### 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 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⋆ 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 language | English |
---|---|

Pages (from-to) | 519-530 |

Number of pages | 12 |

Journal | Positivity |

Volume | 18 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2014 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Theoretical Computer Science
- Mathematics(all)

### Cite this

*Positivity*,

*18*(3), 519-530. https://doi.org/10.1007/s11117-013-0259-z

**On the geometry of von Neumann algebra preduals.** / Martín, Miguel; Ueda, Yoshimichi.

Research output: Contribution to journal › Article

*Positivity*, vol. 18, no. 3, pp. 519-530. https://doi.org/10.1007/s11117-013-0259-z

}

TY - JOUR

T1 - On the geometry of von Neumann algebra preduals

AU - Martín, Miguel

AU - Ueda, Yoshimichi

PY - 2014/9/1

Y1 - 2014/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84957429321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957429321&partnerID=8YFLogxK

U2 - 10.1007/s11117-013-0259-z

DO - 10.1007/s11117-013-0259-z

M3 - Article

VL - 18

SP - 519

EP - 530

JO - Positivity

JF - Positivity

SN - 1385-1292

IS - 3

ER -