Notes on Free Probability Theory

Abstract

Lecture 1. Free Independence and Free Harmonic Analysis. 2 1.1. Probability spaces. 2 1.2. Non-commutative probability spaces. 3 1.3. Classical independence. 4 1.4. Free products of non-commutative probability spaces. 5 1.5. Free Fock space. 6 1.6. Free Central Limit Theorem. 8 1.7. Free Harmonic Analysis. 10 1.8. Further topics. 16 Lecture 2. Random Matrices and Free Probability. 17 2.1. Random matrices. 17 2.2. Asymptotics of random matrices. 18 2.3. An application to random matrix theory. 22 2.4. Applications to von Neumann algebras. 22 Lecture 3. Free Entropy via Microstates. 23 3.1. Definition of free entropy. 23 3.2. Properties of free entropy. 25 3.3. Free entropy dimension. 27 3.4. Properties of free entropy dimension. 28 3.5. Relation with L-Betti numbers. 30 Lecture 4. Non-microstates Approach to Free Entropy. 31

Cite this paper

@inproceedings{Shlyakhtenko2005NotesOF, title={Notes on Free Probability Theory}, author={Dimitri Shlyakhtenko}, year={2005} }