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The Hao-Ng isomorphism theorem for reduced crossed products

Adam Dor-On, Ian Thompson

TL;DR

This work resolves the reduced Hao-Ng isomorphism problem for reduced crossed products by arbitrary locally compact Hausdorff groups by proving that the reduced crossed product functor commutes with the Cuntz-Pimsner construction. The authors develop operator-valued Maharam-type lifting theorems via Hamana's Fubini tensor products to enable lifting arguments in non-separable settings, establishing a key commutation result for C*-envelopes with reduced crossed products. Leveraging this envelope-commutation, they prove ${ olinebreak O}_X times_{r} G \,\cong\, {\nolinebreak O}_{X \rtimes_{r} G}$ for generalized gauge actions, thereby solving the reduced Hao-Ng problem in full generality. The results illuminate the role of the C*-envelope and unique extension properties in crossed-product theory and have potential implications for broader connections between non-self-adjoint and self-adjoint operator algebra frameworks.

Abstract

We prove the Hao-Ng isomorphism for reduced crossed products by locally compact Hausdorff groups. More precisely, for a non-degenerate $\mathrm{C}^*$-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.

The Hao-Ng isomorphism theorem for reduced crossed products

TL;DR

This work resolves the reduced Hao-Ng isomorphism problem for reduced crossed products by arbitrary locally compact Hausdorff groups by proving that the reduced crossed product functor commutes with the Cuntz-Pimsner construction. The authors develop operator-valued Maharam-type lifting theorems via Hamana's Fubini tensor products to enable lifting arguments in non-separable settings, establishing a key commutation result for C*-envelopes with reduced crossed products. Leveraging this envelope-commutation, they prove for generalized gauge actions, thereby solving the reduced Hao-Ng problem in full generality. The results illuminate the role of the C*-envelope and unique extension properties in crossed-product theory and have potential implications for broader connections between non-self-adjoint and self-adjoint operator algebra frameworks.

Abstract

We prove the Hao-Ng isomorphism for reduced crossed products by locally compact Hausdorff groups. More precisely, for a non-degenerate -correspondence and a generalized gauge action by a locally compact Hausdorff group , we prove the commutation of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction.
Paper Structure (6 sections, 6 theorems, 24 equations)

This paper contains 6 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The C*-envelope and injective envelope
  4. C*-correspondences, Cuntz-Pimsner algebras, and their dynamics
  5. Operator-valued Maharam-type lifting theorems
  6. Reduced crossed products and their C*-envelopes

Key Result

Theorem A

Let ${\mathcal{A}}$ be an operator algebra with a self-adjoint contractive approximate identity, and let $G$ be a locally compact Hausdorff group. If $\alpha:G\curvearrowright{\mathcal{A}}$ is an action, then $\mathrm{C}^*_e({\mathcal{A}})\rtimes_{\alpha,r}G\cong\mathrm{C}^*_e({\mathcal{A}}\rtimes_{

Theorems & Definitions (11)

  • Theorem A
  • Theorem B
  • Proposition 3.1
  • proof
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 1 more