论文标题
Von Neumann代数的预测测量学
Geodesics of projections in von Neumann algebras
论文作者
论文摘要
令$ {\ cal a} $为von neumann代数，$ {\ cal p} _ {\ cal a} $ $ {\ cal a} $中的Projections的歧管。在$ {\ cal p} _ {\ cal a} $中有天然线性连接，在有限的维情况下，它与$ \ mathbb {c}^n $的Grassmann歧管的Levi-Civita连接一致。在本文中，我们表明，当$ {\ cal a} $的标准给出的两个预测$ p，q $时，当$ {\ cal a} $给出的指标中，当$ p \ p \ p \ wedge q^\ wedge q^\ perp \ sim perp p^\ sim perp \ perp \ wedge q，$ sim $ \ sim n ne ne n ne ne ne ne ne ne ne ne ne ne ne ne ne ne ne and n ne ne ands n ne ands n ne，结果表明，仅当$ p \ wedge q^\ perp = p^\ perp \ wedge q = 0 $时，最小的大地测量是唯一的。如果$ {\ cal a} $是有限因素，则在$ {\ cal p} _ {\ cal a} $的相同连接组件中的任何一对投影（即具有相同跟踪的痕迹）都可以通过最小的地理位置加入。 我们探索了与琼斯为子因素的索引理论的某些关系。例如，结果表明，如果$ {\ cal n} \ subset {\ cal m} $为{\ bf ii} $ _ 1 $ _ 1 $ rAINTIE索引$ [{\ cal m}：{\ cal n}] = t^{ - 1}诱导的投影$ e _ {\ cal n} $和$ e _ {\ cal m} $是$ d（e _ {\ cal n}，e _ {\ cal m}）= \ arccos（t^{1/2}）$。
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 $d(e_{\cal N},e_{\cal M})=\arccos(t^{1/2})$.
