*(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 $\parallel f\parallel =p\left(f\right)+{\parallel f\parallel}_{\infty}$ is denoted by $\text{Lip}(X,d)$. Here ${\parallel f\parallel}_{\infty}$ is the supremum norm and $p\left(f\right)$ 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 $f\left(x\right)\in \mathbb{R}$ and $f\left(x\right)\ge 0$ 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 $\epsilon >0$, there exists $\delta >0$ such that $0<d(x,y)<\delta $ implies $\left|f\right(x)-f(y\left)\right|/d(x,y)<\epsilon $.

Let $\alpha $ be a real number in $(0,1]$, then by ${d}^{\alpha}$ the authors denote the metric ${d}^{\alpha}(x,y)={\left(d(x,y)\right)}^{\alpha}$. 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 $\text{lip}(Y,{d}_{Y}^{\beta})$ 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 $T\left(f\right)=a\xb7(f\circ h)$ for every $f\in \text{lip}(X,{d}_{X}^{\alpha})$ 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 $T\left(f\right)=a\xb7(f\circ h)$.

##### MSC:

46E05 | Lattices of continuous, differentiable or analytic functions |

46J10 | Banach algebras of continuous functions, function algebras |

47B38 | Operators on function spaces (general) |

47B65 | Positive and order bounded operators |