# Functional identities of degree 2 in CSL algebras

Document Type: Research Paper

Author

School of Mathematics and Information Science‎, ‎Henan Polytechnic University‎, ‎Jiaozuo‎, ‎454000‎, ‎P.R‎. ‎China.

Abstract

‎Let $\mathscr{L}$ be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space $\mathbf{H}$ with ${\rm dim}\hspace{2pt}\mathbf{H}\geq 3$‎, ‎${\rm Alg}\mathscr{L}$‎ ‎the CSL algebra associated with $\mathscr{L}$ and $\mathscr{M}$ be an algebra containing ${\rm Alg}\mathscr{L}$‎. ‎This article is aimed at describing the form of‎ ‎additive mapppings $F_1‎, ‎F_2‎, ‎G_1‎, ‎G_2\colon {\rm Alg}\mathscr{L}\longrightarrow \mathscr{M}$ satisfying functional identity‎ ‎$F_1(X)Y+F_2(Y)X+XG_2(Y)+YG_1(X)=0$ for all $X‎, ‎Y\in {\rm Alg}\mathscr{L}$‎. ‎As an application generalized inner biderivations and commuting‎ ‎additive mappings are determined‎.

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### History

• Receive Date: 10 May 2016
• Revise Date: 05 July 2016
• Accept Date: 12 July 2016