Let $\mathcal{R}$ be a commutative ring with unity, $\mathcal{A}$ be an algebra over $\mathcal{R}$ and $\mathcal{Z}\left(\mathcal{A}\right)$ be the center of $\mathcal{A}$. Recall that an $\mathcal{R}$-linear map $L:\mathcal{A}\to \mathcal{A}$ is called a derivation on $\mathcal{A}$ if for all $\mathfrak{S},\mathfrak{T}\in \mathcal{A},L\left(\mathfrak{S}\mathfrak{T}\right)=L\left(\mathfrak{S}\right)\mathfrak{T}+\mathfrak{S}L\left(\mathfrak{T}\right)$. An $\mathcal{R}$-linear map $L:\mathcal{A}\to \mathcal{A}$ is called a Lie derivation (resp. Lie triple derivation ) on $\mathcal{A}$ if for all $\mathfrak{S},\mathfrak{T},\mathcal{W}\in \mathcal{A},L\left(\right[\mathfrak{S},\mathfrak{T}\left]\right)=\left[L\right(\mathfrak{S}),\mathfrak{T}]+[\mathfrak{S},L(\mathfrak{T}\left)\right](resp.L(\left[\right[\mathfrak{S},\mathfrak{T}],\mathcal{W}])=[\left[L\right(\mathfrak{S}),\mathfrak{T}],\mathcal{W}]+[[\mathfrak{S},L(\mathfrak{T}\left)\right],\mathcal{W}]+[[\mathfrak{S},\mathfrak{T}],L\left(\mathcal{W}\right)]$) , where $[\mathfrak{S},\mathfrak{T}]=\mathfrak{S}\mathfrak{T}-\mathfrak{T}\mathfrak{S}$ is the usual Lie product. Let $\mathbb{N}$ be the set of non-negative integers and $\mathcal{D}={\left\{d_m\right\}}_{m\in \mathbb{N}}$ be a family of additive mappings $d_m:\mathcal{A}\to \mathcal{A}$ such that $d_0=i{d}_{\mathcal{A}}$, the identity map on $\mathcal{A}$. Then $\mathcal{D}$ is called :

In [1], Abdullaev initiated the study of Lie n-derivations. Define the sequence of polynomials: $p_1\left(\mathfrak{S}\right)=\mathfrak{S}$ and $p_n({\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_n)=[p_{n-1}({\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_{n-1}),{\mathfrak{S}}_n]$ for all $n\in \mathbb{Z}$ and $n\ge 2$. Here $p_n({\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_n)$ is known as the ($n-1)$-th commutator. An additive (linear) map $L_m:\mathcal{A}\to \mathcal{A}$ is called a Lie n-higher derivation if for all ${\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_n\in \mathcal{A}$. In particular, by giving different values to n, we obtain Lie higher derivation, Lie triple higher derivation and Lie n-higher derivations. These derivations collectedly are referred as Lie type higher derivations.

Since last few decades, examining the various properties of derivations defined through the most well known rule given by Leibniz under the influence of various algebraic structures is a vast topic of study among the algebraists. In [5], Bresar characterized an additive Lie derivation as the sum of a derivation and an additive map on a prime ring $\mathcal{R}$ with $ch\left(\mathcal{R}\right)\ne 2$. In [10], Johnson worked on Lie derivation on ${\mathbb{C}}^{*}$-algebras and proved that every continous linear Lie derivation from a ${\mathbb{C}}^{*}$-algebra $\mathcal{A}$ into a Banach $\mathcal{A}$-bimodule $\mathcal{M}$ can be written as $\tau +h$ (i.e., every continous linear Lie derivation from a ${\mathbb{C}}^{*}$-algebra $\mathcal{A}$ into a Banach $\mathcal{A}$-bimodule $\mathcal{M}$ is standard), where $\tau :\mathcal{A}\to \mathcal{M}$ is a derivation and $h:\mathcal{A}\to Z\left(\mathcal{M}\right)$ vanishing at each commutators. In [15], Mathieu and Villena proved that on ${\mathbb{C}}^{*}$-algebra, every linear Lie derivation is standard. Qi and Hou [20], worked on nest algebras and proved that additive Lie derivations of nest algebras on Banach spaces is standard.

