Actions of rigid groups on UHF-algebras
Actions of rigid groups on UHF-algebras
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刚性群对 UHF 代数的作用
DOI：
10.1016/j.jfa.2017.12.005
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发表时间：
2016
影响因子：
1.7
通讯作者：
M. Lupini
中科院分区：
文献类型：
--
作者：
Eusebio Gardella;M. Lupini
设Λ是可数无限性质(T)群，D是无限型UHF-代数。我们证明了存在两两非(弱)上循环共轭的连续体，即ΛOne的强外作用。证明在于将Λ在D上的一个强外作用赋给任何第二个可数交换群G，它的(弱)余圈共轭类完全记住群G。我们的构造还证明了：Λ的强外作用的共轭、余圈共轭、弱余圈共轭之间的关系是完备的解析集，特别不是Borel。同样的结论在假设Λ只包含一个具有相对性质(T)的无限子群，并且对于至少有一个迹的可分、核、超高频吸收、自吸收C*-代数上的作用(不一定是简单的)时，同样的结论更普遍地成立。最后，我们利用本文的技巧来构造具有指定上同调的R上的外作用。确切地说，对于每个无限性质(T)群Λ和每个可数交换群Γ，我们构造了Λ在R上的一个外作用，它的1-上同调与Γ同构。
Let Λ be a countably infinite property (T) group, and letDbe UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of Λ onD. The proof consists in assigning, to any second countable abelian pro-pgroupG, a strongly outer action of Λ onDwhose (weak) cocycle conjugacy class completely remembers the groupG. The groupGis reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function.Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of Λ onDare complete analytic sets, and in particular not Borel. The same conclusions hold more generally when Λ is only assumed to contain an infinite subgroup with relative property (T), and for actions on (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebras with at least one trace.Finally, we use the techniques of this paper to construct outer actions onRwith prescribed cohomology. Precisely, for every infinite property (T) group Λ, and for every countable abelian group Γ, we construct an outer action of Λ onRwhose 1-cohomology is isomorphic to Γ.