论文标题
无限尺寸统一组的几何形状：凸和固定点
Geometry of infinite dimensional unitary groups: convexity and fixed points
论文作者
论文摘要
在本文中，我们研究了无限尺寸鳍单一群体中距离函数的凸性特性，例如完整的单一群体，有限的von Neumann代数的身份和单一群体的单一schatten扰动。鳍结构是通过在身份的切线空间上的不同规范的翻译来定义的。我们首先证明了从整个统一组中从操作员规范中得出的度量的凸度结果。我们还证明了Hilbert-Schmidt单一组和有限von Neumann代数的单一群体组和单一组的平方指标的强大凸度结果，在这两种情况下，切线空间都赋予用痕量定义的内部产品。这些结果适用于固定点特性，并在某些刚性问题中的定量度量界限。所有凸度和固定点结果的半径界限显示为最佳。
In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full unitary group, the unitary Schatten perturbations of the identity and unitary groups of finite von Neumann algebras. The Finsler structures are defined by translation of different norms on the tangent space at the identity. We first prove a convexity result for the metric derived from the operator norm on the full unitary group. We also prove strong convexity results for the squared metrics in Hilbert-Schmidt unitary groups and unitary groups of finite von Neumann algebras, in both cases the tangent spaces are endowed with an inner product defined with a trace. These results are applied to fixed point properties and to quantitative metric bounds in certain rigidity problems. Radius bounds for all convexity and fixed point results are shown to be optimal.
