Characterization of derivations on reflexive algebras

Abstract

Let Algℒ be a reflexive algebra on a Hilbert space H. We say that a linear map δ: Algℒ → Algℒ is derivable at Ω ∈ Algℒ if δ(A)B + Aδ(B) = δ(Ω) for every A, B ∈ Algℒ with AB = Ω. In this article, we give a necessary and sufficient condition for a map δ on Algℒ 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 Algℒ is a CSL algebra, and if for some nontrivial projection P ∈ ℒ, PΩP and (I − P)Ω(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 Ω if and only if δ is a derivation.