On an algebraic extension of \(A(E)\). (English. Russian original) Zbl 1029.46074

Batikyan, B. T.; Grigoryan, S. A.

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

Keywords:

Banach algebra; uniform algebra; maximal algebra; unitary polynomial; ramified holomorphic covering; extension of a Banach algebra; sheaf of germs of continuous functions