OP-00692 Solution claimed

Kadison’s similarity problem

Let $\mathcal A$ be a unital complex $C^*$-algebra, $H$ a complex Hilbert space and $\pi : \mathcal A \to \mathcal B(H)$ a bounded unital algebra homomorphism, not assumed to respect the involution. Haagerup proved that $\pi$ is similar to a $*$-homomorphism, in the sense below, whenever $\pi$ is completely bounded, and also whenever $\pi$ has a finite cyclic set: finitely many vectors $\xi_1, \dots, \xi_n \in H$ such that the vectors $\pi(a)\xi_i$, $a \in \mathcal A$, span a dense subspace of $H$.

Problem. Must there exist, for every such $\mathcal A$, $H$ and $\pi$, a bounded invertible $S \in \mathcal B(H)$ for which $$a \longmapsto S^{-1}\pi(a)S$$ is a $*$-homomorphism, that is, $S^{-1}\pi(a^*)S = (S^{-1}\pi(a)S)^*$ for every $a \in \mathcal A$? No complete-boundedness assumption is made on $\pi$.

Source: Kadison (1955), for bounded homomorphisms of C*-algebras; stated for unital algebras and unital homomorphisms by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: OA — Operator Algebras
Secondary: FA
Keywords: AIM Open Applied Problems, C*-algebras, completely bounded maps, similarity problem

References

@article{kadison1955,
  author  = {Kadison, Richard V.},
  title   = {On the orthogonalization of operator representations},
  journal = {Amer. J. Math.},
  volume  = {77},
  number  = {3},
  year    = {1955},
  pages   = {600--620},
  doi     = {10.2307/2372645},
}
@article{haagerup1983,
  author  = {Haagerup, Uffe},
  title   = {Solution of the similarity problem for cyclic representations of {$C^*$}-algebras},
  journal = {Ann. of Math. (2)},
  volume  = {118},
  number  = {2},
  year    = {1983},
  pages   = {215--240},
  doi     = {10.2307/2007028},
}
@article{papapetros2024,
  author        = {Papapetros, E.},
  title         = {A new approach to the similarity problem},
  journal       = {Adv. Oper. Theory},
  volume        = {9},
  number        = {3},
  year          = {2024},
  pages         = {65},
  doi           = {10.1007/s43036-024-00363-4},
  eprint        = {2403.19668},
  archivePrefix = {arXiv},
}
@misc{aim-kadison-similarity,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Kadison’s similarity problem}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/037-kadison-similarity.md},
  note         = {Entry 037 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Kadison's similarity theorem through uniform derivation estimates v1 GitHub ↗

OpenAI

We prove that every bounded complex-linear unital algebra homomorphism from a unital complex $C^*$-algebra into the bounded operators on an arbitrary Hilbert space is similar to a $*$-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for all unital von Neumann algebras on arbitrary complex Hilbert spaces.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00692, "Kadison’s similarity problem", clopen.solutions.