Short Geodesics of Unitaries in the L 2 Metric

Andruchow, Esteban

Canadian mathematical bulletin, Tome 48 (2005) no. 3, pp. 340-354

Résumé

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 ${{L}^{2}}\left( \mathcal{M},\tau\right)$

Détail

DOI : 10.4153/CMB-2005-032-0

Mots-clés : 46L51, 58B10, 58B25, unitary group, short geodesics, infinite dimensional riemannian manifolds