SOME EXTREMELY AMENABLE GROUPS RELATED TO OPERATOR ALGEBRAS AND ERGODIC THEORY
SOME EXTREMELY AMENABLE GROUPS RELATED TO OPERATOR ALGEBRAS AND ERGODIC THEORY
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一些与算子代数和遍历理论相关的极其适用的群
DOI：
10.1017/s1474748006000090
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发表时间：
2004
影响因子：
0.9
通讯作者：
V. Pestov
中科院分区：
文献类型：
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作者：
T. Giordano;V. Pestov
一个拓扑群G称为极顺从群，如果G在紧空间上的每个连续作用都有不动点。这个概念与高维几何（测量集中）有关。我们证明了von Neumann代数是近似有限维的当且仅当它的具有强拓扑的酉群是一个极顺从群与一个紧群的乘积，这加强了de拉哈尔佩pe的一个结果.因此，一个C^\ast$-代数A$是核的当且仅当具有相对弱拓扑的酉群U（A）$在弱拓扑意义下是强顺从的。我们证明了具有非原子测度的勒贝格空间的自同构群是非常服从弱拓扑的，并为非奇异变换群建立了类似的结果。作为结果，我们证明了$L^p（0，1）$，$1\leq p<\infty$的等距群的极端顺从性，推广了Gromov和Milman（$p=2$）的一个经典结果.我们证明了标准Borel空间上的测度类保持等价关系$\mathcal{R}$是顺从的当且仅当配备有一致拓扑的全群$[\mathcal{R}]$是极其顺从的。最后，我们给出了自然的例子，浓度的非平凡空间的意义上格罗莫夫发生在自同构群的内射因子的III型。
A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal{R}$ on a standard Borel space is amenable if and only if the full group $[\mathcal{R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.
DOI：
10.1090/surv/089
发表时间：
2001
期刊：
--
影响因子：
--
作者：
M. Ledoux
通讯作者：
M. Ledoux
DOI：
--
发表时间：
1984
期刊：
--
影响因子：
--
作者：
M. Wodzicki
通讯作者：
M. Wodzicki