Characterization of higher derivations on CSL algebras

Abstract

Let AlgL be a CSL algebra. We say that a family of linear maps δ={δn,δn:AlgL→AlgL,n∈N} is higher derivable at Ω∈AlgL if ∑i+j=nδi(A)δj(B)=δn(Ω) for all A,B∈AlgL with AB=Ω. In this paper, a necessary and sufficient condition for a family of linear maps δ={δn,n∈N} on AlgL to be higher derivable at Ω∈AlgL is given. Moreover, we show that if there is a faithful projection P in L such that PΩP and (I−P)Ω(I−P) are a left or right separating point in PAlgLP and (I−P)AlgL(I−P) respectively, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω if and only if it is a higher derivation. In particular, if AlgL is an irreducible CDCSL algebra or a nest algebra, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω≠0 if and only if it is a higher derivation.