×

Stability of homomorphisms in \(\text{JC}^*\)-algebras. (English) Zbl 1141.17302

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)].

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