Banach spaces with the (strong) Gelfand--Phillips property

TL;DR

This paper introduces a strong version of the Gelfand-Phillips property for Banach spaces, characterizes it in terms of embedding into c_0, and applies it to spaces of continuous functions on compact spaces.

Contribution

It defines the strong Gelfand-Phillips property and characterizes Banach spaces with this property as those embedding into c_0, with applications to C(K) spaces.

Findings

A Banach space has the strong Gelfand-Phillips property iff it embeds into c_0.

C(K) has the strong Gelfand-Phillips property iff K is countable with finite scattered height.

The paper provides new characterizations of the Gelfand-Phillips property.

Abstract

Several new characterizations of the Gelfand-Phillips property are given. We define a strong version of the Gelfand-Phillips property and prove that a Banach space has this stronger property iff it embeds into $c_0$. For an infinite compact space $K$, the Banach space $C(K)$ has the strong Gelfand-Phillips property iff $C(K)$ is isomorphic to $c_0$ iff $K$ is countable and has finite scattered height.