Inverses of disjointness preserving operators (Q2835993)

Let \(A(X)\) and \(A(Y)\) be spaces of functions on Hausdorff topological spaces \(X\) and \(Y\), respectively. A~linear map \(T : A(X) \to A(Y)\) that sends a pair of functions with disjoint support to functions with disjoint support is said to be disjointness preserving. A~linear isomorphism \(T\) such that \(T\) and also \(T^{-1}\) are disjointness preserving is called biseparating. Such maps have been studied in many cases: spaces of continuous functions, vector lattices, some spaces of differentiable or Lipschitz functions, etc. Often, finding a representation of this class of maps is a key step to obtain properties like continuity or to prove that they are biseparating. The main results of this paper are of the former type. For example, it is shown that a disjointness preserving linear isomorphism \(T : A(X) \to A(Y)\) is biseparating when \(A(X)\) is \(\operatorname{Lip}_{\alpha}(X)\), \(\operatorname{lip}_{\alpha}(X) \), for \(0<\alpha<1\), or \(U(X)\), the space of uniformly continuous functions on \(X\), and \(A(Y)\) is a linear subspace of \(C(Y)\) containing the constant functions. This holds under the assumptions that \(X\) is a complete metric space and \(Y\) is a first countable Hausdorff space with finitely many connected components. A~similar result for spaces of differentiable functions is also proved.