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Order isomorphisms of little Lipschitz algebras. (English) Zbl 1169.46010
Let $(X,d_X)$ be a compact metric space, let $\Bbb K$ be the set of complex or real numbers. The Banach space of all Lipschitz functions on $X$ to $\Bbb K$ 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 $f\ge 0$ if and only if $f(x)\in \Bbb R$ and $f(x)\ge 0$ 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 $T(f)=a\cdot (f\circ h)$.

46E05Lattices of continuous, differentiable or analytic functions
46J10Banach algebras of continuous functions, function algebras
47B38Operators on function spaces (general)
47B65Positive and order bounded operators
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