Characterization of derivations on reflexive algebras

摘要

Let AlgL be a reflexive algebra on a Hilbert space H. We say that a linear map δ: AlgL!AlgL is derivable at ?∈ AlgL if δ(A)B+Aδ(B)=δ(?) for every A, B ∈ AlgL with AB=?. In this article, we give a necessary and sufficient condition for a mapδ on AlgL to be derivable at x. In particular, we show that every linear map δ derivable at ?≠0 from an irreducible CDC algebra (in particular, a nest algebra) into itself is a derivation. Moreover, if AlgL is a CSL algebra, and if for some nontrivial projection P∈L, P?P and (I-P)?(I-P) are left or right separating points in PAlgLP and (I-P)AlgL(I-P) respectively, then a linear map δ on AlgL is derivable at ? if and only if δ is a derivation.