Central extensions and almost representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910866594594816 |
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| author | Dadarlat, Marius Glebe, Forrest |
| author_facet | Dadarlat, Marius Glebe, Forrest |
| contents | For a sequence of unital tracial $C^*$-algebras $(A_n,τ_n),$ we construct a canonical central extension of the unitary group $U(\ell^\infty (\mathbb{N},A_n)/c_0(\mathbb{N},A_n))$ by $Q(\mathbb{R})=c_0(\mathbb{N},\mathbb{R})/\mathbb{R}^\infty,$ using de la Harpe-Skandalis pre-determinant. For an asymptotic group homomorphism $ρ_n : Γ\to U(A_n),$ the corresponding pullback of the canonical central extension gives a 2-cohomology class in $H^2(Γ,Q(\mathbb{R}))$ which obstructs the perturbation of $(ρ_n)$ to a sequence of true homomorphisms of groups $π_n:Γ\to GL(A_n)$. The pairing of the obstruction class with elements of $H_2(Γ,\mathbb{Z})$ yields numerical invariants in $τ_{n\,*} (K_0(A_n))$ that subsume the winding number invariants of Kazhdan, Exel and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group $Γ$ into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to $H^2(Γ,Q(\mathbb{C})),$ where $Q(\mathbb{C})=c_0(\mathbb{N},\mathbb{C})/\mathbb{C}^\infty.$ As an application, we show that 2-cohomology obstructs various stability properties under weaker assumptions than those found in existing literature. In particular we show that the full group $C^*$-algebra $C^*(Γ)$ of a discrete group $Γ$ is not $C^*$-stable if $H^2(Γ,\mathbb{R})\neq 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04590 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Central extensions and almost representations Dadarlat, Marius Glebe, Forrest Operator Algebras Group Theory For a sequence of unital tracial $C^*$-algebras $(A_n,τ_n),$ we construct a canonical central extension of the unitary group $U(\ell^\infty (\mathbb{N},A_n)/c_0(\mathbb{N},A_n))$ by $Q(\mathbb{R})=c_0(\mathbb{N},\mathbb{R})/\mathbb{R}^\infty,$ using de la Harpe-Skandalis pre-determinant. For an asymptotic group homomorphism $ρ_n : Γ\to U(A_n),$ the corresponding pullback of the canonical central extension gives a 2-cohomology class in $H^2(Γ,Q(\mathbb{R}))$ which obstructs the perturbation of $(ρ_n)$ to a sequence of true homomorphisms of groups $π_n:Γ\to GL(A_n)$. The pairing of the obstruction class with elements of $H_2(Γ,\mathbb{Z})$ yields numerical invariants in $τ_{n\,*} (K_0(A_n))$ that subsume the winding number invariants of Kazhdan, Exel and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group $Γ$ into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to $H^2(Γ,Q(\mathbb{C})),$ where $Q(\mathbb{C})=c_0(\mathbb{N},\mathbb{C})/\mathbb{C}^\infty.$ As an application, we show that 2-cohomology obstructs various stability properties under weaker assumptions than those found in existing literature. In particular we show that the full group $C^*$-algebra $C^*(Γ)$ of a discrete group $Γ$ is not $C^*$-stable if $H^2(Γ,\mathbb{R})\neq 0$. |
| title | Central extensions and almost representations |
| topic | Operator Algebras Group Theory |
| url | https://arxiv.org/abs/2502.04590 |