标题： 几乎表示 显示英文标题

标题： Almost Representations

摘要： 设$H$为一个无限维可分的希尔伯特空间，$B(H)$为$C^*$-代数，即$H,$ $U(B(H))$ 上所有有界线性算子的代数，单位群为$B(H)$，而${\cal K}\subset B(H)$为紧致算子的理想。 设$G$是一个可数离散的阿贝尔群。 We prove the following: For any $\epsilon>0,$ any finite subset ${\cal F}\subset G,$ and $0<\sigma\le 1,$ there exists $\delta>0,$ finite subsets ${\cal G}\subset G$ and ${\cal S}\subset {\bf C}[G]$ satisfying the following property: For any map $\phi: G\to U(B(H))$ such that $$ \|\phi(fg)-\phi(f)\phi(g)\|<\delta\,\,\,for\,\, all\,\, f,g\in {\cal G}\,\,\, and \,\,\, \|\pi\circ \tilde \phi(x)\|\ge \sigma \|x\|\,\,\, for\,\, all\,\, x\in {\cal S}, $$ there is a group homomorphism $h: G\to U(B(H))$ such that $$ \|\phi(f)-h(f)\|<\epsilon\,\,\, for\,\,\, all\,\,\, f\in {\cal F}, $$ where $\tilde \phi$ is the linear extension of $\phi$ on the group ring ${\bf C}[G]$ and $\pi: B(H)\to B(H)/{\cal K}$ is the quotient map. 给出一个反例说明上述满条件不能被去掉。我们实际上证明了关于可分的阿贝尔$C^*$-代数的一个更一般的结果。 显示英文摘要

摘要： 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 $\epsilon>0,$ any finite subset ${\cal F}\subset G,$ and $0<\sigma\le 1,$ there exists $\delta>0,$ finite subsets ${\cal G}\subset G$ and ${\cal S}\subset {\bf C}[G]$ satisfying the following property: For any map $\phi: G\to U(B(H))$ such that $$ \|\phi(fg)-\phi(f)\phi(g)\|<\delta\,\,\,for\,\, all\,\, f,g\in {\cal G}\,\,\, and \,\,\, \|\pi\circ \tilde \phi(x)\|\ge \sigma \|x\|\,\,\, for\,\, all\,\, x\in {\cal S}, $$ there is a group homomorphism $h: G\to U(B(H))$ such that $$ \|\phi(f)-h(f)\|<\epsilon\,\,\, for\,\,\, all\,\,\, f\in {\cal F}, $$ where $\tilde \phi$ is the linear extension of $\phi$ on the group ring ${\bf C}[G]$ and $\pi: B(H)\to B(H)/{\cal K}$ 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.