Almost Representations

Abstract

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^{\ast }$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and $\mathcal{K}\subset B(H)$ the ideal of compact operators. Let $G$ be a countable discrete amenable group. We prove the following: For any $\hat{I}\mu >0,$ any finite subset $\mathcal{F}\subset G,$ and $0<\ddot{I}ƒ\le 1,$ there exists $\hat{I}´>0,$ finite subsets $\mathcal{G}\subset G$ and $\mathcal{S}\subset \mathbf{C}[G]$ satisfying the following property: For any map $\ddot{I}•:G\to U(B(H))$ such that $$\Vert \ddot{I}•(fg)-\ddot{I}•(f)\ddot{I}•(g)\Vert <\hat{I}´forallf,g\in \mathcal{G}and\Vert \ddot{I}€\circ \tilde{\ddot{I}}•(x)\Vert \ge \ddot{I}ƒ\Vert x\Vert forallx\in \mathcal{S},$$ there is a group homomorphism $h:G\to U(B(H))$ such that $$\Vert \ddot{I}•(f)-h(f)\Vert <\hat{I}\mu forallf\in \mathcal{F},$$ where $\tilde{\ddot{I}}•$ is the linear extension of $\ddot{I}•$ on the group ring $\mathbf{C}[G]$ and $\ddot{I}€:B(H)\to B(H)/\mathcal{K}$ 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 $C^{\ast }$-algebras.