Geodesics of projections in von Neumann algebras

Abstract:

Let ${\mathcal {A}}$ be a von Neumann algebra and ${\mathcal {P}}_{\mathcal {A}}$ the manifold of projections in ${\mathcal {A}}$. There is a natural linear connection in ${\mathcal {P}}_{\mathcal {A}}$, 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 ${\mathcal {A}}$), if and only if \begin{equation*} p\wedge q^\perp \sim p^\perp \wedge q, \end{equation*} 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 ${\mathcal {A}}$ is a finite factor, any pair of projections in the same connected component of ${\mathcal {P}}_{\mathcal {A}}$ (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 ${\mathcal {N}}\subset {\mathcal {M}}$ are II$_1$ factors with finite index $[{\mathcal {M}}:{\mathcal {N}}]={\mathbf {t}}^{-1}$, then the geodesic distance $d(e_{\mathcal {N}},e_{\mathcal {M}})$ between the induced projections $e_{\mathcal {N}}$ and $e_{\mathcal {M}}$ is $d(e_{\mathcal {N}},e_{\mathcal {M}})=\arccos ({\mathbf {t}}^{1/2})$.