Spatial conjugacy of close separable C*-algebras
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Must sufficiently close unital norm-separable C*-algebras on one separable Hilbert space be conjugate by a unitary? The Kadison–Kastler spatial-stability target allows arbitrarily small distance and a common identity, even when the algebras have the same von Neumann closure.
Why it matters and what counts as progress
Why it matters: Spatial stability concerns unitary implementation, a stronger requirement than abstract isomorphism and distinct from the von Neumann-algebra question.
What counts as progress: Prove spatial conjugacy under broader assumptions or construct arbitrarily close algebras with no implementing unitary; this need not show abstract nonisomorphism.
How the problem developed
- 1972 · Conjecture [1]
Richard V. Kadison and Daniel Kastler Proposed that sufficiently close von Neumann algebras on the same Hilbert space should be conjugate by a unitary close to the identity.
- 26 Nov 2012 · Partial result [2]
Jan Cameron, Erik Christensen, Allan M. Sinclair, Roger R. Smith, Stuart White and Alan D. Wiggins Established the spatial-isomorphism stability conjecture for specified classes of nonamenable type II₁ factors; this is a restricted von Neumann algebra result.
- 5 Oct 2026 · Counterexample claim [3]
OpenAI Reported arbitrarily close unital norm-separable C*-subalgebras with common identity that are not ambiently unitarily conjugate, while having the same von Neumann closure.
AI contribution
Close Separable C*-Algebras Without Spatial Conjugacy
For every epsilon>0, the OpenAI-authored manuscript constructs unital norm-separable C*-algebras on a common separable Hilbert space, with common identity and Kadison-Kastler distance below epsilon, which share their von Neumann closure but are not conjugate by any unitary. It does not assert abstract nonisomorphism or a von Neumann counterexample.
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OpenAI reports close separable C*-algebras without spatial conjugacy