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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.
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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