Nearly Jordan ∗-Homomorphisms between Unital 𝐶^∗-Algebras

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^𝑛𝑢𝑦+3^𝑛𝑦𝑢)=ℎ(3^𝑛𝑢)ℎ(𝑦)+ℎ(𝑦)ℎ(3^𝑛𝑢) 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^𝑛𝑢𝑦+3^𝑛𝑦𝑢)=ℎ(3^𝑛𝑢)ℎ(𝑦)+ℎ(𝑦)ℎ(3^𝑛𝑢) holds for all 𝑢∈𝐼₁ (𝐴_sa), 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.

1. Introduction

The stability of functional equations was first introduced by Ulam [1] in 1940. More precisely, he proposed the following problem: given a group 𝐺₁, a metric group (𝐺₂,𝑑) and a positive number 𝜖, does there exist a 𝛿>0 such that if a function 𝑓∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(𝑓(𝑥𝑦),𝑓(𝑥)𝑓(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there exists a homomorphism 𝑇∶𝐺₁→𝐺₂ such that 𝑑(𝑓(𝑥),𝑇(𝑥))<𝜖 for all 𝑥∈𝐺₁?. As mentioned above, when this problem has a solution, we say that the homomorphisms from 𝐺₁ to 𝐺₂ are stable. In 1941, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive mappings under the assumption that 𝐺₁ and 𝐺₂ are Banach spaces. Hyers theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. This phenomenon of stability is called the Hyers-Ulam-Aoki-Rassias stability.

J. M. Rassias [5–7] established the stability of linear and nonlinear mappings with new control functions.

During the last decades, several stability problems of functional equations have been investigated by many mathematicians. A large list of references concerning the stability of functional equations can be found in [8–10].

Bourgin is the first mathematician dealing with the stability of ring homomorphisms. The topic of approximate ring homomorphisms was studied by a number of mathematicians, see [11–19] and references therein.

Jun and Lee [20] proved the following: Let 𝑋 and 𝑌 be Banach spaces. Denote by 𝜙∶𝑋−{0}×𝑌−{0}→[0,∞) a function such that 𝜙(𝑥,𝑦)=∑^∞_𝑛=03^−𝑛𝜙(3^𝑛𝑥,3^𝑛𝑦)<∞ for all 𝑥,𝑦∈𝑋−{0}. Suppose that 𝑓∶𝑋→𝑌 is a mapping satisfying 2𝑓𝑥+𝑦2−𝑓(𝑥)−𝑓𝑦≤𝜙𝑥,𝑦(1.1) for all 𝑥,𝑦∈𝑋−{0}. Then there exists a unique additive mapping 𝑇∶𝑋→𝑌 such that 𝑓(𝑥)(0)(𝑥≤1−𝑓−𝑇)3𝜙(𝑥,−𝑥)+𝜙(−𝑥,3𝑥)(1.2) for all 𝑥∈𝑋−{0}.

Recently, C. Park and W. Park [21] applied the Jun and Lee's result to the Jensen's equation in Banach modules over a 𝐶^∗-algebra. Johnson (Theorem 7.2 of [22]) also investigated almost algebra ∗-homomorphisms between Banach ∗-algebras: Suppose that U and B are Banach ∗-algebras which satisfy the conditions of (Theorem 3.1 of [22]). Then for each positive 𝜖 and 𝐾, there is a positive 𝛿 such that if 𝑇∈𝐿(𝑈,𝐵) with ‖𝑇‖<𝐾,‖𝑇^∨‖<𝛿 and ‖𝑇(𝑥^∗)^∗−𝑇(𝑥)‖<𝛿‖𝑥‖(𝑥∈𝑈), then there is a ∗-homomorphism 𝑇′∶𝑈→𝐵 with ‖𝑇−𝑇′‖<𝜖. Here 𝐿(𝑈,𝐵) is the space of bounded linear maps from U into B, and 𝑇^∨(𝑥,𝑦)=𝑇(𝑥𝑦)−𝑇(𝑥)𝑇(𝑦)(𝑥,𝑦∈𝑈). See [22] for details. Throughout this paper, let A be a unital 𝐶^∗-algebra with unit e, and B a unital 𝐶^∗-algebra. Let 𝑈(𝐴) be the set of unitary elements in A, 𝐴_sa∶={𝑥∈𝐴𝑥=𝑥^∗}, and 𝐼₁(𝐴_sa)={𝑣∈𝐴_sa∣‖𝑣‖=1,𝑣∈𝐼𝑛𝑣(𝐴)}. In this paper, we prove that every almost unital almost linear mapping ℎ∶𝐴→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∈𝑈(𝐴), all 𝑦∈𝐴, and all 𝑛=0,1,2,…, and that for a unital 𝐶^∗-algebra 𝐴 of real rank zero (see [23]), every almost unital almost linear continuous mapping ℎ∶𝐴→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∈𝐼₁(𝐴_sa), 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 point methods.

