All-derivable subsets for nest algebras on Banach spaces

Abstract

Let N be a nest on a complex Banach space X and let AlgN be the associated nest algebra. We say that a subset S ⊂ AlgN is an all-derivable subset of AlgN if every linear map δ from AlgN into itself derivable on S (i.e. δ satisfies that, for each Z ∈ S, δ(A)B + Aδ(B) = δ(Z) for any A, B ∈ AlgN with AB = Z) is a derivation. In this paper, we show that S is an all-derivable subset of AlgN if span{ran(Z) : Z ∈ S} is dense in X or ∩{ker Z : Z ∈ S} = {0}. Mathematics Subject Classification: 47B47, 47L35

Linear Maps Which are Anti-derivable at Zero

“…Burgos, Cabello-Sánchez and the third author of this note explore in [6] those linear maps between C * -algebras which are * -homomorphisms at certain points of the domain, for example, at the unit element or at zero. We refer to [12,22,25,[52][53][54][55][56][57][58] and [60] for additional related results.…”