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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)
##### MSC:
 46H05 General theory of topological algebras 46H10 Ideals and subalgebras
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