Norm or numerical radius attaining polynomials on C(K)

Let C(K, C) be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on C(K, C) attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on C(K, C) can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of C (K, E) is a norming set for every continuous complex polynomial. Similar results can be obtained if "norm" is replaced by "numerical radius." (C) 2004 Elsevier lnc. All rights reserved.