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

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 Tx=Φx and Ty=Φy. We show that under certain conditions on locally compact Hausdorff spaces Q, K and a Banach space E, every 2-local isometry on C0(Q,E) to C0(K,E) is linear and surjective. We also show that every 2-local isometry on ℓp is linear and surjective for 1⩽p<∞, p≠2, but this fails for the Hilbert space ℓ2.