On 2-local diameter-preserving maps between C(X)-spaces

The 2-locality problem of diameter-preserving maps between C ( X )-spaces is addressed in this paper. For any compact Hausdorff space X with at least three points, we give an example of a 2-local diameter-preserving map on C ( X ) which is not linear. However, we show that for first countable compact Hausdorff spaces X and Y , every 2-local diameter-preserving map from C ( X ) to C ( Y ) is linear and surjective up to constants in some sense. This fact yields the 2-algebraic reflexivity of isometries with respect to the diameter norms on the quotient spaces.