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

Published:2005-09-01
Printed: Sep 2005
• Esteban Andruchow
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## Abstract

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 $\l2$
 Keywords: unitary group, short geodesics, infinite dimensional riemannian manifolds.
 MSC Classifications: 46L51 - Noncommutative measure and integration 58B10 - Differentiability questions 58B25 - Group structures and generalizations on infinite-dimensional manifolds [See also 22E65, 58D05]

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