We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(ℳ,τ) of τ-measurable operators associated to a semifinite von Neumann algebra (ℳ,τ) or of the unit ball in the unitary matrix space . We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric space E(ℳ,τ) inherit these properties from their singular value function μ(x) in the unit ball of E with additional necessary requirements on x in the case of complex extreme points. We also obtain the full converse statements for the von Neumann algebra ℳ with a faithful, normal, σ-finite trace τ as well as for the unitary matrix space . Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm201-3-3,
author = {Ma\l gorzata Marta Czerwi\'nska and Anna Kami\'nska},
title = {Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {253-285},
zbl = {1214.46008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm201-3-3}
}
Małgorzata Marta Czerwińska; Anna Kamińska. Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators. Studia Mathematica, Tome 196 (2010) pp. 253-285. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm201-3-3/