Central Extensions and Almost Representations

Abstract For a sequence of unital tracial $C^{*}$-algebras $(A_{n},\tau _{n}),$ 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 $\rho _{n}: \Gamma \to U(A_{n}),$ the corresponding pullback of the canonical central extension gives a 2-cohomology class in $H^{2}(\Gamma ,Q(\mathbb{R})),$ which obstructs the perturbation of $(\rho _{n})$ to a sequence of true homomorphisms of groups $\pi _{n}:\Gamma \to GL(A_{n})$. The pairing of the obstruction class with elements of $H_{2}(\Gamma ,\mathbb{Z})$ yields numerical invariants in $\tau _{n\,*} (K_{0}(A_{n}))$ that subsume the winding number invariants of Kazhdan, Exel, and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group $\Gamma $ into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to $H^{2}(\Gamma ,Q(\mathbb{C})),$ 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 $C^{*}(\Gamma )$ of a discrete group $\Gamma $ is not $C^{*}$-stable if $H^{2}(\Gamma ,\mathbb{R})\neq 0$ and in fact, $\Gamma $ is not stable in operator norm with respect to tracial von Neumann algebras.