Uniformly separation property in vector-valued little Lipschitz space

Suppose that (𝑋, 𝑑) be a compact metric space with a distinguished point𝑒 and𝐸 be a Banach space.Collection of 𝐸 −valued function 𝑓 on 𝑋 such that ℒ(𝑓) = sup ,𝑥≠𝑦 𝑥,𝑦∈𝑋 ‖𝑓(𝑥) − 𝑓(𝑦)‖ 𝑑(𝑥, 𝑦) < ∞ , 𝑓(𝑒) = 0 is called vector-valued Lipschitz space and denoted by 𝐿𝑖𝑝0 (𝑋, 𝐸). The space 𝐿𝑖𝑝0(𝑋, 𝐸) with respect to the point wise operations on functions and the norm ℒ(. ) is a Banach space that separates points of 𝑋. The subset consists of all functions such that lim 𝑑(𝑥,𝑦)→0 ‖𝑓(𝑥) − 𝑓(𝑦)‖ 𝑑(𝑥, 𝑦) = 0 is a closed subspace of 𝐿𝑖𝑝0 (𝑋, 𝐸), denoted by 𝑙𝑖𝑝0(𝑋, 𝐸) and called little vector-valued Lipschitz space. In particular when Banach space 𝐸 coincides with scaler field, 𝐿𝑖𝑝0(𝑋, 𝐸) and 𝑙𝑖𝑝0(𝑋, 𝐸) is denoted by 𝐿𝑖𝑝0(𝑋) and 𝑙𝑖𝑝0(𝑋) respectively. Definition. The space 𝑙𝑖𝑝0(𝑋) separates points of 𝑋 uniformly when there exists 𝐶 > 1 such that for each distinct pair point 𝑥, 𝑦 ∈ 𝑋 there is 𝑓 ∈ 𝑙𝑖𝑝0(𝑋, 𝐸) with 𝑓(𝑦) = 0, ‖𝑓(𝑥)‖ = 𝑑(𝑥, 𝑦), ℒ(𝑓) ≤ 𝐶. Definition.The Banach space 𝐸 has approximation property if for each ε > 0 and compact subset 𝐾 of 𝐸 there exists a finite dimensional bounded operator 𝑇: 𝐸 → 𝐸 such that sup 𝑥∈𝐾 ‖𝑇𝑥 − 𝑥‖ < 𝜀.

In this paper we deal with the uniform separation property of a metric space 𝑋 by the little vector-valued Lipschitz space, namely 𝑙𝑖𝑝0(𝑋, 𝐸).

We show that if 𝑙𝑖𝑝0 (𝑋) has the approximation property and 𝐸 be a topological dual of some Banach space, then there exists a compact metric space 𝑌 with a distinguished point and a non-expansive function 𝜋: 𝑋 → 𝑌 such that 𝑙𝑖𝑝0(𝑌, 𝐸) separates the point of 𝑌 uniformly and 𝐶𝜋, the composition operator induced by 𝜋, is a surjective linear isometry from 𝑙𝑖𝑝0(𝑌, 𝐸) to 𝑙𝑖𝑝0(𝑋, 𝐸).