On the geometry of von Neumann algebra preduals

Miguel Martín, Yoshimichi Ueda

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1 Citation (Scopus)

Abstract

Let M be a von Neumann algebra and let M⋆ be its (unique) predual. We study when for every φ∈M⋆ there exists ψ∈M⋆ solving the equation ‖φ±ψ‖=‖φ‖=‖ψ‖. This is the case when M does not contain type I nor type III1 factors as direct summands and it is false at least for the unique hyperfinite type III1 factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of M⋆ 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.

Original languageEnglish
Pages (from-to)519-530
Number of pages12
JournalPositivity
Volume18
Issue number3
DOIs
Publication statusPublished - Sep 1 2014

All Science Journal Classification (ASJC) codes

  • Analysis
  • Theoretical Computer Science
  • Mathematics(all)

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