Hom-derivations in C *-ternary Algebras
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Hom-derivations in C ∗-ternary Algebras Yuan Feng JIN Department of Mathematics, Yanbian University, Yanji 133001, P. R. China E-mail : [email protected]
Choonkil PARK1) Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea E-mail : [email protected]
Michael Th. RASSIAS Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland E-mail : [email protected] Abstract
In this paper, we introduce and solve the following additive (ρ1 , ρ2 )-functional inequalities f (x + y + z) − f (x) − f (y) − f (z) ≤ ρ1 (f (x + z) − f (x) − f (z)) + ρ2 (f (y + z) − f (y) − f (z)),
where ρ1 and ρ2 are fixed nonzero complex numbers with |ρ1 | + |ρ2 | < 2. Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of the above additive (ρ1 , ρ2 )functional inequality in complex Banach spaces. Furthermore, we prove the Hyers–Ulam stability of hom-derivations in C ∗ -ternary algebras. Keywords Hyers–Ulam stability, additive (ρ1 , ρ2 )-functional inequality, fixed point method, direct method, hom-derivation on C ∗ -ternary algebra MR(2010) Subject Classification
1
16W25, 39B62, 47H10, 39B52, 47B47, 46L57
Introduction and Preliminaries
Let A be a complex Banach algebra and h : A → A be a C-linear mapping. Mirzavaziri and Moslehian [17, 18] introduced the concept of h-derivation as follows: f (xy) = f (x)h(y) + h(x)f (y) for all x, y ∈ A. Let A be a complex Banach algebra, h : A → A a homomorphism and δ : A → A a derivation. Then h ◦ δ(xyz) = h ◦ δ(x)h(y)h(z) + h(x)h ◦ δ(y)h(z) + h(x)h(y)h ◦ δ(z) Received August 3, 2019, accepted March 26, 2020 The first author was supported by National Natural Science Foundation of China (Grant No. 11761074), the second was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant No. NRF-2017R1D1A1B04032937) 1) Corresponding author
Jin Y. F. et al.
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= h ◦ δ(x)h(yz) + h(x)h ◦ δ(y)h(z) + h(xy)h ◦ δ(z), δ ◦ h(xyz) = δ ◦ h(x)h(y)h(z) + h(x)δ ◦ h(y)h(z) + h(x)h(y)δ ◦ h(z) = δ ◦ h(x)h(yz) + h(x)δ ◦ h(y)h(z) + h(xy)δ ◦ h(z) for all x, y, z ∈ A. A C ∗ -ternary algebra is a complex Banach space A, equipped with a ternary product (x, y, z) → [x, y, z] of A3 into A, which is C-linear in the outer variables, conjugate C-linear in the middle variable, and associative in the sense that [x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v], and satisfies [x, y, z] ≤ x · y · z and [x, x, x] = x3 (see [28]). Let A and B be C ∗ -ternary algebras. A C-linear mapping H : A → B is called a C ∗ -ternary homomorphism if H([x, y, z]) = [H(x), H(y), H(z)] for all x, y, z ∈ A. A C-linear mapping δ : A → A is called a C ∗ -ternary derivation if δ([x, y, z]) = [δ(x), y, z] + [x, δ(y), z] + [x, y, δ(z)] for all x, y, z ∈ A (see [1, 19]). In this paper, we introduce and investigate hom-derivations in a C ∗
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