Short Geodesics of Unitaries in the $L^2$ Metric

http://dx.doi.org/10.4153/CMB-2005-032-0
Canad. Math. Bull. 48(2005), 340-354

  Published:2005-09-01
 Printed: Sep 2005

Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$