On Property-$(P_{1})$ in Banach spaces

TL;DR

This paper explores property-$(P_1)$, a set-valued generalization of strong proximinality in Banach spaces, establishing conditions under which subspaces inherit this property and examining its implications for various classes of Banach spaces.

Contribution

It provides new results on property-$(P_1)$ inheritance in Banach subspaces, characterizes strongly proximinal subspaces, and connects property-$(P_1)$ with hyperplanes and Chebyshev centers.

Findings

Property-$(P_1)$ is inherited by subspaces under certain conditions.

Closed unit ball of an $M$-ideal in an $L_{1}$-predual satisfies property-$(P_1)$ for compact sets.

Characterization of strongly proximinal subspaces in terms of property-$(P_1)$ for $C(S)$ and $A(K)$ spaces.

Abstract

We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory, 29(3):223--230, 1980] as property-$(P_1)$. We establish that if the closed unit ball of a closed subspace of a Banach space $X$ possesses property-$(P_1)$ for each of the classes of closed bounded, compact and finite subsets of $X$, then so does the subspace. It is also proved that the closed unit ball of an $M$-ideal in an $L_{1}$-predual space satisfies property-$(P_{1})$ for the compact subsets of the space. For a Choquet simplex $K$, we provide a sufficient condition for the closed unit ball of a finite co-dimensional closed subspace of $A(K)$ to satisfy property-$(P_{1})$ for the compact subsets of $A(K)$. This condition also helps to establish the equivalence of strong proximinality of the closed unit ball of a finite co-dimensional subspace of $A(K)$ and property-$(P_{1})$ of the closed unit ball of the subspace for the compact subsets of $A(K)$. Further, for a compact Hausdorff space $S$, a characterization is provided for a strongly proximinal finite co-dimensional closed subspace of $C(S)$ in terms of property-$(P_{1})$ of the subspace and that of its closed unit ball for the compact subsets of $C(S)$. We generalize this characterization for a strongly proximinal finite co-dimensional closed subspace of an $L_{1}$-predual space. As a consequence, we prove that such a subspace is a finite intersection of hyperplanes such that the closed unit ball of each of these hyperplanes satisfy property-$(P_1)$ for the compact subsets of the $L_1$-predual space and vice versa. We conclude this article by providing an example of a closed subspace of a non-reflexive Banach space which satisfies $1 \frac{1}{2}$-ball property and does not admit restricted Chebyshev centre for a closed bounded subset of the Banach space.