von Neumann代数的消失第一上同调和强1有界性

IF 0.7 2区数学 Q2 MATHEMATICS

Journal of Noncommutative Geometry Pub Date : 2023-10-27 DOI:10.4171/jncg/530

Ben Hayes, David Jekel, Srivatsav Kunnawalkam Elayavalli

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引用次数: 1

摘要

我们得到了Shlyakhtenko结果的一个新的证明，即如果$G$是一个有限呈现的群，且第一个$\ well ^2$-Betti数消失，则$L(G)$是强1有界的。我们对这一结果的证明适应并简化了荣格的技术论证，该论证表明在捕捉关系的矩阵的Fuglede-Kadison行列式的一定条件下强1有界性。我们的证明还包括Jung的一个关键思想，它涉及对微状态空间覆盖数的迭代估计。我们还利用Shlyakhtenko和Shalom的工作给出了一个简短的证明，证明了具有性质(T)的群的von Neumann代数是强有界的，这是作者另一个结果的特例。

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Vanishing first cohomology and strong 1-boundedness for von Neumann algebras

We obtain a new proof of Shlyakhtenko's result which states that if $G$ is a sofic, finitely presented group with vanishing first $\ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proof of this result adapts and simplifies Jung's technical arguments which showed strong 1-boundedness under certain conditions on the Fuglede–Kadison determinant of the matrix capturing the relations. Our proof also features a key idea due to Jung which involves an iterative estimate for the covering numbers of microstate spaces. We also use the works of Shlyakhtenko and Shalom to give a short proof that the von Neumann algebras of sofic groups with Property (T) are strongly 1 bounded, which is a special case of another result by the authors.

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来源期刊

Journal of Noncommutative Geometry 数学-数学

CiteScore

1.60

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Noncommutative Geometry covers the noncommutative world in all its aspects. It is devoted to publication of research articles which represent major advances in the area of noncommutative geometry and its applications to other fields of mathematics and theoretical physics. Topics covered include in particular: Hochschild and cyclic cohomology K-theory and index theory Measure theory and topology of noncommutative spaces, operator algebras Spectral geometry of noncommutative spaces Noncommutative algebraic geometry Hopf algebras and quantum groups Foliations, groupoids, stacks, gerbes Deformations and quantization Noncommutative spaces in number theory and arithmetic geometry Noncommutative geometry in physics: QFT, renormalization, gauge theory, string theory, gravity, mirror symmetry, solid state physics, statistical mechanics.