SOME RESULTS ON 2-BANACH ALGEBRAS Vol. 4(1) Jan. 2016, No. 2, pp. 11-14 R. TAPDIGOGLU U. KOSEM

Abstract

We consider a 2-Banach algebra and prove some new results, including Gelfand-Mazur type theorems. IntroductionIn this article we give some new results for 2-Banach algebras. In particular, we prove an analogy of famous Gelfand-Mazur theorem for 2-Banach algebras.Recall that the concept of 2-Banach algebra, apparently, was introduced by Mohammed and Siddiqui [1]. Following by Mohammed and Siddiqui [1], note that a 2-Banach algebra B is an algebra with dim B > 2 which is a 2-Banach space (with respect to 2-norm topology) and in addition, the following condition being satisfied:Note that the concept of 2-Banach space was introduced by Gähler [2]. Later, the various aspect of this concept have been studied in [3,4,5,6,7,8,9]. In particular, Mohammed and Siddique [1] proved analog of some known results of the usual Banach algebras in 2-Banach algebras.Before giving our results, let us give some necessary definitions and notations. Let X be a vector space of dimension greater than 1 and ∥., .∥ be a real function on X × X satisfying the following conditions:1) ∥a, b∥ = 0 if and only if a and b are linearly dependent; 2) ∥a, b∥ = ∥b, a∥ ; 3) ∥λa, b∥ = |λ| . ∥a, b∥ for any number λ ; 4) ∥a + b, c∥ ≤ ∥a, c∥ + ∥b, c∥ for every a, b, c ∈ X. ∥., .∥ is called a 2-norm and X equipped with ∥., .∥ is a 2-normed space (see [2]). Gähler [2] has proved that ∥., .∥ is a non-negative function.A sequence {x n } in 2-Banach space X is called a Cauchy sequence if there exists y, z ∈ X such that y and z are linearly independent, the lim ∥x n − x m , y∥ = 0 and 2010 Mathematics Subject Classification. 46J15.