Note that a unital 𝐶^∗-algebra is of real rank zero, if the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [23]). We denote the algebric center of algebra 𝐴 by 𝑍(𝐴).

2. Jordan ∗-Homomorphisms on Unital 𝐶^∗-Algebras

By a following similar way as in [24], we obtain the next theorem.

Theorem 2.1. Let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓(0)=0 and that 𝑓3^𝑛𝑢𝑦+3^𝑛𝑦𝑢=𝑓3^𝑛𝑢𝑓𝑦+𝑓𝑦𝑓3^𝑛𝑢(2.1) for all 𝑢∈𝑈(𝐴), all 𝑦∈𝐴, and all 𝑛=0,1,2,…. If there exists a function 𝜙∶(𝐴−{0})²×𝐴→[0,∞) such that 𝜙(𝑥,𝑦,𝑧)=∑^∞_𝑛=03^−𝑛𝜙(3^𝑛𝑥,3^𝑛𝑦,3^𝑛𝑧)<∞ for all 𝑥,𝑦∈𝐴−{0} and all 𝑧∈𝐴 and that 2𝑓𝜇𝑥+𝜇𝑦2−𝜇𝑓(𝑥)−𝜇𝑓𝑦+𝑓𝑢^∗−𝑓(𝑢)^∗≤𝜙𝑥,𝑦,𝑢,(2.2) for all 𝜇∈𝕋 and all 𝑥,𝑦∈𝐴,𝑢∈(𝑈(𝐴)∪{0}). If lim_𝑛(𝑓(3^𝑛𝑒)/3^𝑛)∈𝑈(𝐵)∩𝑍(𝐵), then the mapping 𝑓∶𝐴→𝐵 is a Jordan ∗-homomorphism.

