Geodesics of projections in von Neumann algebras

TL;DR

This paper characterizes when two projections in a von Neumann algebra can be connected by a minimal geodesic, relates this to Murray-von Neumann equivalence, and explores implications for subfactor index theory.

Contribution

It provides a complete characterization of geodesics between projections in von Neumann algebras and links this to subfactor index theory.

Findings

Minimal geodesics exist iff certain projections are Murray-von Neumann equivalent.

Uniqueness of minimal geodesics occurs under specific orthogonality conditions.

Geodesic distance between subfactor projections relates to Jones' index as arccos of the square root of the index.

Abstract

Let ${\cal A}$ be a von Neumann algebra and ${\cal P}_{\cal A}$ the manifold of projections in ${\cal A}$. There is a natural linear connection in ${\cal P}_{\cal A}$, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold of $\mathbb{C}^n$. In this paper we show that two projections $p,q$ can be joined by a geodesic, which has minimal length (with respect to the metric given by the usual norm of ${\cal A}$), if and only if $$ p\wedge q^\perp\sim p^\perp\wedge q, $$ where $\sim$ stands for the Murray-von Neumann equivalence of projections. It is shown that the minimal geodesic is unique if and only if $p\wedge q^\perp= p^\perp\wedge q=0$. If ${\cal A}$ is a finite factor, any pair of projections in the same connected component of ${\cal P}_{\cal A}$ (i.e., with the same trace) can be joined by a minimal geodesic. We explore certain relations with Jones' index theory for subfactors. For instance, it is shown that if ${\cal N}\subset{\cal M}$ are {\bf II}$_1$ factors with finite index $[{\cal M}:{\cal N}]=t^{-1}$, then the geodesic distance $d(e_{\cal N},e_{\cal M})$ between the induced projections $e_{\cal N}$ and $e_{\cal M}$ is $d(e_{\cal N},e_{\cal M})=\arccos(t^{1/2})$.