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The uniform separation property and Banach-Stone theorems for lattice-valued Lipschitz functions. (English) Zbl 1202.46043

This paper contributes to a very active area of ‘local’ Banach-Stone theorems for spaces of vector-valued functions. For an isomorphism T of such spaces, ‘local conditions’ are assumed on T to ensure that the underlying spaces are isomorphic and T is described in terms of these objects in a canonical way (composition operator).

Let (X,d) be a metric space and let E be a nonzero Banach lattice. Let Lip(X,E) denote the Banach space of bounded E-valued Lipschitz functions with pointwise order and with respect to the norm max{Lip(f),f }. For X,Y and E,F in this category, let A(X,E) and A(Y,F) 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 T:A(X,E)A(Y,F) be a vector lattice isomorphism that preserves the nowhere vanishing functions in both the directions. Then there exists a bi-Lipschitz map φ:YX and a Lipschtz map T :YL(E,F) such that T takes values as lattice isomorphisms and T(f)(y)=T (y)(f(φ(y)) for yY and fA(X,E).

MSC:
46E40Spaces of vector- and operator-valued functions
46E05Lattices of continuous, differentiable or analytic functions