Strong Ulam Stability Characterizes Amenability
A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan's amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.
Comments and reviews
No comments yet.
Discussed elsewhere
API
curl /api/v1/records/openai-math/Strong-Ulam-Stability-Characterizes-Amenability
curl -X POST /api/v1/records/openai-math/Strong-Ulam-Stability-Characterizes-Amenability/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/openai-math/Strong-Ulam-Stability-Characterizes-Amenability/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'
100+ reactions to 100+ solutionsProofs and PromptsOct 8, 2026
The solution seems to combine Furstenberg’s boundary theory with quantum circuits from quantum computing, how wild is that? In general, it is impressive to have many conjectures settled in the positive this round. Hugo Duminil-Copin I expected that one day we would be surpassed, and that it would…