On Gelfand-Mazur theorem on a class of F-algebras

Abstract

Abstract A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence(xn ) in A, (xn ) is Cauchy whenever the sequence bn (xn − xn-1 ) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra,then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theoremfor a large class of F-algebras, containing both locally bounded algebras and locally convex algebras withbounded elements.