On property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps in $\ell_{\infty}$-direct sums

Abstract

For a compact Hausdorff space S, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on S, denoted by CpSq, satisfies property-pP1q (the setvalued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of CpSq. Various stability results related to property-pP1q and semi-continuity properties of restricted Chebyshevcenter maps are also established. As a consequence, we derive that if Y is a proximinal finite co-dimensional subspace of c0 then the closed unit ball of Y satisfies property-pP1q for the non-empty closed bounded subsets of ℓ8 and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on the class of non-empty closed bounded subsets of ℓ8. We also investigate a variant of the transitivity property, similar to the one discussed in [C. R. Jayanarayanan and T. Paul, Strong proximinality and intersection properties of balls in Banach spaces,