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Order isomorphisms of little Lipschitz algebras. (English) Zbl 1169.46010

Let $\left(X,{d}_{X}\right)$ be a compact metric space, let $𝕂$ be the set of complex or real numbers. The Banach space of all Lipschitz functions on $X$ to $𝕂$ with the norm $\parallel f\parallel =p\left(f\right)+{\parallel f\parallel }_{\infty }$ is denoted by $\text{Lip}\left(X,d\right)$. Here ${\parallel f\parallel }_{\infty }$ is the supremum norm and $p\left(f\right)$ is the Lipschitz constant of $f$. $\text{Lip}\left(X,d\right)$ 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 $f\left(x\right)\in ℝ$ and $f\left(x\right)\ge 0$ for all $x\in X$. The little Lipschitz algebra $\text{lip}\left(X,d\right)$ is the closed subspace of $\text{Lip}\left(X,d\right)$ consisting of all those functions $f$ in $\text{Lip}\left(X,d\right)$ with the property that for each $\epsilon >0$, there exists $\delta >0$ such that $0 implies $|f\left(x\right)-f\left(y\right)|/d\left(x,y\right)<\epsilon$.

Let $\alpha$ be a real number in $\left(0,1\right]$, then by ${d}^{\alpha }$ the authors denote the metric ${d}^{\alpha }\left(x,y\right)={\left(d\left(x,y\right)\right)}^{\alpha }$. The metric space $\left(X,{d}^{\alpha }\right)$ and the Lipschitz algebras $\text{Lip}\left(X,{d}^{\alpha }\right)$, $\text{lip}\left(X,{d}^{\alpha }\right)$ are considered in the paper.

Let $\left(X,{d}_{X}\right)$ and $\left(Y,{d}_{Y}\right)$ be compact metric spaces, and let $\alpha$ and $\beta$ be real numbers in $\left(0,1\right]$. A linear map $T:\text{lip}\left(X,{d}^{\alpha }\right)\to \text{lip}\left(X,{d}^{\beta }\right)$ is called an order isomorphism if $T$ is bijective and both $T$ and ${T}^{-1}$ are order-preserving. If $a:Y\to \left(0,\infty \right)$ is a function in $\text{lip}\left(Y,{d}_{Y}^{\beta }\right)$ and $h$ is a Lipschitz homeomorphism from $\left(Y,{d}_{Y}^{\beta }\right)$ onto $\left(X,{d}_{X}^{\alpha }\right)$, then the map $T:\left(X,{d}_{X}^{\alpha }\right)\to \left(Y,{d}_{Y}^{\beta }\right)$ defined by $T\left(f\right)=a·\left(f\circ h\right)$ for every $f\in \text{lip}\left(X,{d}_{X}^{\alpha }\right)$ is an order isomorphism. The main result of the paper is the proof that the converse is also true: every order isomorphism $T$ from $\left(X,{d}_{X}^{\alpha }\right)$ onto $\left(Y,{d}_{Y}^{\beta }\right)$ is a weighted composition operator of the form $T\left(f\right)=a·\left(f\circ h\right)$.

##### 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