Proper $CQ^*$-ternary algebras

Abstract

In this paper, modifying the construction of a $C^*$-ternary algebra from a given $C^*$-algebra, we define a proper $CQ^*$-ternary algebra from a given proper $CQ^*$-algebra.

We investigate homomorphisms in proper $CQ^*$-ternary algebras and derivations on proper $CQ^*$-ternary algebras associated with the Cauchy functional inequality \begin{eqnarray*} \| f(x)+f(y)+f(z) \| \le \| f(x+y+z) \| . \end{eqnarray*} We moreover prove the generalized Hyers-Ulam stability of homomorphisms in proper $CQ^*$-ternary algebras and of derivations on proper $CQ^*$-ternary algebras associated with the Cauchy functional equation \begin{eqnarray*} f(x+y+z) = f(x)+f(y)+f(z) . \end{eqnarray*} The concept of generalized Hyers-Ulam stability originated from the Th.M. Rassias` stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. {\bf 72} (1978), 297--300.