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**Nearly Jordan \(\ast\)-homomorphisms between unital \(C^{\ast}\)-algebras.**
*(English)*
Zbl 1223.39015

Summary: Let \(A, B\) be two unital \(C^{\ast}\)-algebras. We prove that every almost unital almost linear mapping \(h : A \rightarrow B\) which satisfies \(h(3^nuy + 3^nyu) = h(3^nu)h(y) + h(y)h(3^nu)\) for all \(u \in U(A)\), all \(y \in A\), and all \(n = 0, 1, 2, \dots\), is a Jordan homomorphism. Also, for a unital \(C^{\ast}\)-algebra \(A\) of real rank zero, every almost unital almost linear continuous mapping \(h : A \rightarrow B\) is a Jordan homomorphism when \(h(3^nuy + 3^nyu) = h(3^nu)h(y) + h(y)h(3^nu)\) holds for all \(u \in I_1 (A_{sa})\), all \(y \in A\), and all \(n = 0, 1, 2, \dots\). Furthermore, we investigate the Hyers-Ulam-Aoki-Rassias stability of Jordan \(\ast\)-homomorphisms between unital \(C^{\ast}\)-algebras by using the fixed points methods.

### MSC:

39B52 | Functional equations for functions with more general domains and/or ranges |

39B82 | Stability, separation, extension, and related topics for functional equations |

46L05 | General theory of \(C^*\)-algebras |

### Keywords:

unital \(C^{\ast}\)-algebras; Jordan homomorphism; Hyers-Ulam-Aoki-Rassias stability; fixed points methods
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\textit{A. Ebadian} et al., Abstr. Appl. Anal. 2011, Article ID 513128, 12 p. (2011; Zbl 1223.39015)

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