Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras

Takeshi Miura,Sin-Ei Takahasi,Go Hirasawa

doi:10.1155/jia.2005.435

Abstract

We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. Using a signum effectively, we can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras. As a direct corollary, we show that to each approximate Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to.

Highlights

Introduction and statement of resultsIt seems that the stability problem of functional equations had been first raised by Ulam: For what metric groups G is it true that an ε-automorphism of G is necessarily near to a strict automorphism?An answer to the above problem has been given as follows
We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism
We show that to each approximate Jordan homomorphism f from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to f

Summary

Introduction

Introduction and statement of resultsIt seems that the stability problem of functional equations had been first raised by Ulam (cf. [11, Chapter VI] and [12]): For what metric groups G is it true that an ε-automorphism of G is necessarily near to a strict automorphism?An answer to the above problem has been given as follows. We prove that a Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra is a ring homomorphism. We can give a simple proof of the Hyers-Ulam-Rassias stability of a Jordan homomorphism between Banach algebras.

Results

Conclusion