Nearly Jordan ∗ -Homomorphisms between Unital ð ¶ âˆ— -Algebras

Let ð ´ , ð µ be two unital ð ¶ âˆ— -algebras. We prove that every almost unital almost linear mapping â„Ž : ð ´ â†’ ð µ which satisfies â„Ž ( 3 ð ‘› ð ‘¢ 𠑦 + 3 ð ‘› 𠑦 ð ‘¢ ) = â„Ž ( 3 ð ‘› ð ‘¢ ) â„Ž ( 𠑦 ) + â„Ž ( 𠑦 ) â„Ž ( 3 ð ‘› ð ‘¢ ) for all ð ‘¢ ∈ 𠑈 ( ð ´ ) , all 𠑦 ∈ ð ´ , and all ð ‘› = 0 , 1 , 2 , … , is a Jordan homomorphism. Also, for a unital ð ¶ âˆ— -algebra ð ´ of real rank zero, every almost unital almost linear continuous mapping â„Ž ∶ ð ´ â†’ ð µ is a Jordan homomorphism when â„Ž ( 3 ð ‘› ð ‘¢ 𠑦 + 3 ð ‘› 𠑦 ð ‘¢ ) = â„Ž ( 3 ð ‘› ð ‘¢ ) â„Ž ( 𠑦 ) + â„Ž ( 𠑦 ) â„Ž ( 3 ð ‘› ð ‘¢ ) holds for all ð ‘¢ ∈ ð ¼ 1 ( ð ´ s a ), all 𠑦 ∈ ð ´ , and all ð ‘› = 0 , 1 , 2 , … . Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan ∗ -homomorphisms between unital ð ¶ âˆ— -algebras by using the fixed points methods.