J. Appl. Anal. ; ():– Research Article Madjid Eshaghi and Sadegh Abbaszadeh Approximate generalized derivations close to derivations in Lie C * -algebras Abstract: We apply a xed point theorem to prove that there exists a unique derivation close to an approxi- mately generalized derivation in Lie C * -algebras. Also, we prove the hyperstability of generalized derivations. In other words, we nd some conditions under which an approximately generalized derivation becomes a derivation. Keywords: Derivation of Lie algebras, generalized derivation, contractively subhomogeneous, expansively superhomogeneous, xed point theorem MSC : W, W, E, B, B DOI: ./jaa-- Received March , ; revised October , ; accepted October , Introduction and preliminaries The theory of nite dimensional complex Lie algebras is an important part of Lie theory. It has several appli- cations in physics and connections with other parts of mathematics. Much of point particle physics can be described in terms of Lie algebras and their representations. With an increasing amount of theory and ap- plications concerning Lie algebras of various dimensions, it is becoming necessary to ascertain which tools are applicable for handling them. The miscellaneous characteristics of Lie algebras constitute such tools and have also found applications: Casimir operators [], derived, lower central and upper central sequences, the Lie algebra of derivations, radicals, nilradicals, ideals, subalgebras [, ] and recently megaideals []. These characteristics are particularly crucial when considering possible connections among Lie algebras. Physically motivated relations between two Lie algebras, namely contractions and deformations, have been extensively studied; see, e.g., [, , ]. In , Fialovski [] considered general questions of deformations of Lie algebras over a eld of charac- teristic zero, and the related problems of computing cohomology with coecients in adjoint representations. In , Bjar and Laudal [] studied deformation of Lie algebras and Lie algebra of deformations. The con- cept of Lie algebra can be expressed in several dierent ways. The most familiar are in terms of generators and relations and in terms of a bilinear bracket on a vector space V satisfying the Jacobi identity. In physical notation, let X a be a basis for V . The bracket [⋅ , ⋅] can be specied by structure constants C c ab via the formula [X a , X b ]= C c ab X c . The structure constants are skew-symmetric in the lower indices a, b. A C * -algebra A endowed with the Lie product μ : A → A on A is called a Lie C * -algebra. See, e.g., [, ] for more information and various aspects on the combinatorics of the Lie C * -algebras. A ℂ-linear mapping D of the Lie C * -algebra A with multiplication μ is called a derivation if μD(x), y+ μx, D(y) = Dμ(x, y) (.) Madjid Eshaghi, Sadegh Abbaszadeh: Department of Mathematics, Semnan University, P.O. Box -, Semnan, Iran, e-mail: madjid.eshaghi@gmail.com, s.abbaszadeh.math@gmail.com Unauthenticated Download Date | 6/8/16 10:46 AM