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 - Sept 1 2014

    All Science Journal Classification (ASJC) codes

    • Analysis
    • Theoretical Computer Science
    • General Mathematics

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