Document Type
Lecture
Publication Date
9-7-2011
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 Hilbert-Schmidt operators ��₂. Let (��, ‖⋅‖) be a Banach space, with the unit sphere and unit ball denoted by ��ₓ and ��ₓ, respectively. An element �� ∈ ��ₓ is an exposed point of ��ₓ if there exists a normalized functional �� ∈ �� which supports ��ₓ exactly at ��, meaning ��(��) = 1 and ��(��) ≠ 1 for every �� ∈ ��ₓ \ {��}. If �� is an exposed point of ��ₓ and �� exposes ��ₓ at ��, and if ��(��ₙ) → 1 implies ‖�� − ��ₙ‖ → 0 for all sequences {��ₙ} ⊆ ��ₓ, then �� is a strongly exposed point of ��ₓ, and �� strongly exposes ��ₓ at ��. We discuss the relationships between exposed and strongly exposed points of the unit ball of an order-continuous symmetric function space �� and of the unit ball of the space of ��-measurable operators ��(��, ��).
Relational Format
presentation
Recommended Citation
Czerwinska, M. M., "Exposed and strongly exposed points in symmetric spaces of measurable operators" (2011). Analysis Seminar. 13.
https://egrove.olemiss.edu/math_analysis/13
