Proper CQ*-ternary algebras

Abstract

In this paper, modifying the construction of a $C^*$-ternary algebra from a given $C^*$-algebra [M. Amyari and M.S. Moslehian, Approximately ternary semigroup homomorphisms, Lett. Math. Phys. {\bf 77} (2006), 1-9], 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*} which was investigated in [C. Park, Y. Cho and M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. {\bf 2007}, Art. ID 41820 (2007)].

Moreover, we 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*}