Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators
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 T-measurable operators associated to a semifinite von Neumann algebra (M, τ) or of the unit ball in the unitary matrix space CE- 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 T as well as for the unitary matrix space CE- Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow. © Instytut Matematyczny PAN, 2010.
Czerwińska, M., & Kamińska, A. (2010). Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators. Studia Mathematica, 201 (3), 253-285. https://doi.org/10.4064/sm201-3-3