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Some conditions under which derivations are zero on Banach *-algebras. (English) Zbl 06764166

Summary: Let \(\mathcal{A}\) be a Banach *-algebra. By \(\mathcal{S}_{\mathcal{A}}\) we denote the set of all self-adjoint elements of \(\mathcal{A}\) and by \(\mathcal{O}_{\mathcal{A}}\) we denote the set of those elements in \(\mathcal{A}\) which can be represented as finite real-linear combinations of mutually orthogonal projections. The main purpose of this paper is to prove the following result:
Suppose that \(\overline {\mathcal{O}_{\mathcal{A}}} = \mathcal{S}_{\mathcal{A}}\) and \(\{d_{n}\}\) is a sequence of uniformly bounded linear mappings satisfying \(d_n(p)=\sum_{k=0}^n d_{n-k}(p)d_k (p)\), where \(p\) is an arbitrary projection in \(\mathcal{A}\). Then \(d_{n}(\mathcal{A})\subseteq\bigcap_{\phi\in\Phi_{\mathcal{A}}} \operatorname{ker}\phi\) for each \(n\geq 1\). In particular, if \(\mathcal{A}\) is semi-prime and further, \(\dim(\bigcap_{\phi\in\Phi_{\mathcal{A}}} \operatorname{ker}\phi)\leq 1\), then \(d_n=0\) for each \(n\geq 1\).

MSC:

47B48 Linear operators on Banach algebras
46Hxx Topological algebras, normed rings and algebras, Banach algebras
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References:

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