Characterization of Hilbert C*-module higher derivations

Abstract

Abstract Let ℳ {\mathcal{M}} be a Hilbert C * {\mathrm{C}^{*}} -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert C * {\mathrm{C}^{*}} -module higher derivations { φ n : ℳ → ℳ } n = 0 ∞ {\{\varphi_{n}:\mathcal{M}\rightarrow\mathcal{M}\}_{n=0}^{\infty}} with φ 0 = I {\varphi_{0}=I} satisfying φ n ( 〈 x , y 〉 z ) = ∑ i + j + k = n 〈 φ i ( x ) , φ j ( y ) 〉 φ k ( z ) ( x , y , z ∈ ℳ , n ∈ ℕ ∪ { 0 } ) \varphi_{n}(\langle x,y\rangle z)=\sum_{i+j+k=n}\langle\varphi_{i}(x),\varphi_% {j}(y)\rangle\varphi_{k}(z)\quad(x,y,z\in\mathcal{M},\,n\in\mathbb{N}\cup\{0\}) and all Hilbert C * {\mathrm{C}^{*}} -module derivations { ψ n : ℳ → ℳ } n = 1 ∞ {\{\psi_{n}:\mathcal{M}\rightarrow\mathcal{M}\}_{n=1}^{\infty}} satisfying ψ n ( 〈 x , y 〉 z ) = 〈 ψ n ( x ) , y 〉 z + 〈 x , ψ n ( y ) 〉 z + 〈 x , y 〉 ψ n ( z ) ( x , y , z ∈ ℳ , n ∈ ℕ ) , \psi_{n}(\langle x,y\rangle z)=\langle\psi_{n}(x),y\rangle z+\langle x,\psi_{n% }(y)\rangle z+\langle x,y\rangle\psi_{n}(z)\quad(x,y,z\in\mathcal{M},\,n\in% \mathbb{N}), and we show that for every Hilbert C * {\mathrm{C}^{*}} -module higher derivation { φ n } n = 0 ∞ {\{\varphi_{n}\}_{n=0}^{\infty}} on ℳ {\mathcal{M}} , there exists a unique sequence of Hilbert C * {\mathrm{C}^{*}} -module derivations { ψ n } n = 1 ∞ {\{\psi_{n}\}_{n=1}^{\infty}} on ℳ {\mathcal{M}} such that ψ n = ∑ k = 1 n ( ∑ ∑ j = 1 k r j = n ( - 1 ) k - 1 ⁢ r 1 ⁢ φ r 1 ⁢ φ r 2 ⁢ … ⁢ φ r k ) \psi_{n}=\sum_{k=1}^{n}\biggl{(}\sum_{\sum_{j=1}^{k}r_{j}=n}(-1)^{k-1}~{}r_{1}% \varphi_{r_{1}}\varphi_{r_{2}}\dots\varphi_{r_{k}}\biggr{)} for all positive integers n , where the inner summation is taken over all positive integers r j {r_{j}} with ∑ j = 1 k r j = n {\sum_{j=1}^{k}r_{j}=n} .