An extension of the disc algebra. I (Q2903279)

Recall that the chordal distance \(\chi\) on \(\mathbb{C}\cup\{\infty\},\) the one point compactification of \(\mathbb{C},\) is given by, for \(z_1,z_2\in\mathbb{C},\) NEWLINE\[NEWLINE\chi(z_1,z_2)=\frac{| z_1-z_2|}{\sqrt{1+| z_1|^2}\sqrt{1+| z_2|^2}}, \quad \chi(z_1,\infty)=\frac{1}{\sqrt{1+| z_1|^2}}, \quad\chi(\infty,\infty)=0.NEWLINE\]NEWLINE Let \(\mathbb{D}=\{z\in\mathbb{C};\;| z|<1\}\) be the open unit disc in the complex plane \(\mathbb{C}\).NEWLINENEWLINEIn this important paper, the author introduces and investigates the set \(\tilde{A}(\mathbb{D})\) of all functions \(f:\overline{\mathbb{D}}\rightarrow \mathbb{C}\cup\{\infty\}\) such that \(f\) is identically equal to \(\infty\) or \(f\) is continuous with respect to the metric \(\chi \), \(f(\mathbb{D})\subset\mathbb{C}\) and \(f_{| {\mathbb{D}}}: \mathbb{D}\rightarrow\mathbb{C}\) is holomorphic. The above class is exactly the set of uniform limits on \(\overline{\mathbb{D}}\) of polynomials with respect to \(\chi.\) It is an extension of the classical disc algebra \(A(\mathbb{D}).\) Thus the larger class \(\tilde{A}(\mathbb{D})\) is naturally endowed with the metric NEWLINENEWLINENEWLINE\[NEWLINE\tilde{\chi}(f,g)=\max_{| z|\leq 1}\chi (f(z),g(z)),\qquad f,g\in \tilde{A}(\mathbb{D}).NEWLINE\]NEWLINENEWLINENEWLINE Some properties of the elements of \(\tilde{A}(\mathbb{D})\) as well as some elegant topological results on \(\tilde{A}(\mathbb{D})\) are discussed. For instance the author proves that:NEWLINENEWLINE - if \(f(\zeta)=g(\zeta)\) for all \(\zeta\in \partial\mathbb{D}\), \(f,g \in \tilde{A}(\mathbb{D})\), then \(f\equiv g\),NEWLINENEWLINE - the set of all \(f\in \tilde{A}(\mathbb{D})\), such that the set \(\{\zeta\in\partial\mathbb{D};\;f(\zeta)\notin f(\mathbb{D})\}\) has zero length, is dense and \(G_{\delta}\) in \(\tilde{A}(\mathbb{D})\).NEWLINENEWLINE Finally the author is interested in an extension of the well-known Mergelyan theorem. The question is whether the set of of uniform limits with respect to the distance \(\chi\) of polynomials on a compact set \(L\subset\mathbb{C}\) with connected complement is the set of continuous functions \(f:L\rightarrow \mathbb{C}\cup\{\infty\}\) such that for every component \(V\) of the interior of \(L\) either \(f_{| V}\equiv\infty\) or \(f(V)\subset\mathbb{C}\) and \(f_{| V}\) is holomorphic. The author gives an affirmative answer in particular cases but the general problem remains open. The paper contains several other open questions.