Derivable Maps and Generalized Derivations on Nest and Standard Algebras

Abstract

Abstract. For an algebra A, an A-bimodule M, and m P M, define a relation on A by RApm, 0q " tpa, bq P AˆA : amb " 0u. We show that generalized derivations on unital standard algebras on Banach spaces can be characterized precisely as derivable maps on these relations. More precisely, if A is a unital standard algebra on a Banach space X then ∆ P LpA, BpXqq is a generalized derivation if and only if ∆ is derivable on RApM, 0q, for some M P BpXq. We give an example to show this is not the case in general for nest algebras. On the other hand, for an idempotent P in a nest algebra A " algN on a Hilbert space H such that P is either left-faithful to N or right-faithful to N K , if δ P LpA, BpHqq is derivable on RApP, 0q then δ is a generalized derivation. IntroductionFor vector spaces U and V, we use LpU, Vq to denote the set of all linear maps from U to V. For a unital algebra A and an A-bimodule M, δ P LpA, Mq is called a derivation if for all a, b P A, δpabq " δpaqb`aδpbq and δ is called a generalized derivation if for all a, b P A, δpabq " δpaqb`aδpbq´aδp1qb. Fix any u, v P M, then δ uv paq " ua´av, @ a P A is a generalized derivation; the study of such maps dates back at least to [13]. By a relation on A, we mean a nonempty subset R A Ď AˆA. We say δ P LpA, Mq is derivable on R A if δpabq " δpaqb`aδpbq for all pa, bq P R A . Derivable maps have garnered interests of many researchers, for example, authors of [2], [4], [8], and [10] have studied maps that are derivable on R A " tpa, bq P AˆA : a " bu, such maps are called Jordan derivations . In [3], [6][7], [9], [11], and [14-16], the authors have studied derivable maps on relations R A pcq " tpa, bq P AˆA : ab " cu, for some c P A. Not all derivable maps are derivations. If a derivable map is not a derivation, it is natural to ask whether it is close to being a derivation, e.g. whether it is a generalized derivation. Every generalized derivation is

Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations

“…Let M be a unital U-bimodule. We say U ij is faithful to M if U i j m = {0} implies e j m = 0 and mU i j = {0} implies me i = 0, for all m ∈ M. Clearly, U 11 and U 22 are always faithful to M. Peirce decomposition can be useful for characterizing derivable maps, see [19,21,22]. Proof.…”