Park, Chun-Gil; Rassias, Themistocles M. Stability of homomorphisms in \(\text{JC}^*\)-algebras. (English) Zbl 1141.17302 Pac.-Asian J. Math. 1, No. 1, 1-17 (2007). Summary: Let \(q=\frac{l(d-1)}{d-l}\) for integers \(l,d\) with \(2\leq l\leq d-1\). It is shown that every almost unital almost linear mapping \(h: \mathcal A\to\mathcal B\) of a \(\text{JC}^*\)-algebra \(\mathcal A\) to a \(\text{JC}^*\)-algebra \(\mathcal B\) is a homomorphism when \(h(3^nu\circ y)=h(3^nu)\circ h(y)\) or \(h(q^nu\circ y)=h(q^nu)\circ h(y)\) holds for all unitaries \(u\in\mathcal A\), all \(y\in\mathcal A\), and all \(n\in\mathbb{Z}\), and that every almost unital almost linear continuous mapping \(h: \mathcal A\to\mathcal B\) of a \(\text{JC}^*\)-algebra \(\mathcal A\) of real rank zero to a \(\text{JC}^*\)-algebra \(\mathcal B\) is a homomorphism when \(h(3^nu\circ y)=h(3^nu)\circ h(y)\) or \(h(q^nu\circ y)=h(q^nu)\circ h(y)\) holds for all \(u\in\{v\in\mathcal A| v=v^*,\|v\|=1, v \text{is invertible}\}\), all \(y\in\mathcal A\), and all \(n\in\mathbb{Z}\).Furthermore, we prove the Cauchy-Rassias stability of homomorphisms in \(\text{JC}^*\)-algebras. This concept of stability of mappings was introduced by Th. M. Rassias [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)]. Cited in 1 Document MSC: 17C65 Jordan structures on Banach spaces and algebras 47C10 Linear operators in \({}^*\)-algebras 39B52 Functional equations for functions with more general domains and/or ranges 46L05 General theory of \(C^*\)-algebras 47B48 Linear operators on Banach algebras 39B82 Stability, separation, extension, and related topics for functional equations Citations:Zbl 0398.47040 PDFBibTeX XMLCite \textit{C.-G. Park} and \textit{T. M. Rassias}, Pac.-Asian J. Math. 1, No. 1, 1--17 (2007; Zbl 1141.17302)