The Hao-Ng isomorphism theorem for reduced crossed products

We prove the Hao-Ng isomorphism for reduced crossed products by locally compact Hausdorff groups. More precisely, for a non-degenerate ${\mathrm{C}}^{\ast }$-correspondence $X$ and a generalized gauge action $G\curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X{⋊}_{r}G}\cong {\mathcal{O}}_X{⋊}_{r}G$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction. This is done by proving that the reduced crossed product of an operator algebra commutes with the C*-envelope, which relies on refined W*-dynamical covers of C*-dynamical systems, unitary implementation of W*-dynamical systems, and an operator-valued extension of Maharam’s lifting theorem.