Proper $CQ^*$-ternary algebras

초록

In this paper, modifying the construction of a $C^{\ast }$-ternary algebra from a given $C^{\ast }$-algebra, we define a proper $C{Q}^{\ast }$-ternary algebra from a given proper $C{Q}^{\ast }$-algebra. We investigate homomorphisms in proper $C{Q}^{\ast }$-ternary algebras and derivations on proper $C{Q}^{\ast }$-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 $C{Q}^{\ast }$-ternary algebras and of derivations on proper $C{Q}^{\ast }$-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.