Document Type
Lecture
Publication Date
12-2-2010
Abstract
Let �� be a semifinite von Neumann algebra with a faithful, normal, semifinite trace ��, and let �� be a symmetric Banach function space on [0, ��(1)). The symmetric space ��(��, ��) of ��-measurable operators consists of all ��-measurable operators �� for which the singular value function ��(��) belongs to �� and is equipped with the norm ‖��‖ₑ(��, ��) = ‖��‖ₑ. Special cases of noncommutative symmetric spaces include Lₚ(��, ��) (noncommutative Lₚ-spaces), unitary matrix spaces ��ₑ, and Schatten spaces ��ₚ, in particular trace-class ��₁ and the class of Hilbert-Schmidt operators ��₂. The noncommutative spaces have been investigated by several mathematicians, including J. Arazy, P. Dodds, B. de Pagter, G. Pisier, Y. Raynaud, F. Sukochev, and Q. Xu. We will discuss some geometric properties of ��(��, ��), such as complex uniform rotundity, smoothness, and Fréchet smoothness. Below are examples of our results. Theorem 1: Let �� be a symmetric space on [0, α), where α = ��(1), and let �� be a semifinite von Neumann algebra with a faithful, normal, σ-finite trace ��. An operator �� is a complex extreme point of ��ₑ(��, ��) if and only if ��(��) is a complex extreme point of ��ₑ and one of the following, not mutually exclusive, conditions holds: (i) ��∞(��) = 0, (ii) ��(��)����(��) = 0 and |��| ≥ ��∞(��)��(��). Theorem 2: Let �� be a semifinite von Neumann algebra with a normal, faithful, semifinite trace ��. The symmetric function space �� on [0, ��(1)) is complex uniformly rotund if and only if ��(��, ��) is complex uniformly rotund.
Relational Format
presentation
Recommended Citation
Czerwinska, M. M., "Noncommutative Symmetric Spaces of Measurable Operators" (2010). Analysis Seminar. 14.
https://egrove.olemiss.edu/math_analysis/14
