Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

Małgorzata Marta Czerwińska, and Anna Kamińska. "Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators." Studia Mathematica 201.3 (2010): 253-285. <http://eudml.org/doc/285891>.

@article{MałgorzataMartaCzerwińska2010,
abstract = {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 $C_\{E\}$. 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 $C_\{E\}$. Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.},
author = {Małgorzata Marta Czerwińska, Anna Kamińska},
journal = {Studia Mathematica},
keywords = {symmetric function space; symmetric sequence space; extreme points; midpoint local uniform rotundity; complex rotundity},
language = {eng},
number = {3},
pages = {253-285},
title = {Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators},
url = {http://eudml.org/doc/285891},
volume = {201},
year = {2010},
}

TY - JOUR
AU - Małgorzata Marta Czerwińska
AU - Anna Kamińska
TI - Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators
JO - Studia Mathematica
PY - 2010
VL - 201
IS - 3
SP - 253
EP - 285
AB - 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 $C_{E}$. 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 $C_{E}$. Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.
LA - eng
KW - symmetric function space; symmetric sequence space; extreme points; midpoint local uniform rotundity; complex rotundity
UR - http://eudml.org/doc/285891
ER -