Lipschitz isometries of compact quantum metric spaces

Lipschitz isomorphisms of compact quantum metric spaces

“…as in [20,Example 3.3]; on the other hand, one can get a better/sharp positive real number in Proposition 3.7 and Theorem 3.9 for asymptotically k-nonexpansive maps.…”

“…If f is invertible and both f and f −1 are Lipschitz, then we say that f is bi-Lipschitz [33]. If there is a bi-Lipschitz map between metric spaces (X, ρ X ) and (Y, ρ Y ), then they are bi-Lipschitz equivalent [20].…”

“…Motivated by [20,Theorem 2.6] and the definition of Lipschitz map between metric spaces, we define a Lipschitz map between order unit spaces with Lipschitz seminorms [10, 12-15, 18-21, 37]. Definition 2.5 ([20]).…”

“…From these definitions we can easily see that (asymptotically) k-nonexpansive maps between metric spaces are uniformly continuous. Note that (asymptotically) 1-nonexpansive maps are the usual (asymptotically) nonexpansive maps [7,20]. A map φ : A → A is said to be asymptotically k-nonexpansive if there is a sequence {k i } i∈N of positive real numbers with…”

“…In [20], the present authors proposed morphisms for the category of compact quantum metric spaces in the framework of order unit spaces, and proved that a unital positive linear map, preserving Lipschitz elements, between compact quantum metric spaces with lower semicontinuous Lip-norms is necessarily Lipschitz. Motivated by the asymptotically expansive (nonexpansive) maps for metric spaces, the authors defined asymptotically expansive (nonexpansive) maps between order unit spaces with Lipschitz seminorms, and showed that these maps are Lipschitz isometries under certain conditions.…”