On 2-local isometries on continuous vector-valued function spaces

Abstract

A (not necessarily linear) mapping Φ from a Banach space X to a Banach space Y is said to be a 2 -local isometry if for any pair x , y of elements of X, there is a surjective linear isometry T : X → Y such that T x = Φ x and T y = Φ y . We show that under certain conditions on locally compact Hausdorff spaces Q, K and a Banach space E, every 2-local isometry on C 0 ( Q , E ) to C 0 ( K , E ) is linear and surjective. We also show that every 2-local isometry on ℓ p is linear and surjective for 1 ⩽ p < ∞ , p ≠ 2 , but this fails for the Hilbert space ℓ 2 .