This paper contributes to a very active area of ‘local’ Banach-Stone theorems for spaces of vector-valued functions. For an isomorphism of such spaces, ‘local conditions’ are assumed on to ensure that the underlying spaces are isomorphic and is described in terms of these objects in a canonical way (composition operator).
Let be a metric space and let be a nonzero Banach lattice. Let Lip denote the Banach space of bounded -valued Lipschitz functions with pointwise order and with respect to the norm . For and in this category, let and be closed sublattices (this correction gets noted in “Correction to ‘The uniform separation property and Banach-Stone theorems for lattice-valued Lipschitz functions”’, ibid. 138, 1535) such that they separate and join points uniformly. Let be a vector lattice isomorphism that preserves the nowhere vanishing functions in both the directions. Then there exists a bi-Lipschitz map and a Lipschtz map such that takes values as lattice isomorphisms and for and .