Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras
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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 . 1. Introduction and statement of results It seems that the stability problem of functional equations had been first raised by Ulam (cf. [11, Chapter VI] and [12]): For what metric groups G is it true that an ε-automorphism of G is necessarily near to a strict automorphism? An answer to the above problem has been given as follows. Suppose E1 and E2 are two real Banach spaces and f : E1 → E2 is a mapping. If there exist δ ≥ 0 and p ≥ 0, p = 1 such that f (x + y) − f (x) − f (y) ≤ δ x p + y p
(1.1)
for all x, y ∈ E1 , then there is a unique additive mapping T : E1 → E2 such that f (x) − T(x) ≤ 2δ x p / |2 − 2 p | for every x ∈ E1 . This result is called the Hyers-Ulam-Rassias stability of the additive Cauchy equation g(x + y) = g(x) + g(y). Indeed, Hyers [5] obtained the result for p = 0. Then Rassias [8] generalized the above result of Hyers to the case where 0 ≤ p < 1. Gajda [4] solved the problem for 1 < p, which was raised by Rassias; In the same paper, Gajda also gave an example that a similar result to the above does not hold for p = 1 (cf. [9]). If p < 0, then x p is meaningless for x = 0; In this case, if we assume that 0 p means ∞, then the proof given in [8] also works for x = 0. Moreover, with minor changes in the proof, the result is also valid for p < 0. Thus, the Hyers-UlamRassias stability of the additive Cauchy equation holds for p ∈ R \ {1}. Here and after, the letter R denotes the real number field and C stands for the complex number field.
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 435–441 DOI: 10.1155/JIA.2005.435
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Stability of Jordan homomorphisms
Suppose A and B are two Banach algebras. We say that a mapping τ : A → B is a Jordan homomorphism if τ(a + b) = τ(a) + τ(b) (a,b ∈ A),
τ a2 = τ(a)2
(a ∈ A).
(1.2)
(a,b ∈ A),
(1.3)
If, in addition, τ is multiplicative, that is τ(ab) = τ(a)τ(b)
we say that τ is a ring homomorphism. The study of ring homomorphisms between Banach algebras A and B is of interest even if A = B = C. For example, the zero mapping, the identity and the complex conjugate are ring homomorphisms on C, which are all continuous. On the other hand, the existence of a discontinuous ring homomorphism on C is well-known (cf. [6]). More explicitly, if G is the set of all surjective ring homomorphisms on C, then G = 2C , where S denotes the cardinal number of a set S. In fact, Charnow [3, Theorem 3] proved that there exist 2k automorphisms for every algebraically closed field k; It is also known that if Ꮽ is a uniform a
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