Journal of Inequalities and Applications 
Volume 2005 (2005), Issue 4, Pages 435-441
doi:10.1155/JIA.2005.435

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

Takeshi Miura,1 Sin-Ei Takahasi,1 and Go Hirasawa2

1Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa 992-8510, Japan
2Department of Mathematics, Nippon Institute of Technology, Miyashiro, Saitama 345-8501, Japan

Received 18 July 2003

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 f from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique ring homomorphism near to f.