A CHARACTERIZATION OF HIGHER DERIVATIONS ON BANACH ALGEBRAS

Abstract

Let \(\mathcal{A}\) be a Banach algebra and let every module-valued derivation from \(\mathcal{A}\) to any Banach \(\mathcal{A}\)-bimodule be continuous.  We show that if {d_m} is a higher derivation from \(\mathcal{A}\) to a Banach algebra \(\mathcal{B}\) with continuous d₀, then there exist a continuous left \(\mathcal{A}\)-module homomorphism U : \(\mathcal{B}\)(\(\mathcal{A}\)₁,\(\mathcal{B}\)) \(\longrightarrow\) \(\mathcal{B}\) and a sequence {D_m} of module-valued derivations from \(\mathcal{A}\) into \(\mathcal{B}\)(\(\mathcal{A}\)₁,\(\mathcal{B}\)) such that d_m = U ◦ D_m (m ≥ 1), and as a consequence {d_m} is automatically continuous.  We also obtain a partial result concerning innerness of higher derivations on W^*-algebras.