Proof. Put 𝑢=0, 𝜇=1 in (2.2), it follows from of [20, Theorem 1] that there exists a unique additive mapping 𝑇∶𝐴→𝐵 such that (𝑥)(𝑥)1‖𝑓−𝑇‖≤3𝜙(𝑥,−𝑥,0)+𝜙(−𝑥,3𝑥,0)(2.3) for all 𝑥∈𝐴−{0}. This mapping is given by 𝑇(𝑥)=lim_𝑛𝑓3^𝑛𝑥3^𝑛(2.4) for all 𝑥∈𝐴. By the same reasoning as the proof of [24, Theorem 1], 𝑇 is ℂ-linear and ∗-preserving. It follows from (2.1) that 𝑇𝑢𝑦+𝑦𝑢=lim_𝑛𝑓3^𝑛𝑢𝑦+3^𝑛𝑦𝑢3^𝑛=lim_𝑛𝑓3^𝑛𝑢3^𝑛𝑓𝑦𝑦𝑓3+𝑓^𝑛𝑢3^𝑛=𝑇(𝑢)𝑓𝑦+𝑓𝑦𝑇(𝑢)(2.5) for all 𝑢∈𝑈(𝐴), all 𝑦∈𝐴. Since 𝑇 is additive, then by (2.5), we have 3^𝑛𝑇𝑢𝑦+𝑦𝑢=𝑇𝑢3^𝑛𝑦+3^𝑛𝑦𝑢=𝑇(𝑢)𝑓3^𝑛𝑦+𝑓3^𝑛𝑦𝑇(𝑢)(2.6) for all 𝑢∈𝑈(𝐴) and all 𝑦∈𝐴. Hence, 𝑇𝑢𝑦+𝑦𝑢=lim_𝑛𝑇(𝑢)𝑓3^𝑛𝑦3^𝑛+𝑓3^𝑛𝑦3^𝑛𝑇(𝑢)=𝑇(𝑢)𝑇𝑦+𝑇𝑦𝑇(𝑢)(2.7) for all 𝑢∈𝑈(𝐴) and all 𝑦∈𝐴. By the assumption, we have 𝑇(𝑒)=lim_𝑛𝑓3^𝑛𝑒3^𝑛∈𝑈(𝐵)𝑍(𝐵)(2.8) hence, it follows by (2.5) and (2.7) that 2𝑇(𝑒)𝑇𝑦=𝑇(𝑒)𝑇𝑦+𝑇𝑦𝑇(𝑒)=𝑇𝑦𝑒+𝑒𝑦=𝑇(𝑒)𝑓𝑦+𝑓𝑦𝑇(𝑒)=2𝑇(𝑒)𝑓𝑦(2.9) for all 𝑦∈𝐴. Since 𝑇(𝑒) is invertible, then 𝑇(𝑦)=𝑓(𝑦) for all 𝑦∈𝐴. We have to show that 𝑓 is Jordan homomorphism. To this end, let 𝑥∈𝐴. By Theorem 4.1.7 of [25], 𝑥 is a finite linear combination of unitary elements, that is, 𝑥=∑^𝑛_𝑗=1𝑐_𝑗𝑢_𝑗(𝑐_𝑗∈ℂ,𝑢_𝑗∈𝑈(𝐴)), and then it follows from (2.7) that 𝑓𝑥𝑦+𝑦𝑥=𝑇𝑥𝑦+𝑦𝑥=𝑇^𝑛_𝑗=1𝑐_𝑗𝑢_𝑗𝑦+^𝑛_𝑗=1𝑐_𝑗𝑦𝑢_𝑗=^𝑛_𝑗=1𝑐_𝑗𝑇𝑢_𝑗𝑦+𝑦𝑢_𝑗=^𝑛_𝑗=1𝑐_𝑗𝑇𝑢_𝑗𝑦+𝑇𝑦𝑢_𝑗=^𝑛_𝑗=1𝑐_𝑗𝑇𝑢_𝑗𝑇𝑦+𝑇𝑦𝑇𝑢_𝑗=𝑇^𝑛_𝑗=1𝑐_𝑗𝑢_𝑗𝑇𝑦+𝑇𝑦𝑇^𝑛_𝑗=1𝑐_𝑗𝑢_𝑗(𝑥)𝑇𝑦𝑦𝑇(𝑥)(𝑥)𝑓𝑦𝑦𝑓(𝑥)=𝑇+𝑇=𝑓+𝑓(2.10) for all 𝑦∈𝐴. And this completes the proof of theorem.

Corollary 2.2. Let 𝑝∈(0,1), 𝜃∈[0,∞) be real numbers. Let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓(0)=0 and that f3^𝑛𝑢𝑦+3^𝑛𝑦𝑢=𝑓3^𝑛𝑢𝑓𝑦+𝑓𝑦𝑓3^𝑛𝑢(2.11) for all 𝑢∈𝑈(𝐴), all 𝑦∈𝐴, and all 𝑛=0,1,2,…. Suppose that 2𝑓𝜇𝑥+𝜇𝑦2−𝜇𝑓(𝑥)−𝜇𝑓𝑦+𝑓𝑧^∗−𝑓(𝑧)^∗≤𝜃‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝(2.12) for all 𝜇∈𝕋 and all 𝑥,𝑦,𝑧∈𝐴. If lim_𝑛(𝑓(3^𝑛𝑒)/3^𝑛)∈𝑈(𝐵)∩𝑍(𝐵), then the mapping 𝑓∶𝐴→𝐵 is a Jordan ∗-homomorphism.

