Pick conditions on a uniform algebra and von Neumann inequalities
type: Item
instance of: scholarly article
title: Pick conditions on a uniform algebra and von Neumann inequalities
zbMATH Open document ID: 0762.46053
DOI: 10.1016/0022-1236(92)90105-R
author: Keith Lewis
author: Brian J. Cole
author: John Wermer
published in: Journal of Functional Analysis
publication date: 1992-09-27T00:00:00Z
review text: The paper is a continuation of a previous paper by the last two authors [``On the theorems of Pick and von Neumann'', Function spaces (K. Jarosz, Ed.), Marcel Dekker, 273-280 (1992)]. Let \(A\) be a uniform algebra and let \(M_ 1,M_ 2,\dots,M_ n\) be maximal ideals of \(A\). The set \({\mathcal D}\) of points \(w=(w_ 1,w_ 2,\dots,w_ n)\) in \({\mathbf C}^ n\), such that for every \(\epsilon>0\) there is an \(f\in A\) with \(| f| \leq 1+\epsilon\) and \(f(M_ j)=w_ j,\;1\leq j\leq n\) is described in this paper. In the case when \(A\) is the disk algebra \(A(D)\) the problem was considered and solved originally by G. Pick in 1916. The authors introduce the following important notion: A compact set \(X\subset {\mathbf C}^ n\) with nonempty interior is called hyperconvex of order \(k\) if for every choice of \(k\) points \(w^{(j)}=(w_ 1^{(j)},w_ 2^{(j)},\dots,w_ n^{(j)}),\;j=1,\dots, k\) in \(X\) the points \((F(w_ 1^{(1)},\dots,w_ 1^{(k)}), F(w_ 2^{(1)},\dots,F(w_ 2^{(k)}),\dots, F(w_ n^{(1)},\dots,w_ n^{(k)}))\) also belong to \(X\), where \(F(z_ 1,\dots,z_ k)\) is any polynomial in \(k\) variables with \(\max_{D^ k}| F|\leq 1\). Hyperconvex sets of order 2 are convex, closed under multiplication in \({\mathbf C}^ n\), circled and contain 0 and 1. The authors show that the set \({\mathcal D}\) from above is hyperconvex of any order \(k\geq 1\). Subsets of \({\mathbf C}^ n\) that are hyperconvex in various orders are described. In particular the subsets of \({\mathbf C}^ n\) that are hyperconvex of any order \(k\geq 1\) are presented as closures of increasing unions of special subsets in \({\mathbf C}^ n\). Theorem. The family of inequalities \(\sum_{\alpha,\beta=1}^ nf_ \alpha{\overline f_ \beta}{\overline w_ \alpha}w_ \beta L_{\alpha\beta}\leq\sum_{\alpha,\beta=1}^ nt_ \alpha{\overline t_ \beta}L_{\alpha\beta}\) (Pick conditions) when \(t\) runs in \({\mathbf C}^ n\) and \(\{L_{\alpha,\beta}\}\) is a special family of semi- definite \(n\times n\) matrices, with compl…
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 65353
zbMATH Keywords: uniform algebra
zbMATH Keywords: maximal ideals
zbMATH Keywords: hyperconvex sets of order 2
zbMATH Keywords: Pick conditions
zbMATH Keywords: von Neumann inequality for contractions on a Hilbert space
zbMATH Keywords: contraction operator
zbMATH Keywords: \(Q\)-algebra
zbMATH Keywords: probability measure
zbMATH Keywords: convex set
reviewed by: Thomas V. Tonev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 65353