Homomorphisms and Derivations in C-Ternary Algebras

and Applied Analysis 3 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 ‖ ‖x‖ see 45, 47 . Every left Hilbert C∗-module is a C∗-ternary algebra via the ternary product x, y, z : 〈x, y〉z. If a C∗-ternary algebra A, ·, ·, · has an identity, that is, an element e ∈ A such that x x, e, e e, e, x for all x ∈ A, then it is routine to verify that A, endowed with x ◦ y : x, e, y and x∗ : e, x, e , is a unital C∗-algebra. Conversely, if A, ◦ is a unital C∗-algebra, then x, y, z : x ◦ y∗ ◦ z makes A into a C∗-ternary algebra. A C-linear mapping H : A → B is called a C∗-ternary algebra homomorphism if H ([ x, y, z ]) [ H x ,H ( y ) ,H z ] 1.6 for all x, y, z ∈ A. If, in addition, the mappingH is bijective, then the mappingH : A → B is called a C∗-ternary algebra isomorphism. 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 ] 1.7 for all x, y, z ∈ A see 23, 45, 48 . Let A, ◦ be a C∗-algebra and x, y, z : x ◦ y∗ ◦ z for all x, y, z ∈ A. The mapping H : A → A defined by H x −ix is a C∗-ternary algebra isomorphism. Let a ∈ A with a∗ a. The mapping δa : A → A defined by δa x i ax − xa is a C∗-ternary derivation. There are some applications, although still hypothetical, in the fractional quantumHall effect, the nonstandard statistics, supersymmetric theory, and Yang-Baxter equation cf. 49–51 . Throughout this paper, assume that p, d are nonnegative integers with p d ≥ 3, and that A and B are C∗-ternary algebras. 2. Stability of Homomorphisms in C∗-Ternary Algebras The stability of homomorphisms in C∗-ternary algebras has been investigated in 31 see also 37 . In this note, we improve some results in 31 . For a given mapping f : A → B,we define Cμf ( x1, . . . , xp, y1, . . . , yd ) : 2f ⎛ ⎝ ∑p j 1 μxj 2 d ∑