Simultaneous similarity, bounded generation and amenability
Simultaneous similarity, bounded generation and amenability
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同时相似性、有限生成和顺从性
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发表时间：
2005
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通讯作者：
G. Pisier
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作者：
G. Pisier
我们证明了一个离散群$G$是顺从的当且仅当它在以下意义下是强么量化的：G$上的每一个么量化的表示$pi$都可以被在由$pi$的值域生成的von Neumann代数中选择的一个可逆元么量化。类似地，一个C^*$-代数A$是核的当且仅当任意有界同态u：A o B（H）$与$*$-同态强相似，因为在由$u$的值域生成的von Neumann代数中存在可逆算子$xi$，使得$a o xi u（a）xi^{-1}$是$*$-同态。一个类似的特征在派生方面成立。我们应用这一点来回答我们以前的工作中遗留下来的几个问题，即两个有单位元的$C^*$-代数的最大张量积$Aotimes_{max} B$的长度$ell（A，B）$，当我们考虑它由子代数$Aotimes 1$和$1otimes B$生成时。证明了如果$ell（A，B）<infty$，无论是对$B=B（ell_2）$，还是当$B$是一个非阿贝尔自由群的$C^*$-代数（满的或约化的），则$A$必是核的.我们还证明了$ell（A，B）le d当且仅当从有单位元的自由积$Aast B$到$Aotimes_{max} B$的典范商映射在长度为$le d$的词的闭跨度内保持完全商映射.
We prove that a discrete group $G$ is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation $pi$ on $G$ can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of $pi$. Analogously a $C^*$-algebra $A$ is nuclear iff any bounded homomorphism $u: A o B(H)$ is strongly similar to a $*$-homomorphism in the sense that there is an invertible operator $xi$ in the von Neumann algebra generated by the range of $u$ such that $a o xi u(a) xi^{-1}$ is a $*$-homomorphism. An analogous characterization holds in terms of derivations. We apply this to answer several questions left open in our previous work concerning the length $ell(A,B)$ of the maximal tensor product $Aotimes_{max} B$ of two unital $C^*$-algebras, when we consider its generation by the subalgebras $Aotimes 1$ and $1otimes B$. We show that if $ell(A,B)<infty$ either for $B=B(ell_2)$ or when $B$ is the $C^*$-algebra (either full or reduced) of a non Abelian free group, then $A$ must be nuclear. We also show that $ell(A,B)le d$ iff the canonical quotient map from the unital free product $Aast B$ onto $Aotimes_{max} B$ remains a complete quotient map when restricted to the closed span of the words of length $le d$.