Large scale geometry of Banach-Lie groups
Large scale geometry of Banach-Lie groups
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DOI：
10.1090/tran/8576
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发表时间：
2020-11
影响因子：
1.3
通讯作者：
Hiroshi Ando;M. Doucha;Yasumichi Matsuzawa
中科院分区：
文献类型：
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作者：
Hiroshi Ando;M. Doucha;Yasumichi Matsuzawa
我们启动了 Banach-Lie 群的大规模几何研究，特别是线性 Banach-Lie 群。我们证明指数长度最初由 Ringrose 为 $C^*$-代数酉群引入，定义了任何连通的 Banach-Lie 群的拟等距类型。作为一个说明性的例子，我们考虑可分离阿贝尔酉$C^*$-代数的酉群，其谱具有有限多个分量，我们将其分类为拓扑同构和准等距，以突出差异。主要结果涉及 Haagerup 性质以及性质 (T) 和 (FH)。我们提出了第一个具有 Haagerup 性质的非平凡非阿贝尔和非局部紧群，其中大多数是不服从的。这些是群 $\mathcal{U}_2(M,\tau)$，其中 $M$ 是半有限冯诺依曼代数，具有正常的忠实半有限迹 $\tau$。最后，我们研究群 $\mathrm{E}_n(A)$，它们是由初等矩阵生成的 $\mathrm{GL}(n,A)$ 的闭子群，其中 $A$ 是单位巴拿赫代数。我们证明，对于 $n\geq 3$，所有这些群都具有性质 (T) 并且它们是无界的，因此它们具有非平凡的性质 (FH)。另一方面，如果 $A$ 是无限维单位 $C^*$-代数，则 $\mathrm{E}_2(A)$ 不具有 Haagerup 性质。如果 $A$ 而且是阿贝尔且可分的，则 $\mathrm{SL}(2,A)$ 不具有 Haagerup 性质。
We initiate the large scale geometric study of Banach-Lie groups, especially of linear Banach-Lie groups. We show that the exponential length, originally introduced by Ringrose for unitary groups of $C^*$-algebras, defines the quasi-isometry type of any connected Banach-Lie group. As an illustrative example, we consider unitary groups of separable abelian unital $C^*$-algebras with spectrum having finitely many components, which we classify up to topological isomorphism and up to quasi-isometry, in order to highlight the difference. The main results then concern the Haagerup property, and Properties (T) and (FH). We present the first non-trivial non-abelian and non-localy compact groups having the Haagerup property, most of them being non-amenable. These are the groups $\mathcal{U}_2(M,\tau)$, where $M$ is a semifinite von Neumann algebra with a normal faithful semifinite trace $\tau$. Finally, we investigate the groups $\mathrm{E}_n(A)$, which are closed subgroups of $\mathrm{GL}(n,A)$ generated by elementary matrices, where $A$ is a unital Banach algebra. We show that for $n\geq 3$, all these groups have Property (T) and they are unbounded, so they have Property (FH) non-trivially. On the other hand, if $A$ is an infinite-dimensional unital $C^*$-algebra, then $\mathrm{E}_2(A)$ does not have the Haagerup property. If $A$ is moreover abelian and separable, then $\mathrm{SL}(2,A)$ does not have the Haagerup property.