Proof. Setting 𝜙(𝑥,𝑦,𝑧)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝+‖𝑧‖^𝑝) for all 𝑥,𝑦,𝑧∈𝐴. Then by Theorem 2.1, we get the desired result.

Theorem 2.3. Let 𝐴 be a 𝐶^∗-algebra of real rank zero. Let 𝑓∶𝐴→𝐵 be a continuous mapping such that 𝑓(0)=0 and that 𝑓3^𝑛𝑢𝑦+3^𝑛𝑦𝑢=𝑓3^𝑛𝑢𝑓𝑦+𝑓𝑦𝑓3^𝑛𝑢(2.13) for all 𝑢∈𝐼₁(𝐴_sa), all 𝑦∈𝐴, and all 𝑛=0,1,2,…. Suppose that there exists a function 𝜙∶(𝐴−{0})²×𝐴→[0,∞) satisfying (2.2) and 𝜙(𝑥,𝑦,𝑧)<∞ for all 𝑥,𝑦∈𝐴−{0} and all 𝑧∈𝐴. If lim_𝑛(𝑓(3^𝑛𝑒)/3^𝑛)∈𝑈(𝐵)∩𝑍(𝐵), then the mapping 𝑓∶𝐴→𝐵 is a Jordan ∗-homomorphism.

Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive ℂ-linear mapping 𝑇∶𝐴→𝐵 satisfying (2.3). It follows from (2.13) that 𝑇𝑢𝑦+𝑦𝑢=lim_𝑛𝑓3^𝑛𝑢𝑦+3^𝑛𝑦𝑢3^𝑛=lim_𝑛𝑓3^𝑛𝑢3^𝑛𝑓𝑦𝑦𝑓3+𝑓^𝑛𝑢3^𝑛=𝑇(𝑢)𝑓𝑦+𝑓𝑦𝑇(𝑢)(2.14) for all 𝑢∈𝐼₁(𝐴_sa), and all 𝑦∈𝐴. By additivity of 𝑇 and (2.14), we obtain that 3^𝑛𝑇𝑢𝑦+𝑦𝑢=𝑇𝑢3^𝑛𝑦+3^𝑛𝑦𝑢=𝑇(𝑢)𝑓3^𝑛𝑦+𝑓3^𝑛𝑦𝑇(𝑢)(2.15) for all 𝑢∈𝐼₁(𝐴_sa) and all 𝑦∈𝐴. Hence, 𝑇𝑢𝑦+𝑦𝑢=lim_𝑛𝑇(𝑢)𝑓3^𝑛𝑦3^𝑛+𝑓3^𝑛𝑦3^𝑛𝑇(𝑢)=𝑇(𝑢)𝑇𝑦+𝑇𝑦𝑇(𝑢)(2.16) for all 𝑢∈𝐼₁(𝐴_sa) and all 𝑦∈𝐴. By the assumption, we have 𝑇(𝑒)=lim_𝑛𝑓3^𝑛𝑒3^𝑛∈𝑈(𝐵)∩𝑍(𝐵).(2.17) Similar to the proof of Theorem 2.1, it follows from (2.14) and (2.16) that 𝑇=𝑓 on 𝐴. So 𝑇 is continuous. On the other hand, 𝐴 is real rank zero. One can easily show that 𝐼₁(𝐴_sa) is dense in {𝑥∈𝐴_sa∶‖𝑥‖=1}. Let 𝑣∈{𝑥∈𝐴_sa∶‖𝑥‖=1}. Then there exists a sequence {𝑧_𝑛} in 𝐼₁(𝐴_sa) such that lim_𝑛𝑧_𝑛=𝑣. Since 𝑇 is continuous, it follows from (2.16) that 𝑇𝑣𝑦+𝑦𝑣=𝑇lim_𝑛𝑧_𝑛𝑦+𝑦𝑧_𝑛=lim_𝑛𝑇𝑧_𝑛𝑦+𝑦𝑧_𝑛=lim_𝑛𝑇𝑧_𝑛𝑇𝑦+lim_𝑛𝑇𝑦𝑇𝑧_𝑛=𝑇lim_𝑛𝑧_𝑛𝑇𝑦+𝑇𝑦𝑇lim_𝑛𝑧_𝑛=𝑇(𝑣)𝑇𝑦+𝑇𝑦𝑇(𝑣)(2.18) for all 𝑦∈𝐴. Now, let 𝑥∈𝐴. Then we have 𝑥=𝑥₁+𝑖𝑥₂, where 𝑥₁∶=(𝑥+𝑥^∗)/2 and 𝑥₂=(𝑥−𝑥^∗)/2𝑖 are self adjoint.
First consider 𝑥₁=0,𝑥₂≠0. Since 𝑇 is ℂ-linear, it follows from (2.18) that 𝑓𝑥𝑦+𝑦𝑥=𝑇𝑥𝑦+𝑦𝑥=𝑇𝑖𝑥₂𝑦+𝑦𝑖𝑥₂=𝑇𝑖𝑥₂𝑥₂𝑥₂𝑦+𝑦𝑖𝑥₂𝑥₂𝑥₂=𝑇𝑖𝑥₂𝑥₂𝑥₂𝑇𝑦+𝑇𝑦𝑇𝑖𝑥₂𝑥₂𝑥₂=𝑇𝑖𝑥₂𝑇𝑦+𝑇𝑦𝑇𝑖𝑥₂(𝑥)𝑇𝑦𝑦𝑇(𝑥)(𝑥)𝑓𝑦𝑦𝑓(𝑥)=𝑇+𝑇=𝑓+𝑓(2.19) for all 𝑦∈𝐴.
If 𝑥₂=0, 𝑥₁≠0, then by (2.18), we have 𝑓𝑥𝑦+𝑦𝑥=𝑇𝑥𝑦+𝑦𝑥=𝑇𝑥₁𝑦+𝑦𝑥₁=𝑇𝑥₁𝑥₁𝑥₁𝑦+𝑦𝑥₁𝑥₁𝑥₁=𝑇𝑥₁𝑥₁𝑥₁𝑇𝑦+𝑇𝑦𝑇𝑥₁𝑥₁𝑥₁=𝑇𝑥₁𝑇𝑦+𝑇𝑦𝑇𝑥₁=𝑇(𝑥)𝑇𝑦+𝑇𝑦𝑇(𝑥)=𝑓(𝑥)𝑓𝑦+𝑓𝑦𝑓(𝑥)(2.20) for all 𝑦∈𝐴.
Finally, consider the case that 𝑥₁≠0, 𝑥₂≠0. Then it follows from (2.18) that 𝑓𝑥𝑦+𝑦𝑥=𝑇𝑥𝑦+𝑦𝑥=𝑇𝑥₁𝑦+𝑖𝑥₂𝑦+𝑦𝑥₁+𝑦𝑖𝑥₂=𝑇𝑥₁𝑥₁𝑥₁𝑦+𝑦𝑥₁𝑥₁𝑥₁+𝑇𝑖𝑥₂𝑥₂𝑥₂𝑦+𝑦𝑖𝑥₂𝑥₂𝑥₁=𝑥₁𝑇𝑥₁𝑥₁𝑇𝑦𝑦𝑇𝑥+𝑇¹𝑥₁+𝑖𝑥₂𝑇𝑥₂𝑥₂𝑇𝑦𝑦𝑇𝑥+𝑇²𝑥₁=T𝑥₁+𝑇𝑖𝑥₂𝑇𝑦+𝑇𝑦𝑇𝑥₁+𝑇𝑖𝑥₂=𝑇(𝑥)𝑇𝑦+𝑇𝑦𝑇(𝑥)=𝑓(𝑥)𝑓𝑦+𝑓𝑦𝑓(𝑥)(2.21) for all 𝑦∈𝐴. Hence, 𝑓(𝑥𝑦+𝑦𝑥)=𝑓(𝑥)𝑓(𝑦)+𝑓(𝑦)𝑓(𝑥) for all 𝑥,𝑦∈𝐴, and 𝑓 is Jordan ∗-homomorphism.

