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An approach to joint spectra. (English) Zbl 0967.46033
Let \(A\) be a unital Banach algebra. For every subset \(C\subset A\) let \(C^k\) denote the kth Cartesian product of \(C\), \(C_{\text{com}}^k\) the set of all \(k\)-tuples of mutually commuting elements from \(C\), \(C_{\text{com}} = \cup_{k\in N}C_{\text{com}}^k\) and \(C^\infty = \cup_{k\in N}C^k\). Let \({\mathbb E}(A)\) be the set of all spectral linear subspaces in \(A\) and \(\sigma _{\mathcal U}(a) = \{\lambda \in {\mathbb C}^k: \exists E\in {\mathcal U}, a - \lambda e\in E^k\}\) for each \({\mathcal U}\subset {\mathbb E}(A)\). On \(\sigma _{\mathcal U}(a)\) is defined such a topology that \(\sigma _{\mathcal{ \overline {U}}}(a) = {\overline {\sigma _{\mathcal U}}}(a)\) (here \({\overline {U}}\) denote the closure of \(U\)) for each \({\mathcal U}\subset {\mathbb E} (A)\) and \(a\in A_{\text{com}}\). It is shown that
a) if \(\tau \) is a joint spectrum and \({\mathcal U}\) is the set of all linear subspaces \(E\in {\mathbb E} (A)\) such that \(\tau (a)\) contains zero for all \(a\in E_{\text{com}}^k\) then \(\tau (a) = \sigma _{\mathcal U}(a)\) for each \(a\in A_{\text{com}}\);
b) a joint spectrum \(\tau \) satisfies the one-way spectral mapping property for all \(a\in A^\infty \) if and only if there exists a subset \({\mathcal U}\subset{\mathbb E}(A)\) consisting of subalgebras of \(A\) such that \(\tau (a) = \sigma _{\mathcal U}(a)\) for each \(a\in A^\infty \);
c) a joint spectrum \(\sigma _{\mathcal U}(a)\) defined by a closed family \({\mathcal U}\subset {\mathbb E} (A)\) is nonvoid for all \(a\in A^\infty \) if and only if there exists on \(A\) a linear multiplicative functional \(\varphi \) such that \(\ker \varphi \in {\mathcal U}\);
d) if the joint spectrum \(\tau \) has the projection property then \(\overline {\tau }(a) = \overline {\tau (a)}\) is a joint spectrum with the projection property.
Moreover, the properties of spectrum \(\sigma _{\mathcal U}(a)\), of spectrum of those single elements defining regularity in \(A\) and of spectral systems are considered in the end of the article.
Reviewer: Mati Abel (Tartu)
46H05 General theory of topological algebras
46H10 Ideals and subalgebras
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