During the recent question of finding the condition under which a linear map becomes a Lie derivation or simply a derivation influenced observations of so many researchers (see [2,4,9,12,18,19] and references therein). The purpose of the above studies in the most of cases, was to obtain the restrictions under which Lie derivations or derivations can be completely determined by the action on some subsets of the algebras. There are several articles on the study of local actions of Lie derivations of operator algebras. In [14], Lu and Jing proved that for a Banach space $\mathcal{X}$ of dimension greater than two and a linear map $L:\mathcal{B}\left(\mathcal{X}\right)\to \mathcal{B}\left(\mathcal{X}\right)$ such that $L\left(\right[\mathfrak{S},\mathfrak{T}\left]\right)=\left[L\right(\mathfrak{S}),\mathfrak{T}]+[\mathfrak{S},L(\mathfrak{T}\left)\right]$ for all $\mathfrak{S},\mathfrak{T}\in \mathcal{B}\left(\mathcal{X}\right)\mathrm{with}\mathfrak{S}\mathfrak{T}=0$ (resp. $\mathfrak{S}\mathfrak{T}=P$, where $\mathfrak{Q}$ is a fixed nontrivial idempotent), then there exist an operator $\mathcal{T}\in \mathcal{B}\left(\mathcal{X}\right)$ and a linear map $\varphi :\mathcal{B}\left(\mathcal{X}\right)\to \mathbb{C}I$ vanishing at all the commutators $[\mathfrak{S},\mathfrak{T}]$ with $\mathfrak{S}\mathfrak{T}=0(\mathrm{resp}.\mathfrak{S}\mathfrak{T}=P)$ such that $L\left(\mathfrak{S}\right)=\mathcal{T}\mathfrak{S}+\varphi \left(\mathfrak{S}\right)$ for all $\mathfrak{S}\in \mathcal{B}\left(\mathcal{X}\right)$. In [8], Ji and Qi proved that if $\mathcal{T}$ is a triangular algebra over a commutative ring $\mathcal{R}$ then under certain restrictions on $\mathcal{T}$, if $L:\mathcal{T}\to \mathcal{T}$ is an $\mathcal{R}$-linear map satisfying $L\left(\right[\mathfrak{S},\mathfrak{T}\left]\right)=\left[L\right(\mathfrak{S}),V]+[\mathfrak{S},L(V\left)\right]\mathrm{for}\mathrm{all}\mathfrak{S},\mathfrak{T}\in \mathcal{T}\mathrm{with}\mathfrak{S}\mathfrak{T}=0(\mathrm{resp}.\mathfrak{S}\mathfrak{T}=\mathfrak{Q}$, where $\mathfrak{Q}$ is the standard idempotent of $\mathcal{T}$ ), then $L=d+\varphi$, where $d:\mathcal{T}\to \mathcal{T}$ is a derivation and $\varphi :\mathcal{T}\to Z\left(\mathcal{T}\right)$ is an $\mathcal{R}$-linear map vanishing at all the commutators $[\mathfrak{S},\mathfrak{T}]$ with $\mathfrak{S}\mathfrak{T}=0(\mathrm{resp}.\mathfrak{S}\mathfrak{T}=\mathfrak{Q})$. Qi and Hou [19] characterized Lie derivation on von Neumann algebra $\mathcal{A}$ without central summands of type $I_1$. In [21], Qi and Ji proved the same result for $\mathfrak{S}\mathfrak{T}=\mathfrak{Q}$, where $\mathfrak{Q}$ is a core-free projection. In [18], Qi characterized Lie derivation on $\mathcal{J}$-subspace lattice algebras and proved the same result due to Lu and Jing [14] on $\mathcal{J}$-subspace lattice algebra Alg$\mathcal{L}$, where $\mathcal{L}$ is a $\mathcal{J}$-subspace lattice on a Banach space $\mathcal{X}$ over the real or complex field with dimension greater than 2. Liu [13] studied the characterization of Lie triple derivation on von Neumann algebra with no central abelian projections. For further references (see [7,23,24], and references therein). Recently, M. Ashraf and A. Jabeen[3] characterized the Lie type derivations on von Neumann Algebra with no central summands of type $I_1$, where they showed that every Lie type derivation on von Neumann algebra has standard form at zero product as well as at projection product.

