{"paper":{"title":"Strict comparison holds in the uniform Roe algebra of a discrete amenable group","license":"http://creativecommons.org/licenses/by/4.0/","headline":"If d_τ(a) < d_τ(b) for all traces τ then a is Cuntz subequivalent to b in A ⊗ K where A is the uniform Roe algebra or minimal crossed product of a countable discrete amenable group.","cross_cats":["math.DS"],"primary_cat":"math.OA","authors_text":"Chun Guang Li, George A. Elliott, Jianguo Zhang, Zhuang Niu","submitted_at":"2026-05-01T19:34:15Z","abstract_excerpt":"Let $\\Gamma$ be a countable discrete amenable group, and let $A=l^\\infty(\\Gamma) \\rtimes \\Gamma$. It is shown that if $a, b \\in A \\otimes \\mathcal K$ are positive elements such that $$\\mathrm{d}_\\tau(a) < \\mathrm{d}_\\tau(b),\\quad \\tau \\in \\mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$.\n  Moreover, consider the universal minimal set $(M, \\Gamma)$. The simple C*-algebra $\\mathrm{C}(M)\\rtimes\\Gamma$ is shown to be AH in the strong sense that there is an increasing net of unital sub-C*-algebras $A_\\lambda \\subseteq A$, $\\lambda \\in \\Lambda$, such that each $A_\\lambda$ is a simple (separab"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"It is shown that if a, b ∈ A ⊗ K are positive elements such that d_τ(a) < d_τ(b), τ ∈ T(A), then a is Cuntz subequivalent to b, where A = l^∞(Γ) ⋊ Γ or A = C(M) ⋊ Γ with (M, Γ) the universal minimal set of the countable discrete amenable group Γ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that Γ is amenable is load-bearing, as the proof relies on approximation properties and Følner sequences available only for amenable groups; the result is stated specifically for these crossed products and may fail without amenability or for other choices of A.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For countable discrete amenable groups, strict comparison holds in A ⊗ K where A is l^∞(Γ) ⋊ Γ or C(M) ⋊ Γ with M the universal minimal set: d_τ(a) < d_τ(b) for all traces τ implies a is Cuntz subequivalent to b.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If d_τ(a) < d_τ(b) for all traces τ then a is Cuntz subequivalent to b in A ⊗ K where A is the uniform Roe algebra or minimal crossed product of a countable discrete amenable group.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"017d2fae3a4a3f70a92ba0c2077ae426bc7ec067e3c520039496d83c0ae05350"},"source":{"id":"2605.01053","kind":"arxiv","version":2},"verdict":{"id":"01457def-53e5-4aa2-8181-f69f545e4950","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T14:28:09.106010Z","strongest_claim":"It is shown that if a, b ∈ A ⊗ K are positive elements such that d_τ(a) < d_τ(b), τ ∈ T(A), then a is Cuntz subequivalent to b, where A = l^∞(Γ) ⋊ Γ or A = C(M) ⋊ Γ with (M, Γ) the universal minimal set of the countable discrete amenable group Γ.","one_line_summary":"For countable discrete amenable groups, strict comparison holds in A ⊗ K where A is l^∞(Γ) ⋊ Γ or C(M) ⋊ Γ with M the universal minimal set: d_τ(a) < d_τ(b) for all traces τ implies a is Cuntz subequivalent to b.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that Γ is amenable is load-bearing, as the proof relies on approximation properties and Følner sequences available only for amenable groups; the result is stated specifically for these crossed products and may fail without amenability or for other choices of A.","pith_extraction_headline":"If d_τ(a) < d_τ(b) for all traces τ then a is Cuntz subequivalent to b in A ⊗ K where A is the uniform Roe algebra or minimal crossed product of a countable discrete amenable group."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.01053/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T18:39:21.919513Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T17:39:38.682256Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"b993494290c9596f20bc8d7fbf139cb201a15a9e1f20e07f35e4913e3d39edd8"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}