Beyond its conceptual impact on noncommutative Choquet theory, the counterexample to Arveson's hyperrigidity we discovered with Bilich informs us on how to prove functoriality properties for the C*-envelope in non-commutative dynamical contexts, which then leads to insight on structural and classification problems through the use of the C*-envelope. For instance, for the seemingly unrelated, long-standing Hao-Ng isomorphism problem, which asks whether the Cuntz-Pimsner C*-algebra construction commutes with taking the full or reduced crossed product by a locally compact Hausdorff group action, beyond the amenable case of Hao and Ng. Building on ideas that emerge from the discovery of the counterexample to Arveson's hyperrigidity conjecture with Bilich, together with ideas arising from a recent amendment to Arveson's hyperrigidity conjecture by Clouatre and Thompson, we resolve the reduced version of the Hao-Ng isomorphism problem in full generality. More precisely, for a non-degenerate C*-correspondence $X$ and a generalized gauge action $G \curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X\rtimes_rG}\cong {\mathcal{O}}_X\rtimes_rG$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction.

*Based on joint work with Ian Thompson.