Noncommutative convexity

Noncommutative convexity

 

Lectures

Mondays and Wednesdays

Time:  14:15-15:45

Classroom: SR 6

This is a half-semester course.

It will run until early December.

It is worth 4.5 ECTS.

 

Contents

Convexity is a simple but powerful concept originating from geometry. A subset of a vector space is convex if it entirely contains the line segment between (i.e., any convex combination of) any two of its points. A well-studied noncommutative notion of convexity is called matrix convexity. A matrix convex set is composed of a sequence of levels, each containing matrices of a specific size. These levels are interconnected by two fundamental operations: direct sums, which move to higher levels (larger matrices), and compressions, which move to lower ones.

Parallel to the classical theory of convexity, the theory of matrix convexity is also heavily supported by the study of extreme points. An extreme point of a convex set K (in the classical sense) is a point that cannot be written as a nontrivial convex combination of the other points of K. A cornerstone of functional analysis is the Krein-Milman theorem, stating that the extreme points generate a compact convex set via (limits of) convex combinations. This property of extreme points motivated the development of their analogs in the noncommutative setting. In fact, there are two concurring notions: matrix extreme points and free extreme points. 

This course provides an introduction to the theory of noncommutative (matrix) convexity. In particular, the following topics will be covered:

• the noncommutative Hahn-Banach separation theorem,
• categorical duality between matrix convex sets and operator systems,
• various notions of extreme points for matrix convex sets,
• the noncommutative Krein-Milman theorem.

 

Language: The course will be taught in English.

Prerequisites: A solid foundation in functional analysis (e.g., knowledge of the Hahn-Banach theorem is assumed). 

 

Exam

Oral exams will be held in December.

In order to be eligible for the oral exam, you have to achieve at least 50% of the available points on the assignments.

 

Literature

Will be added throughout the course.