Mini Course: Noncommutative Lp-spaces (5 Lectures)

Guixiang Hong ºé¹ğÏé

*(Wuhan University)*

13:00-14:30, Oct 30 - Nov 3, 2017 Science Building A1510

__Abstract:__

1st lecture£º Definition. Modulo some preliminaries on operator algebra, we shall define noncommutative Lp spaces rigorously. We will focus on the proof of Holder and Minkowski inequalities.
2nd lecture£ºDuality. This lecture is devoted to the proof of duality result between noncommutative Lp spaces. The key ingredient in the proof is a noncommutative version of Clarkson inequality, which is in turn based on noncommutative Riesz-Thorin interpolation.
3rd lecture£ºInterpolation. In this lecture, we mainly introduce real interpolation between nc Lp spaces, which is based on a reduction theorem: relate nc Lp spaces with commutative Lp spaces.
4th lecture: Geometric inequalities. Some geometric properties of nc Lp spaces---uniform covexity, uniform smoothness, type and cotype, will be introduced in the present lecture. All these properties are direct consequences of some Clarkson type inequalities.
5th lecture: Vector-valued Lp spaces. In the lecture, modulo some preliminaries on operator spaces, we will introduce the column (or row) and \ell_\infty-valued nc Lp spaces. These vector-valued Lp spaces find many applications to quantum information and nc harmonic analysis.

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