Real parts of functional algebras
Let \(n\geq 2\), \(T_ r^ n\) be the torus with vector-radius \(r\) in \(\mathbb{C}^ n\) and \(S^ n\) be the unit sphere in \(\mathbb{C}^ n\). Let \(A_ 1\) be the algebra of such \(\mathbb{C}\)-valued continuous functions on \(T_ r^ n\) which are analytic in the open ball (with vector-radius \(r\)) of \(\mathbb{C}^ n\), \(A_ 2\) be the algebra of such \(\mathbb{C}\)-valued continuous functions on \(S^ n\) which are analytic in the open unit ball of \(\mathbb{C}^ n\) and \(B_ 1(B_ 2)\) be an algebra of \(\mathbb{C}\)-valued functions on \(T_ r^ n\) (respectively on \(S^ n\)) which contains constants and separates the points of \(T_ r^ n\) (respectively of \(S^ n\)). For each algebra \(A\) of \(\mathbb{C}\)-valued functions on a subset of \(\mathbb{C}^ n\) let \(\text{Re }A=\{\text{Re }f\): \(f\in A\}\) and \(\bar A=\{\bar f\): \(f\in A\}\) where \(\text{Re }f\) denotes the real part of \(f\) and \(\bar f\) denotes the conjugate function of \(f\). It is proved that either \(B_ k=A_ k\) or \(B_ k=\bar A_ k\) if \(\text{Re }B_ k=\text{Re }A_ k\) with \(k=1,2\).
scientific article; zbMATH DE number 22973