The Non-selfadjoint Approach to the Hao–Ng Isomorphism

Abstract

Abstract In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.