Weak Riemannian manifolds from finite index subfactors

TL;DR

This paper explores the geometric structure of a homogeneous space derived from finite index subfactors of II$_1$ factors, establishing it as a weak Riemannian manifold with properties akin to classical Hilbert manifolds.

Contribution

It introduces a weak Riemannian metric on the orbit of the Jones projection, analyzing its geometric properties and partial geodesic existence results in an infinite-dimensional setting.

Findings

Metric completeness of geodesic distance

Uniqueness of geodesics in small neighborhoods

Partial results on minimal geodesics

Abstract

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $$ {\cal O}(p) =\{u p u^*: u\in U_M\} $$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p) $ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\|<r\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\cal O}(p)\subset {\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$.