Abstract
We point out the deep relation between the Poisson geometry and the standard form representation of any von Neumann algebra. This is done via a canonical presymplectic groupoid structure of the representation Hilbert space \(\mathcal {H}\) endowed with a suitable Banach manifold structure \(\widetilde {\mathcal {H}}\) for which the identity mapping is a bijective weak immersion \(\widetilde {\mathcal {H}}\to \mathcal {H}\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Beltiţă, D., Goliński, T., Jakimowicz, G., Pelletier, F.: Banach-Lie groupoids and generalized inversion. J. Funct. Anal. 276(5), 1528–1574 (2019). MR 3912784
Beltiţă, D., Odzijewicz, A.: Poisson geometrical aspects of the Tomita-Takesaki modular theory. Preprint (2019). arXiv:1910.14466
Bourbaki, N.: Variétés différentielles et analytiques. Fascicule de résultats. Actualités scientifiques et industrielles, no. pts. 8–15. Hermann, Paris (1975)
Bursztyn, H., Crainic, M., Weinstein, A., Zhu, C.: Integration of twisted Dirac brackets. Duke Math. J. 123(3), 549–607 (2004). MR 2068969
Haagerup, U.: The standard form of von Neumann algebras. Math. Scand. 37(2), 271–283 (1975). MR 407615
Odzijewicz, A., Jakimowicz, G., Sliżewska, A.: Banach-Lie algebroids associated to the groupoid of partially invertible elements of a W ∗-algebra. J. Geom. Phys. 95, 108–126 (2015). MR 3357825
Odzijewicz, A., Jakimowicz, G., Sliżewska, A.: Fiber-wise linear Poisson structures related to W ∗-algebras. J. Geom. Phys. 123, 385–423 (2018). MR 3724794
Odzijewicz, A., Ratiu, T.S.: Banach Lie-Poisson spaces and reduction. Commun. Math. Phys. 243(1), 1–54 (2003). MR 2020219
Odzijewicz, A., Sliżewska, A.: Banach-Lie groupoids associated to W ∗-algebras. J. Symplectic Geom. 14(3), 687–736 (2016). MR 3548483
Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23(3–4), 379–394 (1997). MR 1484598
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Beltiţă, D., Odzijewicz, A. (2020). Standard Groupoids of von Neumann Algebras. In: Kielanowski, P., Odzijewicz, A., Previato, E. (eds) Geometric Methods in Physics XXXVIII. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-53305-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-53305-2_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-53304-5
Online ISBN: 978-3-030-53305-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)