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On an algebraic extension of $$A(E)$$. (English. Russian original) Zbl 1029.46074
Math. Notes 72, No. 5, 600-604 (2002); translation from Mat. Zametki 72, No. 5, 649-653 (2002).
Summary: An algebraic extension of the algebra $$A(E)$$, where $$E$$ is a compactum in $$\mathbb{C}$$ with nonempty connected interior, leads to a Banach algebra $$B$$ of functions that are holomorphic on some analytic set $$K^0\subset \mathbb{C}^2$$ with boundary $$bK$$ and continuous up to $$bK$$. The singular points of the spectrum of $$B$$ and their defects are investigated. For the case in which $$B$$ is a uniform algebra, the depth of $$B$$ in the algebra $$C(bK)$$ is estimated. In particular, conditions under which $$B$$ is maximal on $$bK$$ are obtained.
##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces
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