Stability of \(J^*\)-derivations (Q2920425)

Assume that \(H\) and \(K\) are Hilbert spaces and \(\mathcal{A}\) is a closed subalgebra of \(B(H, K)\). If for each \(x\in \mathcal A\), we have \(xx^*x\in \mathcal A\), then \(\mathcal{A}\) is called a \(J^*\)-algebra. A \(J^*\)-derivation on a \(J^*\)-algebra \(\mathcal{A}\) is a \(\mathbb{C}\)-linear mapping \(d:\mathcal{A}\to \mathcal{A}\) such that \(d(aa^*a)=d(a)a^*a+a(d(a))^*a+aa^*d(a)\) for all \(a\in A\). In this paper, the authors prove the Hyers-Ulam stability and the superstability of \(J^*\)-derivations in \(J^*\)-algebras for the generalized Jensen type functional equation NEWLINE\[NEWLINErf\left(\frac{x+y}{r}\right)+rf\left(\frac{x-y}{r}\right)=2f(x),NEWLINE\]NEWLINE where \(r\neq 0\) by using the direct method and the fixed point method.