Almost Representations
Huaxin Lin
TL;DR
This work analyzes when almost representations of separable amenable C*-algebras and countable amenable groups can be promoted to genuine representations. It develops a robust framework based on fullness, regularity properties, and absorption techniques, leveraging the Choi–Effros lifting theorem and Voiculescu’s Weyl–von Neumann theory within the quasidiagonal/UCT context. The main result shows that for separable quasidiagonal algebras in the UCT class ${\cal N}$, a contractive positive map that is almost multiplicative on a finite set and nontrivial modulo the compact ideal can be approximated by a faithful, full representation on a finite subset; analogous results hold for amenable groups under a fullness condition, with counterexamples proving the necessity of fullness. The paper also provides a broad array of regularity properties (P1–P3) and absorbs techniques, offering a unified approach to almost representations in both operator-algebraic and group-theoretic settings, with implications for the Elliott program and macroscopic observables.
Abstract
Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a countable discrete amenable group. We prove the following: For any $ε>0,$ any finite subset ${\cal F}\subset G,$ and $0<σ\le 1,$ there exists $δ>0,$ finite subsets ${\cal G}\subset G$ and ${\cal S}\subset {\bf C}[G]$ satisfying the following property: For any map $φ: G\to U(B(H))$ such that $$ \|φ(fg)-φ(f)φ(g)\|<δ\,\,\,for\,\, all\,\, f,g\in {\cal G}\,\,\, and \,\,\, \|π\circ \tilde φ(x)\|\ge σ\|x\|\,\,\, for\,\, all\,\, x\in {\cal S}, $$ there is a group homomorphism $h: G\to U(B(H))$ such that $$ \|φ(f)-h(f)\|<ε\,\,\, for\,\,\, all\,\,\, f\in {\cal F}, $$ where $\tilde φ$ is the linear extension of $φ$ on the group ring ${\bf C}[G]$ and $π: B(H)\to B(H)/{\cal K}$ is the quotient map. A counterexample is given that the fullness condition above cannot be removed. We actually prove a more general result for separable amenable $C^*$-algebras.