On hyperrigidity and non-degenerate C*-correspondences

We revisit the results of Kim, and of Katsoulis and Ramsey concerning hyperrigidity for non-degenerate C*-correspondences. We show that the tensor algebra is hyperrigid, if and only if Katsura’s ideal acts non-degenerately, if and only if Katsura’s ideal acts non-degenerately under any representation. This gives a positive answer to the question of Katsoulis and Ramsey, showing that their necessary condition and their sufficient condition for hyperrigidity of the tensor algebra are equivalent. Non-degeneracy of the left action of Katsura’s ideal was also shown by Kim to be equivalent to hyperrigidity for the selfadjoint operator space associated with the C*-correspondence, and our approach provides a simplified proof of this result as well. In the process we study unitisations of selfadjoint operator spaces in the sense of Werner, and revisit Arveson’s criterion connecting maximality with the unique extension property and hyperrigidity, in conjunction with the work of Salomon on generating sets.

💡 Research Summary

This paper revisits and unifies several recent results on hyperrigidity for non‑degenerate C*‑correspondences, providing a definitive answer to a question raised by Katsoulis and Ramsey. Let (X) be a non‑degenerate C*‑correspondence over a C*‑algebra (A). Denote by (\mathcal{T}_X) the Toeplitz–Pimsner algebra, by (\mathcal{O}X) the Cuntz–Pimsner algebra, and by (J_X) Katsura’s ideal in (A). The left action of (A) on (X) is (\varphi_X). The authors consider three closely related objects: the tensor algebra (\mathcal{T}+(X)), the self‑adjoint operator space (S(A,X)) generated by (A) and (X) inside (\mathcal{T}_X), and the generating set (\pi_X(A)\cup t_X(X)\subseteq\mathcal{O}_X).

The main theorem (Theorem 4.4) establishes the equivalence of nine statements, among which are:

1. (\pi_X(A)\cup t_X(X)) is hyperrigid in (\mathcal{O}_X);
2. ((\pi_X\times t_X)(S(A,X))) is hyperrigid in (\mathcal{O}_X);
3. ((\pi_X\times t_X)(\mathcal{T}_+(X))) is hyperrigid in (\mathcal{O}_X);
4. For every Cuntz–Pimsner covariant representation ((H,\pi,\psi)) with ((H,\pi)) non‑degenerate, (