Central Extensions and Almost Representations

Abstract For a sequence of unital tracial C^*-algebras (A₍, ₍), we construct a canonical central extension of the unitary group U (^ (N, A₍) /c₀ (N, A₍) ) by Q (R) =c₀ (N, R) /R^, using de la Harpe–Skandalis pre-determinant. For an asymptotic group homomorphism ₍: U (A₍), the corresponding pullback of the canonical central extension gives a 2-cohomology class in H^2 (, Q (R) ), which obstructs the perturbation of (₍) to a sequence of true homomorphisms of groups ₍: GL (A₍). The pairing of the obstruction class with elements of H₂ (, Z) yields numerical invariants in ₍\, * (K₀ (A₍) ) that subsume the winding number invariants of Kazhdan, Exel, and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to H^2 (, 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^* () of a discrete group is not C^*-stable if H^2 (, R) 0 and in fact, is not stable in operator norm with respect to tracial von Neumann algebras.