Geometry of infinite dimensional unitary groups: convexity and fixed points

TL;DR

This paper investigates convexity properties of distance functions in various infinite dimensional Finsler unitary groups, establishing new convexity results and fixed point properties with optimal bounds.

Contribution

It introduces new convexity results for distance functions in infinite dimensional unitary groups, including operator norm and Hilbert-Schmidt cases, with applications to fixed points.

Findings

Convexity of the operator norm metric on the full unitary group.

Strong convexity of squared metrics in Hilbert-Schmidt and von Neumann algebra unitary groups.

Optimal radius bounds for convexity and fixed point results.

Abstract

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.