{"paper":{"title":"Asymptotic unitary equivalence in $C^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Huaxin Lin, Zhuang Niu","submitted_at":"2012-06-28T09:50:02Z","abstract_excerpt":"Let $C=C(X)$ be the unital $C^*$-algebra of all continuous functions on a finite CW complex $X$ and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. We show that two unital monomorphisms $\\phi, \\psi: C\\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\\{u_t: t\\in [0,1)\\}\\subset A$ such that\n  $$\n  \\lim_{t\\to 1} u_t^*\\phi(f)u_t=\\psi(f) {\\rm for all} \\in C(X),\n  $$ if and only if \\beq [\\phi]&=&[\\psi] {\\rm in} KK(C, A), \\tau\\circ \\phi&=&\\tau\\circ \\psi {\\rm for all} \\tau\\in T(A), and \\phi^{\\dag}&=&\\psi^{\\dag}, \\eneq where $T(A)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.6610","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}