Some applications of conditional expectations to convergence for the quantum Gromov–Hausdorff propinquity

We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov–Hausdorff propinquity. We also show that given an action of a compact metrizable group $G$ on a quasi-Leibniz quantum compact metric space $(\mathfrak A,\sf{L})$, the function associating any closed subgroup of $G$ group to its fixed point C*-subalgebra in $A$ is continuous from the topology of the Hausdorff distance to the topology induced by the propinquity. Our techniques are inspired from our work on AF algebras as quantum metric spaces, as they are based on the use of various types of conditional expectations.