On the embeddings of selfadjoint operator spaces
Alexandros Chatzinikolaou, Evgenios T. A. Kakariadis, Se-Jin Kim, Ioannis Apollon Paraskevas
TL;DR
The paper develops a comprehensive framework for when a map on a selfadjoint operator space $E$ is an embedding, presenting multiple equivalent characterizations in terms of extending functionals and cp maps, and a gauge/distance formulation. It introduces the automatic embedding property and proves that completely approximately 1-generated spaces yield automatic embeddings, with extensions of cp maps preserved under unitality assumptions in many cases. The results connect embedding theory with positive generation, one-dimensional generation, and hyperrigidity, and they show how the C*-envelope of $E$ can be identified with $C^*(E)$ in several contexts. Overall, the work unifies intrinsic and extrinsic approaches to positivity and paves the way for applying hyperrigidity to identify C$^*$-envelopes in non-unital settings.
Abstract
We investigate when a map on a selfadjoint operator space $E$ is an embedding, i.e., when its unitisation in the sense of Werner is completely isometric. Combining with results of Russell, of Ng, and of Dessi, the second and the last author, it is shown that this is equivalent to: (a) extending bounded positive functionals on each matrix level with the same norm; (b) extending quasistates to quasistates in each matrix level; (c) extending completely bounded completely positive maps with the same cb-norm; and (d) the map being a gauge maximal isometry in the sense of Russell. If $E$ is approximately positively generated and $\mathrm{C}^*(E)$ is unital, or if $E_{sa}$ is singly generated, then completely positive maps on $E\subseteq\mathcal{B}(H)$ have completely positive extensions on $\mathrm{C}^*(E)$, but possibly not with the same cb-norm; and this is not enough for the inclusion $E \subseteq \mathrm{C}^*(E)$ to be an embedding. We show that the inclusion $E \subseteq \mathrm{C}^*(E)$ is always an embedding when $E$ is completely approximately 1-generated, and we fully resolve the case when $E_{sa}$ is singly generated. Combining with the works of Salomon, Humeniuk--Kennedy--Manor, and previous work of the third author, we show that if the inclusion $E \subseteq \mathrm{C}^*(E)$ is an embedding, then rigidity at zero, in the sense of Salomon, coincides with $E$ being approximately positively generated. Consequently, we show that $E$ is approximately positively generated if and only if $M_n(E)$ is approximately positively generated for all $n\in \mathbb{N}$, thus extending a previous result of Humeniuk--Kennedy--Manor to the approximation setting. As an application we show that hyperrigidity of $E$ in $\mathrm{C}^*(E)$ allows to identify $\mathrm{C}^*(E)$ as the C*-envelope of $E$ in several (non-unital) contexts.