2025/07/31 by Tatiana Shulman, Shulman, Tatiana
Mathematics · #46L05 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L05
paper · pdf · doi:10.48550/arxiv.2508.00125
New results are added. They state that every homotopy symmetric C*-algebra is MF and that the C*-algebra qA is always quasidiagonal. Also some new statements on extension groups are added
arxiv created 2026/07/31 · arxiv updated 2026/08/03
The following homotopy lifting theorem is proved: Let ϕ, ψ: B → D/I be homotopic ∗-homomorphisms and suppose ψ lifts to a (discrete) asymptotic homomorphism. Then ϕ lifts to a (discrete) asymptotic homomorphism. Moreover the whole homotopy lifts. We also prove a cp version of this theorem and a version where ϕ is replaced by an asymptotic homomorphism. We obtain a lifting characterization of several important properties of C*-algebras and use them together with the lifting theorem to get the following applications: 1) MF-property is homotopy invariant; 2) If either A or B is exact, A is homotopy dominated by B and all amenable traces on B are quasidiagonal, then all amenable traces on A are quasidiagonal; 3) If a C*-algebra A is homotopy dominated by a nuclear C*-algebra B and all (hyperlinear) traces on B are MF, then all hyperlinear traces on A are MF. 4) Some of the extension groups introduced by Manuilov and Thomsen coincide. 5) The C*-algebra qA from Cuntz's picture of KK-theory is always quasidiagonal. 6) Every homotopy symmetric C*-algebra is MF.