A generalization of Gelfand‐Mazur theorem

Sung Guen Kim

doi:10.1155/s0161171293000481

Abstract

In this paper, we show that ifAis a unital semisimple complex Banach algebra with only the trivial idempotents and ifσA(x)is countable for eachx∈Fr(G(A)), thenA≅C, this generalizes the Gelfand-Mazur theorem.

Highlights

In this paper, we show that if A is a unital semisimple complex Banach algebra with only the trivial idempotents and if aA(x is countable for each x Fr(G(A)), A C; this generalizes the Gelfand-Mazur theorem
Let x Fr(G(A)), aA(X is a connected subset of C by Lemma 3
By Proposition 5, it suffices to show that A is simisimple and that A has only the trivial idempotents and that aA(X is countable for each x Fr(G(A))

Summary

Introduction

We show that if A is a unital semisimple complex Banach algebra with only the trivial idempotents and if aA(x is countable for each x Fr(G(A)), A C; this generalizes the Gelfand-Mazur theorem. From the theory of Sanach algebras, we know that Rad(A) is a closed two sided ideal. Let A be a tmital Banach algebra. Rad(A)= {x A[PA(xy 0 for all yeA}.

Results

Conclusion