Almost Representations

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 K⊂ 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 F⊂ G, and 0<σ≤ 1, there exists δ>0, finite subsets G⊂ G and S⊂ C[G] satisfying the following property: For any map φ: G→ U(B(H)) such that \|φ(fg)-φ(f)φ(g)\|<δ\,\,\,for\,\, all\,\, f,g∈ G\,\,\, and \,\,\, \|π∘ φ(x)\|≥ σ \|x\|\,\,\, for\,\, all\,\, x∈ S, there is a group homomorphism h: G→ U(B(H)) such that \|φ(f)-h(f)\|<ε\,\,\, for\,\,\, all\,\,\, f∈ F, where φ is the linear extension of φ on the group ring C[G] and π: B(H)→ B(H)/ 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.