The objective of this paper is to investigate Lie type higher derivations on von Neumann algebras with no central summands of type $I_1$ and prove that on a von Neumann algebra every Lie type higher derivation has standard form at zero product as well as at projection product. Precisely, we prove that every additive map $L_m:\mathcal{A}\to \mathcal{A}$ satisfying $L_m\left(p_n({\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_n)\right)={\sum }_{l_1+l_2+···+l_n=m}{p}_n\left(L_{l_1}\left({\mathfrak{S}}_1\right),L_{l_2}\left({\mathfrak{S}}_2\right),L_{l_3}\left({\mathfrak{S}}_3\right),\cdots ,L_{l_n}\left({\mathfrak{S}}_n\right)\right)$ for all ${\mathfrak{S}}_1,{\mathfrak{S}}_2,\cdots ,{\mathfrak{S}}_n\in \mathcal{A}$ with ${\mathfrak{S}}_1{\mathfrak{S}}_2=0$ is of the form $L_m\left(\mathfrak{S}\right)={\varphi }_m\left(\mathfrak{S}\right)+{\zeta }_m\left(\mathfrak{S}\right)$ for all $\mathfrak{S}\in \mathcal{A},$ where ${\varphi }_m:\mathcal{A}\to \mathcal{A}$ is an additive higher derivation and ${\zeta }_m:\mathcal{A}\to Z\left(\mathcal{A}\right)$ an additive higher map whose range is in $Z\left(\mathcal{A}\right).$ Further, we discuss some more related results.

In this section, we discuss the characterization of Lie type higher derivation on von Neumann algebras having no central summands of type $I_1$ at zero product. In proving our main results, we use the following known lemmas:

The first main result of this paper is the following theorem.

For projections ${\mathfrak{Q}}_p,{\mathfrak{Q}}_q\in \mathcal{A}$, let ${\mathfrak{Q}}_0={\mathfrak{Q}}_p{L}_m\left({\mathfrak{Q}}_p\right){\mathfrak{Q}}_q-{\mathfrak{Q}}_q{L}_m\left({\mathfrak{Q}}_p\right){\mathfrak{Q}}_p=0$ and let us define a map ${\pi }_m:\mathcal{A}\to \mathcal{A}$ as an inner higher derivation ${\pi }_m\left(\mathfrak{S}\right)=[\mathfrak{S},{\mathfrak{Q}}_0]$ for all $\mathfrak{S}\in \mathcal{A}.$ Clearly $L_m=L_m^{\prime }-{\pi }_m$ is a Lie n-higher derivation. Since One easily gets ${\mathfrak{Q}}_p{L}_m^{\prime }\left({\mathfrak{Q}}_p\right){\mathfrak{Q}}_q={\mathfrak{Q}}_q{L}_m^{\prime }\left({\mathfrak{Q}}_p\right){\mathfrak{Q}}_p=0.$ Accordingly, it suffices to consider only those Lie n-higher derivations, which satisfy ${\mathfrak{Q}}_p{L}_m\left({\mathfrak{Q}}_p\right){\mathfrak{Q}}_q={\mathfrak{Q}}_q{L}_m\left({\mathfrak{Q}}_p\right){\mathfrak{Q}}_p=0.$

We give proof of Theorem 2.1 in series of lemmas. We begin with the following lemmas:

Now define two maps ${\varphi }_m:\mathcal{A}\to \mathcal{A}$ by ${\varphi }_m\left(\mathfrak{S}\right)=L_m\left(\mathfrak{S}\right)-{\zeta }_m\left(\mathfrak{S}\right)$ and ${\zeta }_m:\mathcal{A}\to Z\left(\mathcal{A}\right)$ by ${\zeta }_m\left(\mathfrak{S}\right)={\zeta }_{m_1}\left({\mathfrak{Q}}_p\mathfrak{S}{\mathfrak{Q}}_p\right)+{\zeta }_{m_2}\left({\mathfrak{Q}}_q\mathfrak{S}{\mathfrak{Q}}_q\right)$ for all $\mathfrak{S}\in \mathcal{A}$. Then one can easily observe that ${\varphi }_m\left({\mathfrak{Q}}_i\right)=0,{\varphi }_m\left({\mathcal{A}}_{ij}\right)\subseteq {\mathcal{A}}_{ij};i,j=1,2and{\varphi }_m\left({\mathfrak{S}}_{ij}\right)=L_m\left({\mathfrak{S}}_{ij}\right);forall{\mathfrak{S}}_{ij}\in {\mathcal{A}}_{ij};i,j=1,2(i\ne j).$