In this paper, we introduce the concept of pexider Hilbert C∗-module higher {An, Bn, Dn}-derivations. Specifically, we focus on a Hilbert C∗-module M and provide a comprehensive characterization of these pexider Hilbert C∗-module higher {An, Bn, Dn}-derivations {Φn}∞ n=0 on M in relation to pexider Hilbert C∗-module {αn, βn, δn}-derivations {φn}∞ n=1 on M. We demonstrate that for every pexider Hilbert C∗-module higher {An, Bn, Dn}-derivation {Φn}∞ n=0 on M, there exists a unique sequence of pexider Hilbert C∗-module {αn, βn, δn}-derivations {φn}∞ n=1 on M such that $$\left\{\begin{array}{lr}\varphi_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1\Phi_{r_1}\Phi_{r_2}\ldots \Phi_{r_k}\Big), \\\alpha_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1A_{r_1}A_{r_2}\ldots A_{r_k}\Big), \\\beta_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1B_{r_1}B_{r_2}\ldots B_{r_k}\Big), \\\delta_n=\sum_{k=1}^n\Big(\sum_{\sum_{j=1}^k r_j=n}(-1)^{k-1}~r_1D_{r_1}D_{r_2}\ldots D_{r_k}\Big), \end{array}\right. $$ for all positive integers n, where the inner summation is taken over all positive integers rj with Σk j=1 rj = n.

In this paper we study higher derivations from JB*-algebras into Banach Jordan algebras. We show that every higher derivation {dm} from a JB*-algebra A into a JB*-algebra B is continuous provided that dO is a *-homomorphism. Also it is proved that every Jordan higher derivation from a commutative C*-algebra or from a C*-algebra which has minimal idempotent and is the closure of its socle is continuous.

‎A sequence of continuous linear mappings $\{\Phi_n\}_{n=0}^\infty$ form a Hilbert C* -‎module‎ M into M is called a local higher derivation if for each $a\in\mathfrak{M}$ there is a continuous higher derivation $\{\varphi_{a,n}\}_{n=0}^\infty$ ‎on‎ M such that $\Phi_n(a)=\varphi_{a,n}(a)$ for each non-negative integer n‎. ‎In this paper w‎e show that if M is a Hilbert C* -‎module‎ such that every local derivation on M is a derivation, then each local higher derivation on M‎ is a higher derivation‎. Also, we prove that each local higher derivation on a unital C*-algebra is automatically continuous‎.

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Let A be a Banach algebra and X be an arbitrary Banach A-module. In this paper, we study the second transpose of derivations with value in dual Banach A-module X : Indeed, for a continuous derivation D: A 􀀀 ! X we obtain a necessary and su cient condition such that the bounded linear map   D00: A  􀀀 ! X   to be a derivation, where  is composition of restriction and canonical injection maps. This characterization generalizes some well known results in [2].

Let S be an inverse semigroup with an upward directed set of idempotents E. In this paper we prove that if S is amenable, then l1(S) Ä^ l1(S) is module amenable as an l1(E)-module. Also we show that L1(S) Ä^ l1 (S) is module super-amenable if an appropriate group homomorphic image of S is finite.