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Order isomorphisms of little Lipschitz algebras. (English) Zbl 1169.46010
Let \((X,d_X)\) be a compact metric space, let \(\mathbb K\) be the set of complex or real numbers. The Banach space of all Lipschitz functions on \(X\) to \(\mathbb K\) with the norm \(\|f\|=p(f)+\|f\|_\infty\) is denoted by \(\text{Lip}(X,d)\). Here \(\|f\|_\infty\) is the supremum norm and \(p(f)\) is the Lipschitz constant of \(f\). \(\text{Lip}(X,d)\) is a commutative Banach algebra with respect to pointwise multiplication, but it also is an ordered vector space with respect to the pointwise order defined by \(f\geq 0\) if and only if \(f(x)\in \mathbb R\) and \(f(x)\geq 0\) for all \(x\in X\). The little Lipschitz algebra \(\text{lip}(X,d)\) is the closed subspace of \(\text{Lip}(X,d)\) consisting of all those functions \(f\) in \(\text{Lip}(X,d)\) with the property that for each \(\varepsilon>0\), there exists \(\delta>0\) such that \(0<d(x,y)<\delta\) implies \(|f(x)-f(y)|/d(x,y)<\varepsilon\).
Let \(\alpha\) be a real number in \((0,1]\), then by \(d^\alpha\) the authors denote the metric \(d^\alpha(x,y)=(d(x,y))^\alpha\). The metric space \((X,d^\alpha)\) and the Lipschitz algebras \(\text{Lip}(X,d^\alpha)\), \(\text{lip}(X,d^\alpha)\) are considered in the paper.
Let \((X,d_X)\) and \((Y,d_Y)\) be compact metric spaces, and let \(\alpha\) and \(\beta\) be real numbers in \((0,1]\). A linear map \(T:\text{lip}(X,d^\alpha)\to \text{lip}(X,d^\beta)\) is called an order isomorphism if \(T\) is bijective and both \(T\) and \(T^{-1}\) are order-preserving. If \(a:Y\to (0,\infty)\) is a function in \(\text{lip}(Y,d^\beta_Y)\) and \(h\) is a Lipschitz homeomorphism from \((Y,d_Y^\beta)\) onto \((X,d_X^\alpha)\), then the map \(T:(X,d_X^\alpha)\to (Y,d_Y^\beta)\) defined by \(T(f)=a\cdot (f\circ h)\) for every \(f\in \text{lip}(X,d^\alpha_X)\) is an order isomorphism. The main result of the paper is the proof that the converse is also true: every order isomorphism \(T\) from \((X,d_X^\alpha)\) onto \((Y,d_Y^\beta)\) is a weighted composition operator of the form \(T(f)=a\cdot (f\circ h)\).

46E05 Lattices of continuous, differentiable or analytic functions
46J10 Banach algebras of continuous functions, function algebras
47B38 Linear operators on function spaces (general)
47B65 Positive linear operators and order-bounded operators
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