Obstructions to the full Hao-Ng isomorphism

Abstract

We show that the Hao-Ng isomorphism for full crossed products fails to hold. More precisely, for a non-degenerate $C^{\ast }$-correspondence $X$ over a $C^{\ast }$-algebra $\mathcal{A}$ and a generalized gauge action $G\curvearrowright X$ by a locally compact group $G$, we isolate three structural obstructions for the canonical commutation ${\mathcal{O}}_X⋊G\cong {\mathcal{O}}_{X⋊G}$ between full crossed product and Cuntz-Pimsner $C^{\ast }$-algebra. The first concerns hyperrigidity, the second concerns WEP of the coefficient $C^{\ast }$-algebra $\mathcal{A}$ for trivial actions, and the third concerns the coincidence of full and reduced crossed products by the action of $G$ on $\mathcal{A}$.

Minor revision. Reordered material on tensor algebras. Added acknowledgements. 18 pages including references