Proper \(CQ^*\)-ternary algebras (Q2926402)

Let \(A\) be a Banach module over a \(C^*\)-algebra \(A_0\) with involution \(*\) and \(C^*\)-norm \(\|\cdot\|_0\) such that \(A_0\subseteq A\). We say that \((A ,A_0)\) is a proper \(CQ^*\)-algebra if (i) \(A_0\) is dense in \(A\) with respect to its norm \(\|\cdot\|_0\), (ii) \((ab)^*= b^*a^*\), whenever the multiplication is defined, (iii) \(\|y\|= \sup_{a\in , \|a\| \leq 1}= \|ay\|\) for all \(y \in A_0\). A proper \(CQ^*\)-algebra \((A ,A_0)\) endowed with the triple product \([\cdot , \cdot, \cdot] : A_0 \times A \times A_0 \to A\), which is \(\mathbb C\)-linear in the outer variables, conjugate \(\mathbb C\)-linear in the middle variable and satisfies that \([w_0, w, w_1]\in A_0\) for all \(w_0, w, w_1 \in A_0\), is called a proper \(CQ^*\)-ternary algebra, and denoted by \((A, A_0, [\cdot , \cdot, \cdot])\).NEWLINENEWLINE The author investigates homomorphisms and derivations in proper \(CQ^*\)-ternary algebras associated with the Cauchy functional inequality \(\|f(x) + f(y) + f(z)\| \leq \|f(x + y + z)\|\). He also proves the Hyers-Ulam stability of homomorphisms in proper \(CQ^*\)-ternary algebras and that of derivations on proper \(CQ^*\)-ternary algebras associated with the Cauchy functional equation \(f(x + y + z) = f(x) + f(y) + f(z)\) .