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Short Geodesics of Unitaries in the L2 Metric

Published online by Cambridge University Press:  20 November 2018

Esteban Andruchow
Esteban Andruchow*
Affiliation:
Instituto de Ciencias Universidad Nacional de Gral. Sarmiento, J. M. Gutierrez entre J.L. Suarez y Verdi, (1613) Los Polvorines, Argentina e-mail: eandruch@ungs.edu.ar
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Abstract

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Let $\mathcal{M}$ be a type $\text{I}{{\text{I}}_{1}}$ von Neumann algebra, $\tau$ a trace in $\mathcal{M}$, and ${{L}^{2}}\left( \mathcal{M},\tau \right)$ the GNS Hilbert space of $\tau$. We regard the unitary group ${{U}_{\mathcal{M}}}$ as a subset of ${{L}^{2}}\left( \mathcal{M},\tau \right)$ and characterize the shortest smooth curves joining two fixed unitaries in the ${{L}^{2}}$ metric. As a consequence of this we obtain that ${{U}_{\mathcal{M}}}$, though a complete (metric) topological group, is not an embedded riemannian submanifold of ${{L}^{2}}\left( \mathcal{M},\tau \right)$

Keywords

46L5158B1058B25unitary groupshort geodesicsinfinite dimensional riemannian manifolds
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 3 , 01 September 2005 , pp. 340 - 354
Copyright
Copyright © Canadian Mathematical Society 2005

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