Central extensions and almost representations
Marius Dadarlat, Forrest Glebe
TL;DR
The work constructs a canonical central extension of corona-unitary groups from a sequence of tracial algebras using the de la Harpe–Skandalis determinant, producing a cohomology class [ρ] in H^2(Γ, Q(C)) that obstructs lifting asymptotic representations to true homomorphisms. The Kronecker pairing of [ρ] with H_2(Γ, Z) yields concrete K_0-invariants in τ_{n*}(K_0(A_n)), unifying winding-number obstructions with new index-theoretic data. The core result is that nonzero H^2(Γ, R) implies non-stability of C*(Γ) and related stability notions under various asymptotic regimes, including MF and twisted group C*-algebras; the framework generalizes to bounded asymptotic maps into GL of tracial Banach algebras and to matrix stability analyses. Overall, the paper provides a cohomological obstruction mechanism for almost representations, linking 2-cocycles to K-theory and index theoretic invariants, with broad stability implications in operator algebras.
Abstract
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 $H^2(Γ,\mathbb{R})\neq 0$.