2025/02/07 by Dadarlat, Marius, Glebe, Forrest · 2 citations
#FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2502.04590
For a sequence of unital tracial C^*-algebras (An,τn), we construct a canonical central extension of the unitary group U(ℓ^∞ (ℕ,An)/c0(ℕ,An)) by Q(ℝ)=c0(ℕ,ℝ)/ℝ^∞, using de la Harpe-Skandalis pre-determinant. For an asymptotic group homomorphism ρn : 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 (ρn) to a sequence of true homomorphisms of groups πn:Γ→ GL(An). The pairing of the obstruction class with elements of H2(Γ,ℤ) yields numerical invariants in τn * (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.