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Harte theorem for Waelbroeck algebras. (English) Zbl 1099.46034
Summary: Let $$B$$ be a locally convex Waelbroeck algebra. Let $$a_1,\dots a_k\in B$$ be an arbitrary $$k$$-tuple of mutually commuting elements. The joint spectrum $$\sigma_B(a_1,\dots, a_k)$$ is defined as the set of those $$(\lambda_1,\dots,\lambda_k)\in \mathbb C^k$$ for which the elements $$a_1-\lambda_1,\dots, a_k-\lambda_k$$ generate a proper (left or right) ideal. Let $$p:\mathbb C^k\to\mathbb C^m$$ be a polynomial mapping. The spectral mapping formula $p(\sigma_B(a_1,\dots,a_k))= \sigma_B(p(a_1,\dots, a_k))$ is proved.
##### MSC:
 46H30 Functional calculus in topological algebras 47A13 Several-variable operator theory (spectral, Fredholm, etc.)
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