Arveson’s hyperrigidity conjecture

Let $A$ be a unital operator algebra generating the $C^*$ -algebra $B=C^*(A)$ . If every irreducible $*$ -representation $\pi:B\to B(H_\pi)$ has the unique completely positive extension property relative to $A$ —that is, the only unital completely positive map $\phi:B\to B(H_\pi)$ satisfying $\phi|_A=\pi|_A$ is $\phi=\pi$ —then $A$ is hyperrigid: every $*$ -representation $\rho:B\to B(H_\rho)$ has the unique completely positive extension property relative to $A$ .

Progress summary

Partially solved

The original conjecture has a counterexample, while a new paper proposes a state-space replacement rather than restoring the failed claim.

Arveson’s conjecture concerned whether unique completely positive extension properties for irreducible $*$ -representations force hyperrigidity of a unital operator algebra. Bilich and Dor-On showed that this implication fails, and later work develops reformulations rather than reviving the original statement.

April 2024 counterexample and August 2026 amendment

In April 2024, Bilich and Dor-On gave an explicit counterexample: an algebra generated by a rank-one perturbation of a unitary is not hyperrigid in its generated type-I $C^*$ -algebra, despite the stated unique-extension condition. Later papers record this as a negative solution.
On August 19, 2026, Hridoyananda Saikia proposed a state-space-level amendment, relating Choquet-type maximality orders and deriving corresponding maximality implications. This is a structural replacement, not a proof of the original conjecture.

Current status (as of August 2026): the original conjecture is settled false by the Bilich–Dor-On counterexample, while Saikia’s state-space amendment remains a recent preprint-level claim requiring further independent assessment.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Choquet-Type Relations and a State Space Level Amendment of Arveson's Hyperrigidity Conjecture — arXiv — Saikia, Hridoyananda

functional-analysis operator-algebras

2 days ago