论文标题
超级Urysohn空间及其统一表示的异构体组
Groups of isometries of ultrametric Urysohn spaces and their unitary representations
论文作者
论文摘要
我们认为超级Urysohn Spaces $ \ Mathbb {u} $的异构体的$ \ Mathbb {i} $。这样的空间$ \ mathbb {u} $允许透明的实现作为某些$ r $ -trees的边界和$ \ mathbb {i} $是这些$ r $ -trees的自动形态的组。用$ \ mathbb {i} [x] \ subset \ mathbb {i} $有限子空间的稳定器$ x \ subset \ mathbb {u} $。 double cosets $ \ mathbb {i} [x] \ cdot g \ cdot \ mathbb {i} [y] $，其中$ g \ in \ mathbb {i} $枚举了ustrics in of Space $ x \ cup x \ cup y $。我们在双cost空间上构建自然关联乘法$ \ mathbb {i} [x] \ backslash \ mathbb {i}/\ mathbb {i} [x] [x] $，更一般而言，乘法$ \ mathbb {i} \ Mathbb {i}/\ Mathbb {i} [y] \，\ times \，\ Mathbb {i} [y] \ Mathbb {i}/\ Mathbb {i} [z] $。这些操作是一种超规模空间的规范合并。另一方面，可以根据某些$ r $ -Trees的部分同构（尤其是我们的反向类别）来解释该产品。这使我们能够对组的所有统一表示形式进行分类$ \ mathbb {i} $，并证明$ \ mathbb {i} $具有类型$ i $。我们还描述了$ \ mathbb {i} $的通用半群紧凑型，其在$ \ mathbb {i} $的任何单一表示中的映像是紧凑的。
We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of these $R$-trees. Denote by $\mathbb{I}[X]\subset \mathbb{I}$ stabilizers of finite subspaces $X\subset \mathbb{U}$. Double cosets $\mathbb{I}[X]\cdot g\cdot \mathbb{I}[Y]$, where $g\in \mathbb{I}$, are enumerated by ultrametrics on union of spaces $X\cup Y$. We construct natural associative multiplications on double coset spaces $\mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[X]$ and, more generally, multiplications $\mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[Y]\,\times\, \mathbb{I}[Y]\backslash \mathbb{I}/\mathbb{I}[Z]\to \mathbb{I}[X]\backslash \mathbb{I}/\mathbb{I}[Z]$. These operations are a kind of canonical amalgamations of ultrametric spaces. On the other hand, this product can be interpreted in terms of partial isomorphisms of certain $R$-trees (in particular, we come to an inverse category). This allows us to classify all unitary representations of the groups $\mathbb{I}$ and to prove that groups $\mathbb{I}$ have type $I$. We also describe a universal semigroup compactification of $\mathbb{I}$ whose image in any unitary representation of $\mathbb{I}$ is compact.
