Kawa, A.H.; Alsuraiheed, T.; Hasan, S.N.; Ali, S.; Wani, B.A. Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions. Mathematics2023, 11, 4770.
Kawa, A.H.; Alsuraiheed, T.; Hasan, S.N.; Ali, S.; Wani, B.A. Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions. Mathematics 2023, 11, 4770.
Kawa, A.H.; Alsuraiheed, T.; Hasan, S.N.; Ali, S.; Wani, B.A. Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions. Mathematics2023, 11, 4770.
Kawa, A.H.; Alsuraiheed, T.; Hasan, S.N.; Ali, S.; Wani, B.A. Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions. Mathematics 2023, 11, 4770.
Abstract
Let $m$ and $n$ be the fixed positive integers. Suppose $\mathcal{A}$ is a von Neuman algebra with no central summands of type $I_{1}$ and $L_{m}$ be a Lie type higher derivation i.e., an additive (linear) map $L_{m} :\mathcal{A}\to \mathcal{A}$ such that
\begin{equation}\label{def2}
\[L_{m}(p_{n}(\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{S}_{n}))=\sum_{l_1+l_2+\cdots+l_n=m}p_{n}\big(L_{l_1}(\mathfrak{S}_{1}),L_{l_2}(\mathfrak{S}_{2}),\cdots,L_{l_n}(\mathfrak{S}_{n})\big)\] %L_{m}(p_{n}(\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{S}_{n}))=\sum_{l_1+l_2+\cdots+l_n=m}p_{n}\big(L_{l_1}(\mathfrak{S}_{1}),L_{l_2}(\mathfrak{S}_{2}),\cdots,L_{l_n}(\mathfrak{S}_{n})\big)\nonumber
\end{equation}
for all $\mathfrak{S}_{1},\mathfrak{S}_{2},\cdots,\mathfrak{S}_{n}\in \mathcal{A}$. In the present paper, we study Lie type higher derivations on von Neuman algebras and prove that every additive Lie type higher derivation on $\mathcal{A}$ has a standard form at zero product as well as at projection product. Further, we discuss some more related results.
Copyright:
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