Central Extensions and Almost Representations

For a sequence of unital tracial C*-algebras (Aₙ,τ ₙ), we construct a canonical central extension of the unitary group U( ^∞ (N,Aₙ)/c₀(N,Aₙ)) by Q(R)=c₀(N,R)/R^∞ , using de la Harpe–Skandalis pre-determinant. For an asymptotic group homomorphism ρ ₙ: Γ → U(Aₙ), the corresponding pullback of the canonical central extension gives a 2-cohomology class in H²(Γ ,Q(R)), which obstructs the perturbation of (ρ ₙ) to a sequence of true homomorphisms of groups π ₙ:Γ → GL(Aₙ). The pairing of the obstruction class with elements of H₂(Γ ,Z) yields numerical invariants in τ n\,* (K₀(Aₙ)) 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 H²(Γ ,Q(C)), where Q(C)=c₀(N,C)/C^∞ . 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 H²(Γ ,R)≠ 0 and in fact, Γ is not stable in operator norm with respect to tracial von Neumann algebras.