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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 $$T(f)=a\cdot (f\circ h)$$.

##### MSC:
 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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