tailieunhanh - Generalization of some properties of Banach algebras to fundamental locally multiplicative topological algebras

In this article we generalized some properties of Banach algebras, to a new class of topological algebras namely fundamental and fundamental locally multiplicative topological algebras (abbreviated by F LM ). Also the new notion of sub-multiplicatively metrizable topological algebra is given and some well known spectral properties of Banach algebras are generalized to such kind of algebras. | Turk J Math 36 (2012) , 445 – 451. ¨ ITAK ˙ c TUB doi: Generalization of some properties of Banach algebras to fundamental locally multiplicative topological algebras Ali Zohri and Ali Jabbari Abstract In this article we generalized some properties of Banach algebras, to a new class of topological algebras namely fundamental and fundamental locally multiplicative topological algebras (abbreviated by F LM ). Also the new notion of sub-multiplicatively metrizable topological algebra is given and some well known spectral properties of Banach algebras are generalized to such kind of algebras. Key Words: F LM algebras, fundamental topological algebras, holomorphic function, multiplicative linear functionals, semi-simple algebras, spectral radius 1. Introduction The notion of fundamental topological spaces (also algebras) has been introduced in [1] in 1990 extending the meaning of both local convexity and local boundedness. A topological linear space A is said to be fundamental one if there exists b > 1 such that for every sequence (xn ) of A, the convergence of bn (xn − xn−1 ) to zero in A implies that (xn ) is Cauchy. A fundamental topological algebra is an algebra whose underlying topological linear space is fundamental. The famous Cohen factorization theorem for complete metrizable fundamental topological algebras is proved in [1] and the nth roots and quasi square roots in fundamental topological algebras are studied in [4]. The fundamental locally multiplicative topological algebras (abbreviated by F LM ) with a property very similar to the normed algebras is also introduced in [2]. A fundamental topological algebra is called locally multiplicative if there exists a neighborhood U0 of zero such that, for every neighborhood V of zero, the sufficiently large powers of U0 lie in V . Also in [2] a topological structure is defined on the algebraic dual space of an F LM algebra to make it a normed space, and some of the famous theorems of Banach .