Characterizations of generalized derivations associated with Hochschild 2-cocycles and higher derivations

Abstract

Let A be a unital Banach algebra and M be a unital A-bimodule. A bilinear mapping α : A X A → M is called a Hochschild 2-cocycle if xα(y; z) - α(xy; z) + α(x; yz) - α(x; y)z = 0 for any x; y; z ∈ A. We show that if δ is a linear mapping from A into M satisfying δ(xy) = δ(x)y + xδ(y) + α(x,y) for any x, y ∈ A with xy = W, where W ∈ A is a left or right separating point of M, then δ is a generalized Jordan derivation associated with a Hochschild 2-cocycle α. We also find the relation of higher derivations and generalized derivations associated with Hochschild 2-cocycles.

Keywords: Generalized derivation, Hochschild 2-cocycle, higher derivation, CSL algebra, C*-algebra