technology , Taiyuan 030024 , P.R. China runlingan@yahoo.com.cn;3. Department of Mathematics , Taiyuan University of Technology , Taiyuan 030024 , P.R. China |
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Abstract: | Let Alg? be a reflexive algebra on a Hilbert space H. We say that a linear map δ: Alg??→?Alg? is derivable at Ω?∈?Alg? if δ(A)B?+?Aδ(B)?=?δ(Ω) for every A, B?∈?Alg? with AB?=?Ω. In this article, we give a necessary and sufficient condition for a map δ on Alg? to be derivable at Ω. In particular, we show that every linear map δ derivable at Ω?≠?0 from an irreducible CDC algebra (in particular, a nest algebra) into itself is a derivation. Moreover, if Alg? is a CSL algebra, and if for some nontrivial projection P?∈??, PΩP and (I???P)Ω(I???P) are left or right separating points in PAlg?P and (I???P)Alg?(I???P) respectively, then a linear map δ on Alg? is derivable at Ω if and only if δ is a derivation. |
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Keywords: | derivations derivable maps reflexive algebras |
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