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We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces *C*(*K*, *X*) and *L*_{∞}(μ, *X*) have the same polynomial numerical index as the complex Banach space *X* for every compact Hausdorff space *K* and every σ-finite measure μ, which does not hold any more in the real case. We give an example of a complex Banach space *X* such that, for every *k* ≥ 2, the polynomial numerical index of order *k* of *X* is the greatest possible, namely 1, while the one of *X*** is the least possible, namely *k*^{k/(1−k)}. We also give new examples of Banach spaces with the polynomial Daugavet property, namely *L*_{∞}(μ, *X*) when μ is atomless, and *C*_{w}(*K*, *X*), *C*_{w*}(*K*, *X**) when *K* is perfect.

*Journal Article.*
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*Subjects: *
Pure Mathematics

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