Corollary 2.4. Let 𝐴 be a 𝐶^∗-algebra of rank zero. Let 𝑝∈(0,1), 𝜃∈[0,∞) be real numbers. Let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓(0)=0 and that 𝑓3^𝑛𝑢𝑦+3^𝑛𝑦𝑢=𝑓3^𝑛𝑢𝑓𝑦+𝑓𝑦𝑓3^𝑛𝑢(2.22) for all 𝑢∈𝐼₁(𝐴_sa), all 𝑦∈𝐴, and all 𝑛=0,1,2,…. Suppose that 2𝑓𝜇𝑥+𝜇𝑦2−𝜇𝑓(𝑥)−𝜇𝑓𝑦+𝑓𝑧^∗−𝑓(𝑧)^∗≤𝜃‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝(2.23) for all 𝜇∈𝕋 and all 𝑥,𝑦,𝑧∈𝐴. If lim_𝑛𝑓(3^𝑛𝑒)/3^𝑛∈𝑈(𝐵)∩𝑍(𝐵), then the mapping 𝑓∶𝐴→𝐵 is a Jordan ∗-homomorphism.

Proof. Setting 𝜙(𝑥,𝑦,𝑧)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝+‖𝑧‖^𝑝) for all 𝑥,𝑦,𝑧∈𝐴. Then by Theorem 2.3, we get the desired result.

3. Stability of Jordan ∗-Homomorphisms: A Fixed Point Approach

We investigate the generalized Hyers-Ulam-Aoki-Rassias stability of Jordan ∗-homomorphisms on unital 𝐶^∗-algebras by using the alternative fixed point.

Recently, C𝑎̆dariu and Radu applied the fixed point method to the investigation of the functional equations. (See also [18, 26–43]).

Theorem 3.1. Let 𝑓∶𝐴→𝐵 be a mapping with 𝑓(0)=0 for which there exists a function 𝜙∶𝐴⁵→[0,∞) satisfying 𝑓𝜇𝑥+𝜇𝑦+𝜇𝑧3+𝑓𝜇𝑥−2𝜇𝑦+𝜇𝑧3+𝑓𝜇𝑥+𝜇𝑦−2𝜇𝑧3(𝑥)()(𝑣)𝑓(𝑢)(𝑢)𝑓(𝑣)𝑤−𝜇𝑓+𝑓𝑢𝑣+𝑢𝑣−𝑓−𝑓+𝑓^∗−𝑓(𝑤)^∗≤𝜙𝑥,𝑦,𝑧,𝑢,𝑣,𝑤,(3.1) for all 𝜇∈𝕋, and all 𝑥,𝑦,𝑧,𝑢,𝑣∈𝐴,𝑤∈𝑈(𝐴)∪{0}. If there exists an 𝐿<1 such that 𝜙(𝑥,𝑦,𝑧,𝑢,𝑣,𝑤)≤3𝐿𝜙(𝑥/3,𝑦/3,𝑧/3,𝑢/3,𝑣/3,𝑤/3) for all 𝑥,𝑦,𝑧,u,𝑣,𝑤∈𝐴, then there exists a unique Jordan ∗-homomorphism ℎ∶𝐴→𝐵 such that 𝑓(𝑥)(𝑥≤𝐿−ℎ)𝜙()1−𝐿𝑥,0,0,0,0,0(3.2) for all 𝑥∈𝐴.

