Abstract and Applied Analysis
Volume 2011 (2011), Article ID 513128, 12 pages
http://dx.doi.org/10.1155/2011/513128
Research Article

Nearly Jordan ∗ -Homomorphisms between Unital 𝐶 ∗ -Algebras

A. Ebadian,1 S. Kaboli Gharetapeh,1 and M. Eshaghi Gordji2,3

1Department of Mathematics, Urmia University, Urmia, Iran
2Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran
3Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran

Received 26 February 2011; Revised 7 April 2011; Accepted 10 April 2011

Academic Editor: Irena Lasiecka

Copyright © 2011 A. Ebadian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐴 , 𝐵 be two unital 𝐶 ∗ -algebras. We prove that every almost unital almost linear mapping ℎ : 𝐴 → 𝐵 which satisfies ℎ ( 3 𝑛 𝑢 𝑦 + 3 𝑛 𝑦 𝑢 ) = ℎ ( 3 𝑛 𝑢 ) ℎ ( 𝑦 ) + ℎ ( 𝑦 ) ℎ ( 3 𝑛 𝑢 ) for all 𝑢 ∈ 𝑈 ( 𝐴 ) , all 𝑦 ∈ 𝐴 , and all 𝑛 = 0 , 1 , 2 , … , is a Jordan homomorphism. Also, for a unital 𝐶 ∗ -algebra 𝐴 of real rank zero, every almost unital almost linear continuous mapping ℎ ∶ 𝐴 → 𝐵 is a Jordan homomorphism when ℎ ( 3 𝑛 𝑢 𝑦 + 3 𝑛 𝑦 𝑢 ) = ℎ ( 3 𝑛 𝑢 ) ℎ ( 𝑦 ) + ℎ ( 𝑦 ) ℎ ( 3 𝑛 𝑢 ) holds for all 𝑢 ∈ 𝐼 1 ( 𝐴 s a ), all 𝑦 ∈ 𝐴 , and all 𝑛 = 0 , 1 , 2 , … . Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan ∗ -homomorphisms between unital 𝐶 ∗ -algebras by using the fixed points methods.