Norm or numerical radius attaining polynomials on C(K)

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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, C ) is a norming set for every continuous complex polynomial. Similar results can be obtained if “norm” is replaced by “numerical radius.”