Proof. It follows from 𝜙(𝑥,𝑦,𝑧,𝑢,𝑣,𝑤)≤3𝐿𝜙(𝑥/3,𝑦/3,𝑧/3,𝑢/3,𝑣/3,𝑤/3) that lim_𝑗3^−𝑗𝜙3^𝑗𝑥,3^𝑗𝑦,3^𝑗𝑧,3^𝑗𝑢,3^𝑗𝑣,3^𝑗𝑤=0(3.3) for all 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤∈𝐴.
Put 𝑦=𝑧=𝑤=𝑢=0 and 𝜇=1 in (3.1) to obtain 𝑥3𝑓3−𝑓(𝑥)≤𝜙(𝑥,0,0,0,0,0)(3.4) for all 𝑥∈𝐴. Hence, 13𝑓()(𝑥≤13𝑥−𝑓)3𝜙(3𝑥,0,0,0,0,0)≤𝐿𝜙(𝑥,0,0,0,0,0)(3.5) for all 𝑥∈𝐴.
Consider the set 𝑋∶={𝑔∣𝑔∶𝐴→𝐵} and introduce the generalized metric on X: 𝑑ℎ,𝑔∶=inf𝐶∈ℝ⁺∶𝑔(𝑥)−ℎ(𝑥)≤𝐶𝜙(𝑥,0,0,0,0,0)∀𝑥∈𝐴.(3.6) It is easy to show that (𝑋,𝑑) is complete. Now we define the linear mapping 𝐽∶𝑋→𝑋 by 𝐽(ℎ𝑥)=1)(3ℎ(3𝑥)(3.7) for all 𝑥∈𝐴. By Theorem 3.1 of [44], 𝑑𝐽𝑔,𝐽(ℎ)≤𝐿𝑑𝑔,ℎ(3.8) for all 𝑔,ℎ∈𝑋.
It follows from (3.5) that 𝑑𝑓,𝐽𝑓≤𝐿.(3.9) Now, from the fixed point alternative [45], 𝐽 has a unique fixed point in the set 𝑋₁∶={ℎ∈𝑋∶𝑑(𝑓,ℎ)<∞}. Let ℎ be the fixed point of 𝐽. ℎ is the unique mapping with ℎ(3𝑥)=3ℎ(𝑥)(3.10) for all 𝑥∈𝐴 satisfying there exists 𝐶∈(0,∞) such that ℎ(𝑥)−𝑓(𝑥)≤𝐶𝜙(𝑥,0,0,0,0,0)(3.11) for all 𝑥∈𝐴. On the other hand, we have lim_𝑛𝑑(𝐽^𝑛(𝑓),ℎ)=0. It follows that lim_𝑛13^𝑛𝑓3^𝑛𝑥=ℎ(𝑥)(3.12) for all 𝑥∈𝐴. It follows from 𝑑(𝑓,ℎ)≤(1/(1−𝐿))𝑑(𝑓,𝐽(𝑓)), that 𝑑≤𝐿𝑓,ℎ.1−𝐿(3.13) This implies the inequality (3.2). It follows from (3.1), (3.3), and (3.12) that ℎ𝑥+𝑦+𝑧3+ℎ𝑥−2𝑦+𝑧3+ℎ𝑥+𝑦−2𝑧3−ℎ(𝑥)=lim_𝑛13^𝑛𝑓3^𝑛−1𝑥+𝑦+𝑧+𝑓3^𝑛−1𝑥−2𝑦+𝑧+𝑓3^𝑛−1𝑥+𝑦−2𝑧−𝑓3^𝑛𝑥≤lim_𝑛13^𝑛𝜙3^𝑛𝑥,3^𝑛𝑦,,3^𝑛𝑧,0,0,0=0(3.14) for all 𝑥,𝑦,𝑧∈𝐴. So ℎ𝑥+𝑦+𝑧3+ℎ𝑥−2𝑦+𝑧3+ℎ𝑥+𝑦−2𝑧3=ℎ(𝑥)(3.15) for all 𝑥,𝑦,𝑧∈𝐴. Put 𝑤=(𝑥+𝑦+𝑧)/3, 𝑡=(𝑥−2𝑦+𝑧)/3 and 𝑠=(𝑥+𝑦−2𝑧)/3 in above equation, we get ℎ(𝑤+𝑡+𝑠)=ℎ(𝑤)+ℎ(𝑡)+ℎ(𝑠) for all 𝑤,𝑡,𝑠∈𝐴. Hence, ℎ is Cauchy additive. By putting 𝑦=𝑧=𝑥, 𝑣=𝑤=0 in (3.1), we have 𝑓𝜇𝑥−𝜇𝑓(𝑥)≤𝜙(𝑥,𝑥,,𝑥,0,0,0)(3.16) for all 𝜇∈𝕋 and all 𝑥∈𝐴. It follows that ℎ𝜇𝑥−𝜇ℎ(𝑥)=lim_𝑚13^𝑚𝑓𝜇3^𝑚𝑥−𝜇𝑓3^𝑚𝑥≤lim_𝑚13^𝑚𝜙3^𝑚𝑥,3^𝑚𝑥,3^𝑚𝑥,0,0,0=0(3.17) for all 𝜇∈𝕋, and all 𝑥∈𝐴. One can show that the mapping ℎ∶𝐴→𝐵 is ℂ-linear. By putting 𝑥=𝑦=𝑧=𝑢=𝑣=0 in (3.1), it follows that ℎ𝑤^∗−(ℎ(𝑤))^∗=lim_𝑚13^𝑚𝑓3^𝑚𝑤^∗−13^𝑚𝑓3^𝑚𝑤^∗≤lim_𝑚13^𝑚𝜙0,0,0,0,0,3^𝑚𝑤=0(3.18) for all 𝑤∈𝑈(𝐴). By the same reasoning as the proof of Theorem 2.1, we can show that ℎ∶𝐴→𝐵 is ∗-preserving.
Since ℎ is ℂ-linear, by putting 𝑥=𝑦=𝑧=𝑤=0 in (3.1), it follows that ‖ℎ(𝑢𝑣+𝑣𝑢)−ℎ(𝑢)ℎ(𝑣)−ℎ(𝑣)ℎ(𝑢)‖=lim_𝑚19^𝑚𝑓9^𝑚()−1𝑢𝑣+𝑣𝑢9^𝑚𝑓3^𝑚𝑢𝑓3^𝑚𝑣+𝑓3^𝑚𝑣𝑓3^𝑚𝑢≤lim_𝑚19^𝑚𝜙0,0,0,3^𝑚u,3^𝑚𝑣,0≤lim_𝑚13^𝑚𝜙0,0,0,3^𝑚𝑢,3^𝑚𝑣,0=0(3.19) for all 𝑢,𝑣∈𝐴. Thus ℎ∶𝐴→𝐵 is Jordan ∗-homomorphism satisfying (3.2), as desired.

