Central extensions and almost representations

TL;DR

This paper constructs a central extension of unitary groups associated with sequences of tracial C*-algebras, using it to identify obstructions to approximating asymptotic group homomorphisms by true homomorphisms, with applications to stability properties.

Contribution

It introduces a canonical central extension framework for asymptotic homomorphisms, linking cohomology obstructions to stability and approximation problems in operator algebras.

Findings

Obstruction classes in cohomology prevent perturbation of asymptotic homomorphisms to genuine homomorphisms.

The pairing of cohomology classes yields numerical invariants related to winding numbers.

The full group C*-algebra is not C*-stable if its second cohomology is non-zero.

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 $H^2(\Gamma,\mathbb{R})\neq 0$.