Rotation algebras and Exel trace formula

2013/08/22 by Jiajie Hua, Hua, Jiajie, Huaxin Lin +1
Materials Science · Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Lanthanide and Transition Metal Complexes #Operator Algebras (math.OA) #math.OA

paper · pdf · doi:10.48550/arxiv.1308.4775

openalex publication_date 2013/08/22 · arxiv created 2014/02/04 · arxiv updated 2014/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We found that if u and v are any two unitaries in a unital C^*-algebra with ‖uv-vu‖<2 such that uvu^*v^* commutes with u and v, then the \SCA A_u,v generated by u and v is isomorphic to a quotient of the rotation algebra A_θ provided that A_u,v has a unique tracial state. We also found that the Exel trace formula holds in any unital C^*-algebra. Let θ∈ (-1/2, 1/2) be a rational number. We prove the following: For any \ep>0, there exists \dt>0 satisfying the following: if u and v are two unitary matrices such that ‖uv-e^2πiθvu‖<\dt\andeqn 1\over2πiτ(log(uvu^*v^*))=θ, then there exists a pair of unitary matrices u and v such that uv=e^2πiθ vu, ‖u-u‖<\ep\andeqn ‖v-v‖<\ep. Furthermore, a generalization of this for all real θ is obtained for unitaries in unital infinite dimensional simple C^*-algebras of tracial rank zero.

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