We prove the following Hyers-Ulam-Aoki-Rassias stability problem for Jordan ∗-homomorphisms on unital 𝐶^∗-algebras.

Corollary 3.2. Let 𝑝∈(0,1), 𝜃∈[0,∞) be real numbers. Suppose 𝑓∶𝐴→𝐴 satisfies 𝑓𝜇𝑥+𝜇𝑦+𝜇𝑧3+𝑓𝜇𝑥−2𝜇𝑦+𝜇𝑧3+𝑓𝜇𝑥+𝜇𝑦−2𝜇𝑧3(𝑥)()(𝑣)𝑓(𝑢)(𝑢)𝑓(𝑣)𝑤−𝜇𝑓+𝑓𝑢𝑣+𝑢𝑣−𝑓−𝑓+𝑓^∗−𝑓(𝑤)^∗≤𝜃‖𝑥‖^𝑝+𝑦^𝑝+‖𝑧‖^𝑝+‖𝑢‖^𝑝+‖𝑣‖^𝑝+‖𝑤‖^𝑝,(3.20) for all 𝜇∈𝕋 and all 𝑥,𝑦,𝑧,𝑢,𝑣∈𝐴,𝑤∈𝑈(𝐴)∪{0}. Then there exists a unique Jordan ∗-homomorphism ℎ∶𝐴→𝐵 such that such that 𝑓(𝑥)(𝑥≤3−ℎ)^𝑝𝜃3−3^𝑝‖𝑥‖^𝑝(3.21) for all 𝑥∈𝐴.

Proof. Setting 𝜙(𝑥,𝑦,𝑧,𝑢,𝑣,𝑤)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝+‖𝑧‖^𝑝+‖𝑢‖^𝑝+‖𝑣‖^𝑝+‖𝑤‖^𝑝) all 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤∈𝐴. Then by 𝐿=3^𝑝−1 in Theorem 3.2, one can prove the result.