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Positivity

, Volume 18, Issue 3, pp 519–530 | Cite as

On the geometry of von Neumann algebra preduals

  • Miguel Martín
  • Yoshimichi UedaEmail author
Article

Abstract

Let \(M\) be a von Neumann algebra and let \(M_\star \) be its (unique) predual. We study when for every \(\varphi \in M_\star \) there exists \(\psi \in M_\star \) solving the equation \(\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert \). This is the case when \(M\) does not contain type I nor type III\(_1\) factors as direct summands and it is false at least for the unique hyperfinite type III\(_1\) factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of \(M_\star \) of length \(4\). An approximate result valid for all diffuse von Neumann algebras allows to show that the equation has solution for every element in the ultraproduct of preduals of diffuse von Neumann algebras and, in particular, the dual von Neumann algebra of such ultraproduct is diffuse. This shows that the Daugavet property and the uniform Daugavet property are equivalent for preduals of von Neumann algebras.

Keywords

von Neumann algebra Predual Factor Diffuseness Daugavet property Extreme point Girth curve Flat space Ultraproduct Ultrapower 

Mathematics Subject Classification (2000)

Primary 46L10 Secondary 46B04 46B20 46L30 

Notes

Acknowledgments

We thank Professor Gilles Godefroy for his comments to the first version of this paper, which gave us a motivation to provide Proposition 2.10 and Corollary 2.11. We also thank the referee for his or her careful reading of this paper and for suggesting us to emphasize the equivalence (1) \(\Leftrightarrow \) (2) in Corollary 3.2.

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoFacultad de Ciencias, Universidad de GranadaGranadaSpain
  2. 2.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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