Lipschitz isometries of compact quantum metric spaces
A real vector space \(A\) is called an \textit{order unit space} if there exists a partial order \(\leq\) on \(A\) and a distinguished element \(1_A\) such that (i) for each \(a\) in \(A\) there is a real number \(r\) with \(a \leq r1_A\) and (ii) if \(a \leq r1_A\) for all positive real numbers \(r\), then \(a \leq 0\). An order unit space admits a norm \(\Vert a\Vert := \inf\{r \in \mathbb{R}^+ : -r1_A \leq a \leq r1_A\}\); if \(A\) is complete in this norm, we say that \(A\) is a \textit{complete order unit space}. The prototypical example of a complete order unit space is the real vector space of self-adjoint elements of a unital \(C^*\)-algebra. A \textit{state} on \(A\) is a continuous linear functional \(\mu\) on \(A\) with the property that \(\mu(1_A) = 1 = \Vert\mu\Vert\); we denote the set of states on \(A\) by \(\mathcal{S}(A)\). A \textit{Lipschitz seminorm} on an order unit space \(A\) is a seminorm \(L\) on \(A\) with the features that (i) the set \(\{a \in A: L(a) < \infty\}\) is dense in \(A\) and (ii) the equation \(L(a) = 0\) holds if and only if a is a real multiple of \(1\). We call \(L\) a \textit{Lip-norm} if the metric \(\rho_L(\mu, \nu) := \sup\{\vert\mu(a) - \nu(a)\vert : a \in A$, $L(a) \leq 1\}\) has an induced topology that is the same as the weak\(*\)-topology on \(\mathcal{S}(A)\). A complete order unit space equipped with a Lip-norm is called a \textit{compact quantum metric space}. Given compact quantum metric spaces \((A, L_A)\) and \((B, L_B)\), a map \(\phi\) from \(A\) to \(B\) is a \textit{morphism} if \(\phi\) is (i) unital, (ii) maps positive elements of \(A\) to positive elements of \(B\), and (iii) there exists a positive real constant \(\lambda\) such that \(L_B(\phi(a)) \leq \lambda L_A(a)\) for every \(a\) in \(A\). In the paper under review, the authors study the category of compact quantum metric spaces. In particular, the authors consider the question of when a morphism of compact quantum metric spaces is a Lipschitz isometry.
scientific article; zbMATH DE number 7217642
scientific article