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Pervasive algebras of analytic functions. (English) Zbl 0981.46044
Let $$X$$ be a compact Hausdorff space and $$S$$ a complex or real closed subspace of $$C(X,\mathbb{C})$$ or $$C(X,\mathbb{R})$$ respectively, and let $$Y$$ be a closed subset of $$X$$. $$S$$ is said to be complex or real pervasive on $$Y$$ if the functions of $$S$$ restricted to $$E$$ are dense in $$C(E,\mathbb{C})$$ or $$C(E,\mathbb{R})$$, respectively for each proper closed subset $$E$$ of $$Y$$. These properties are investigated for the case where $$X$$ is an open proper subset $$U$$ of the Riemann sphere $$\widehat{\mathbb{C}}$$ and $$S$$ is the algebra $$A(U)$$ of all complex valued functions continuous on $$\widehat{\mathbb{C}}$$ and analytic on $$U$$, or $$S= \text{Re }A(U)$$, respectively, and it is supposed that $$U$$ has no inessential boundary points – i.e. points where all functions of $$A(U)$$ can be extended analytically. Then $$A(U)$$ is complex pervasive on $$\partial U$$ if and only if $$\partial U_i=\partial U$$ for each component $$U_i$$ of $$U$$. If $$\text{Re }A(U)$$ is real pervasive on $$\partial U$$ then $$U$$ has at most one component $$U_k$$ that is not simply connected, and in this case $$\partial U_k=\partial U$$. If on the other hand $$U$$ has at least one component $$U_k$$ such that $$\partial U_k=\partial U$$ then $$\text{Re }A(U)$$ is real pervasive on $$\partial U$$. In case that all components $$U_i$$ of $$U$$ are simply connected and $$\partial U_i\neq\partial U$$, then the real pervasiveness is characterized in terms of properties of the boundary points.
##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 30H05 Spaces of bounded analytic functions of one complex variable
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##### References:
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