# On the geometry of von Neumann algebra preduals

On the geometry of von Neumann algebra preduals Let $$M$$ M be a von Neumann algebra and let $$M_\star$$ M ⋆ be its (unique) predual. We study when for every $$\varphi \in M_\star$$ φ ∈ M ⋆ there exists $$\psi \in M_\star$$ ψ ∈ M ⋆ solving the equation $$\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert$$ ‖ φ ± ψ ‖ = ‖ φ ‖ = ‖ ψ ‖ . This is the case when $$M$$ M does not contain type I nor type III $$_1$$ 1 factors as direct summands and it is false at least for the unique hyperfinite type III $$_1$$ 1 factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $$M_\star$$ M ⋆ of length $$4$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On the geometry of von Neumann algebra preduals

, Volume 18 (3) – Oct 30, 2013
12 pages

/lp/springer_journal/on-the-geometry-of-von-neumann-algebra-preduals-j08Z5eCnnE
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0259-z
Publisher site
See Article on Publisher Site

### Abstract

Let $$M$$ M be a von Neumann algebra and let $$M_\star$$ M ⋆ be its (unique) predual. We study when for every $$\varphi \in M_\star$$ φ ∈ M ⋆ there exists $$\psi \in M_\star$$ ψ ∈ M ⋆ solving the equation $$\Vert \varphi \pm \psi \Vert =\Vert \varphi \Vert =\Vert \psi \Vert$$ ‖ φ ± ψ ‖ = ‖ φ ‖ = ‖ ψ ‖ . This is the case when $$M$$ M does not contain type I nor type III $$_1$$ 1 factors as direct summands and it is false at least for the unique hyperfinite type III $$_1$$ 1 factor. We also characterize this property in terms of the existence of centrally symmetric curves in the unit sphere of $$M_\star$$ M ⋆ of length $$4$$ 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.

### Journal

PositivitySpringer Journals

Published: Oct 30, 2013

### References

• The geometry of flat Banach spaces
Harrell, RE; Karlovitz, LA

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