Characterization of Hilbert C^*-module higher derivations

Abstract

Let ℳ be a Hilbert C * -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert C * -module higher derivations { φ n : ℳ → ℳ } n = 0 ∞ with φ 0 = I satisfying

and all Hilbert C * -module derivations { ψ n : ℳ → ℳ } n = 1 ∞ satisfying

and we show that for every Hilbert C * -module higher derivation { φ n } n = 0 ∞ on ℳ , there exists a unique sequence of Hilbert C * -module derivations { ψ n } n = 1 ∞ on ℳ such that

for all positive integers n, where the inner summation is taken over all positive integers r j with ∑ j = 1 k r j = n .