Boundaries for algebras of holomorphic functions on the preduals of the Lorentz sequence spaces
For a given subalgebra \(A\) of the algebra of continuous and bounded complex functions defined on a topological Hausdorff space \(\Omega,\) a boundary \(\Gamma\) is a norming subset of \(\Omega,\) that is, \(\|f\|=\sup_{x\in \Gamma}|f(x)|\) for all \(f\in A.\) This general notion was introduced by J. Globevnik in 1979. Since then interest has arisen in the study of minimal boundaries, called Bishop boundaries, and minimal closed boundaries, called Shilov boundaries, particularly for algebras of holomorphic functions.NEWLINENEWLINEThe present article deals with such matters in case that \(\Omega\) is the closed unit ball \(B_E\) of the predual \(E=\chi_p,\; p\in \mathbb{N}\cup \{0\},\) of the Lorentz sequence spaces or of their finite dimensional subspaces \(E=\chi_{p,F}\) of the sequences in \(\chi_p\) that vanish outside a (given) finite subset \(F \subset \mathbb{N}\cup \{0\},\) and \(A\) is the algebra \(A_p(B_E)\) of bounded and continuous \(\mathbb{C}\)-valued functions defined on \(B_E\) that are analytic in its interior. A direct antecedent of this work is the paper [\textit{M. D. Acosta, L. A. Moraes} and \textit{L. R. Grados}, J. Math. Anal. Appl. 336, No. 1, 470--479 (2007; Zbl 1127.46036)] that includes a characterization of the boundaries of the corresponding subalgebra of uniformly continuous functions.NEWLINENEWLINEIt is proved, among other things, that for \(E=\chi_{p,F}\) the Shilov and Bishop boundaries of \(A_p(B_E)\) coincide with the so-called torus of the space \(\chi_{p,F},\) while for the whole space \(\chi_{p}\) suitable collections of such tori are found so that their union is a closed boundary for \(A_p(B_E),\) and that as a consequence, once more, there is no Shilov boundary for this algebra.
scientific article; zbMATH DE number 6023035