Mini Course: Noncommutative Lp-spaces (5 Lectures)

Guixiang Hong 洪桂祥  (Wuhan University)

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

Abstract:

1st lecture： 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：Duality. 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：Interpolation. 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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