Fuglede--Kadison determinants and entropy for actions of discrete amenable groups
Fuglede--Kadison determinants and entropy for actions of discrete amenable groups
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DOI：
10.1090/s0894-0347-06-00519-4
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发表时间：
2005-02
影响因子：
3.9
通讯作者：
C. Deninger
中科院分区：
文献类型：
--
作者：
C. Deninger
考虑一个离散群T和一个元/在整群环ZT中。然后T通过群的自同构从左作用于离散加性群ZT/ZTf。对偶，利用群的连续自同构，得到紧态Pontrjagin对偶群Xf = ZT/ZTf上的左t作用。根据定义，Xf是（S1）1*的闭子移。对于可服从群的可测量或拓扑行为，一个好的熵理论是可用的，[j], [M]。因此，从现在开始，设T是一个有限生成的离散可服从群。我们感兴趣的是确定t作用在Xf上的拓扑熵hf。它与具有哈尔概率测度的Xf上t作用的测量理论熵一致。在他们1952年的论文[FK]中，Fuglede和Kadison引入了有限因子von Neumann代数中单位的行列式。最近，L？ck将他们的理论推广到非单位和群von Neumann代数，这些代数不一定是因子[L2]。他需要这个来定义一般覆盖空间的l2 -扭转。冯·诺伊曼代数A？可服从群T的T是Cr的一定完成，因此我们可以考虑Fuglede-Kadison-L？关于对数强f0序列的定义，我们参考第6节。已知有限生成的虚幂零群具有对数强的f0ner序列。我们的主要结果是
Consider a discrete group T and an element / in the integral group ring ZT. Then T acts from the left on the discrete additive group ZT/ZTf by automorphisms of groups. Dualizing, we obtain a left T-action on the compact Pontrjagin dual group Xf = ZT/ZTf by continuous automorphisms of groups. By definition, Xf is a closed subshift of (S1)1*. For measurable or topological actions of amenable groups a good entropy theory is available, [OW], [M]. Thus from now on let T be a finitely generated, discrete amenable group. We are interested in determining the topological entropy hf of the T-action on Xf. It agrees with the measure theoretic entropy of the T-action on Xf equipped with its Haar probability measure. In their paper [FK] from 1952, Fuglede and Kadison introduced a determinant for units in finite factor von Neumann algebras. More recently, L?ck has general ized their theory to nonunits and to group von Neumann algebras which are not necessarily factors [L2]. He needed this to define L2-torsion for general covering spaces. The von Neumann algebra A?T of the amenable group T is a certain completion of Cr and hence we may consider the Fuglede-Kadison-L?ck determinant detjvr / of/ We refer to Section 6 for the definition of a log-strong F0lner sequence. It is known that finitely generated virtually nilpotent groups have log-strong F0lner sequences. Our main result is