An asymptotic homotopy lifting property
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910383635169280 |
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| author | Carrión, José R. Schafhauser, Christopher |
| author_facet | Carrión, José R. Schafhauser, Christopher |
| contents | A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $π\colon E \rightarrow B$ and for every $^*$-homomorphism $ϕ\colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $πϕ$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $ϕ$. Blackadar has shown that this property holds for all semiprojective $C^*$-algebras.
We show that a version of the homotopy lifting property for asymptotic morphisms holds for separable $C^*$-algebras that are sequential inductive limits of semiprojective $C^*$-algebras. It also holds for any separable $C^*$-algebra if the quotient map $π$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_06677 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An asymptotic homotopy lifting property Carrión, José R. Schafhauser, Christopher Operator Algebras 46L05, 46L85 A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $π\colon E \rightarrow B$ and for every $^*$-homomorphism $ϕ\colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $πϕ$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $ϕ$. Blackadar has shown that this property holds for all semiprojective $C^*$-algebras. We show that a version of the homotopy lifting property for asymptotic morphisms holds for separable $C^*$-algebras that are sequential inductive limits of semiprojective $C^*$-algebras. It also holds for any separable $C^*$-algebra if the quotient map $π$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions. |
| title | An asymptotic homotopy lifting property |
| topic | Operator Algebras 46L05, 46L85 |
| url | https://arxiv.org/abs/2311.06677 |