Proper CQ^*-ternary algebras

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 properCQ^*-ternary algebras associated with the Cauchy functional inequality[\|f(x) + f(y) + f(z)\| \leq\| f(x + y + z)\|.\] We moreover prove the HyersUlam stability of homomorphisms in proper CQ^*-ternary algebras and of derivations on proper CQ^*-ternary algebras associated with the Cauchy functional equation \[f(x + y + z) = f(x) + f(y) + f(z).\]