Functional identities of degree 2 in CSL algebras

Document Type : Research Paper


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


‎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.


Main Subjects

R.L. An and J.C. Hou, Characterization of derivations on reexive algebras, Linear Multilinear Algebra 61 (2013), no. 1, 1408--1418.
W. Arveson, Operator algebras and invariant subspaces, Ann. Math. (2) 100 (1974) 438--532.
K.I. Beidar, M. Brešar and M.A. Chebotar, Functional identities on upper triangular matrix algebras, J. Math. Sci. (New York) 102 (2000), no. 6, 4557--4565.
M. Brešar, On generalized biderivations and related maps, J. Algebra 172 (1995), no. 3, 764--786.
M. Brešar, M.A. Chebotar and W.S. Martindale, Functional Identities, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2007.
Y.H. Chen and J.K. Li, Mappings on some reexive algebras characterized by action on zero products or Jordan zero products, Studia Math. 206 (2011), no. 2, 121--134.
W.S. Cheung, Commuting maps of triangular algebras, J. Lond. Math. Soc. (2) 63 (2001), no. 1, 117--127.
K.R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series 191, Longman Scientific & Technical, Harlow; Copublished in the United States with John Wiley & Sons, New York, 1988.
D. Eremita, Functional identities of degree 2 in triangular rings, Linear Algebra Appl. 438 (2013) 584--597.
D. Eremita, Functional identities of degree 2 in triangular rings revisited, Linear Multi-linear Algebra 63 (2015), no. 3, 534--553.
F. Gilfeather, Derivations on certain CSL algebras, J. Operator Theory 11 (1984) 145--156.
F. Gilfeather and D.R. Larson, Commutants modulo the compact operators of certain CSL algebras, in: Topics in Modern Operator Theory (Timişoara/Herculane, 1980), pp. 105--120, Operator Theory: Adv. Appl. 2, Birkhauser, Basel-Boston, 1981.
J.K. Li and J.B. Guo, Characterizations of higher derivations and Jordan higher derivations on CSL algebras, Bull. Aust. Math. Soc. 83 (2011), no. 3, 486--499.
W.Y. Yu and J.H. Zhang, Lie triple derivations of CSL algebras, Internat. J. Theoret. Phys. 52 (2013), no. 6, 2118--2127.
J.H. Zhang, S. Feng, H.X. Li and R.H. Wu, Generalized biderivations of nest algebras, Linear Algebra Appl. 418 (2006), no. 1, 225--233.