On property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps in $\ell_{\infty}$-direct sums
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 $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ satisfies property-$(P_1)$ for the non-empty closed bounded subsets of $\ell_{\infty}$ 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 $\ell_{\infty}$. 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, J. Math. Anal. Appl., 426(2):1217--1231, 2015], for property-$(P_1)$.