Pick conditions on a uniform algebra and von Neumann inequalities
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 complex entries depending on the initial algebra \(A\), is necessary and sufficient for the point \((w_ 1,\dots,w_ n)\in{\mathbf C}^ n\) to belong to \({\mathcal D}\). In the case when \(A\) is the disk algebra, the family \(\{L_{\alpha,\beta}\}\) restricts to a single matrix with entries \(L_{\alpha,\beta}= (1-{\overline z_ \alpha z_ \beta})^{-1}\) for all \(\alpha,\beta\) and the above condition becomes the classical Pick's condition. The paper is a rich sourse of other interesting results, the description of the set \({\mathcal D}_{\mathcal A}=\{w\in{\mathbf C}^ n:| \sum_{k=1}^ nw_ kP_ k| \leq 1\}\), where \({\mathcal A}\) is an \(n\)-dimensional commutative, semi-simple Banach algebra and \(\{P_ 1,\dots,P_ n\}\) is the set of the minimal idempotents of \({\mathcal A}\), among others. The connections with generalizations of the classical von Neumann inequality for contractions on a Hilbert space are studied as well.
von Neumann inequality for contractions on a Hilbert space
scientific article; zbMATH DE number 65353