A Characterization of Jordan Homomorphism on Banach Algebras

Abbas Zivari-Kazempour

doi:10.1155/2014/698621

Abstract

Let 𝒜 and ℬ be Banach algebras and let φ:𝒜→ℬ be a Jordan homomorphism. We show that, under special hypotheses, φ is ring homomorphism. Some related results are given as well.

Highlights

Let A and B be Banach algebras and let φ : A → B be a linear map
It is obvious that ring homomorphisms are Jordan, but the converse is false, in general
The converse is true under certain conditions

Summary

Introduction

Let A and B be Banach algebras and let φ : A → B be a linear map. φ is called Jordan homomorphism if φ (ab + ba) = φ (a) φ (b) + φ (b) φ (a) , (a, b ∈ A) , (1)or equivalently, φ(a2) = φ(a)2 for all a ∈ A. Φ is called Jordan homomorphism if φ (ab + ba) = φ (a) φ (b) + φ (b) φ (a) , (a, b ∈ A) , (1) Each Jordan homomorphism from a commutative Banach algebra A into C is a ring homomorphism. In [1], Zelazko proved that each Jordan homomorphism of Banach algebra A into a semisimple commutative Banach algebra B is ring homomorphism (see Theorem 1.1 of [2]).

Results

Conclusion