×

Homomorphisms between \(JC^*\)-algebras. (English) Zbl 1199.47170

In the paper under review, the authors study some approximate-homomorphisms in \(JC^*\)-algebras. Assume that if \(h : \mathcal A \to \mathcal B\) is a map between two \(JC^*\)-algebras satisfying the following conditions:
(i) There exists a function \(\varphi: \mathcal A \times \mathcal A \to [0, \infty)\) such that \[ \widetilde{\varphi} (x,y)= \sum_{j=0}^ \infty \frac{1}{2^j} \varphi (2^jx, 2^j y) < \infty, \]
\[ \| h( \nu x +\nu y )- \nu h(x) - \nu h(y) \| \leq \varphi (x,y) \] for all \(\nu \in \{ \lambda \in \mathbb C: |\lambda| =1 \}\) and for all \(x,y \in \mathcal A\).
(ii) \(h(0)=0\) and \(\lim \frac{h(2^n e)}{2^n} =e'\), where \(e\) (resp., \(e'\)) is the unit of \(\mathcal A\) (resp., \(\mathcal B\)).
(iii) \(h(2^n u \circ y) = h(2^n u) \circ h(y)\) for all \( u \in \mathcal A\) with \(uu^*=u^*u=e\), for all \(y \in \mathcal A\) and for all \(n \in \mathbb Z\).
Then \(h\) must be a homomorphism. It should be pointed out that the first assumption on \(h\) refers to P.Găvruţă’s generalisation [J. Math.Anal.Appl.184, No.3, 431–436 (1994; Zbl 0818.46043)] of the Cauchy-Rassias inequality [Th.M.Rassias, Proc.Am.Math.Soc.72, 297–300 (1978; Zbl 0398.47040)]. The authors also investigate some continuous approximate homomorphisms \(h: \mathcal A \to \mathcal B\), where \(\mathcal A \) is a \(JC^*\)-algebra of real rank zero and \(\mathcal B\) is an arbitrary \(JC^*\)-algebra. The paper is concluded with a result about the Hyers-Ulam stability of homomorphisms in \(JC^*\)-algebras.

MSC:

47B48 Linear operators on Banach algebras
39B52 Functional equations for functions with more general domains and/or ranges
46L05 General theory of \(C^*\)-algebras
17C99 Jordan algebras (algebras, triples and pairs)
PDFBibTeX XMLCite