Homotopy lifting, asymptotic homomorphisms, and traces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917470155046912 |
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| author | Shulman, Tatiana |
| author_facet | Shulman, Tatiana |
| contents | The following homotopy lifting theorem is proved: Let $ϕ, ψ: B \to D/I$ be homotopic $\ast$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00125 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy lifting, asymptotic homomorphisms, and traces Shulman, Tatiana Operator Algebras Functional Analysis 46L05 The following homotopy lifting theorem is proved: Let $ϕ, ψ: B \to D/I$ be homotopic $\ast$-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. |
| title | Homotopy lifting, asymptotic homomorphisms, and traces |
| topic | Operator Algebras Functional Analysis 46L05 |
| url | https://arxiv.org/abs/2508.00125 |