Norm or numerical radius attaining polynomials on \(C(K\))
A real Banach space X has the Badé property if the set of extreme points \(\operatorname{ext} B_X\) of the closed ball \(B_X\) is a norming subset of \(X^\ast.\) Also, \(X\) has the \(E\)-property if every norm attaining linear form \(x^\ast\) attains its norm at a point in \(\text{ext} B_X.\) Finally, \(X\) is said to have the \(\lambda\)-property if \(B_X\) coincides with the sequential convex hull of \(\operatorname{ext} B_X.\) \textit{A. Aizpuru} [Arch. Math. 69, 333--337 (1997; Zbl 0917.46010)] proved the following two results. Theorem 1. The following are equivalent for \(X = {\mathcal C}(K,\mathbb C): \) \(X\) has the E-property; \(X\) has the \(\lambda\)-property; \(K\) is \(0\) or \(1\) dimensional. Theorem 2. The following are equivalent for \(X = {\mathcal{C}}(K,\mathbb R):\) \(\text{ext} B_X\) is a norming set for \(X^\ast;\) \(X\) has the Badé property; \(X\) has the \(\lambda\)-property; \(K\) is \(0\)-dimensional. The two main questions posed here are the following. Question 1. Given \(K,\) a compact Hausdorff space such that \(X = {\mathcal C}(K,\mathbb K)\) has the E-property, \(n \in \mathbb N\), \(n \geq 2,\) and a norm attaining \(n\)-homogeneous polynomial \(P:{\mathcal C}(K,\mathbb K) \to \mathbb K,\) does it follow that \(\| P\| \) is attained at some point of \(\operatorname{ext} B_X?\) Question 2. Suppose now that \(X = {\mathcal C}(K,\mathbb K)\) has the Badé property. Is it true that \(\operatorname{ext} B_X\) is a norming set for the space of \(n\)-homogeneous polynomials \(P: X \to \mathbb K\), \(n\geq 2?\) The authors show that Question 1 has an affirmative answer if \(\mathbb K = \mathbb C\) and \(K\) is compact of dimension at most 1. If \(K\) is also metrizable, the authors show that the conditions in Theorem 1 are also equivalent to the statement that every norm attaining \(n\)-homogeneous polynomial \(P:{\mathcal C}(K,\mathbb C) \to \mathbb C\) attains its norm at an extreme point. Also, if \(K\) is scattered (dispersed) [\textit{A. Pełczyński} and \textit{Z. Semadeni}, Stud. Math. 18, 211--222 (1959; Zbl 0091.27803)], the answer to Question 2 is positive. On the other hand, in the case of real scalars, the answers to Questions 1 and 2 are negative whenever \(n \geq 2.\) Several examples are also provided. In the second half of this paper, the authors examine norming sets as well as denseness of norm attaining and numerical radius attaining polynomials at extreme points. One consequence of their work is that if \(K\) is scattered and \(T:B_{\mathcal C}(K,\mathbb C)\to \mathbb C\) is holomorphic and uniformly continuous, then \(\| T\| = \sup\{ | T(f)| : f \in \text{ext} B_{{\mathcal C}(K,\mathbb C)}\}.\)
scientific article; zbMATH DE number 2093794
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