论文标题
Banach-Lie群体的大规模几何形状
Large scale geometry of Banach-Lie groups
论文作者
论文摘要
我们启动了Banach-Lie群体的大规模几何研究，尤其是线性Banach-Lie群体。我们表明，最初由Ringrose介绍的指数长度针对$ C^*$ - 代数的统一组定义了任何连接的Banach-Lie组的准等级类型。作为一个说明性的例子，我们考虑了可分离的Abelian Unital $ C^*$ - 具有有限的组件的代数，我们将其分类为拓扑同构和准等级法，以突出差异。然后，主要结果涉及Haagerup属性以及属性（T）和（FH）。我们介绍了具有Haagerup特性的第一个非平凡的非亚洲和非元素紧凑型群体，其中大多数是不可分割的。这些是$ \ MATHCAL {U} _2（M，τ）$的组，其中$ m $是半蛋白的von neumann代数，具有正常的忠实的半决赛trace $τ$。最后，我们调查了$ \ mathrm {e} _n（a）$的组，它们是由$ \ mathrm {gl}的封闭子组（n，a）$由基础矩阵生成的，其中$ a $是一个Unitalital Banach代数。我们表明，对于$ n \ geq 3 $，所有这些组都有属性（t），并且它们是不受限制的，因此它们具有非属性（FH）。另一方面，如果$ a $是无限尺寸的unital $ c^*$ - 代数，则$ \ mathrm {e} _2（a）$没有haagerup属性。如果$ a $是Abelian且可分开的，则$ \ Mathrm {sl}（2，a）$没有Haagerup属性。
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 $\mathcal{U}_2(M,τ)$, where $M$ is a semifinite von Neumann algebra with a normal faithful semifinite trace $τ$. 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.
