Seminar on Free Probability
and Random Matrices

Fall 2018

It is very natural to study separability or PPT property of quantum states in view of quantum information theory and the need to study those properties for so-called Jones-Wenzl projections has emerged in recent years. In this talk, I am going to introduce the notions of separability, PPT property and Jones-Wenzl projections.

We will continue the discussion from last week and show an analytic representation of $A$-valued laws.

In this talk, we will introduce the $A$-valued probability space $(B,E)$ and the way to define the $A$-valued law $\mu_b$ for self-adjoint element $b$ in $B$. Also, we will introduce the generalized law, and establish the relation between $\mu_b$ and the generalized law.

For the last talk, we gave some motivating examples and the definition of right $A$-module. We will begin this talk by showing some properties of  $A$-valued pre inner product, and then we will deduce the process to construct a right Hilbert $A$-module. Next, we will introduce the operators on  right Hilbert $A$-modules, and proving some properties of the corresponding adjoint operators. 

Last week, we gave a sketch of proving the GNS construction. We will start to introduce the matrices over a C*-algebra this week, which is an application of the GNS construction. Then, we will begin the new topic: Right Hilbert $A$-modules. We will deduce the process to construct the inner product on a right Hilbert $A$-module $H$. As long as we have the inner product, we can consider the orthogonality on $H$ and compare the different between the scalar case and $A$-valued case..

In order to study operator-valued probability, we need some basic knowledge of C*-algebras. In this talk, I am trying to review what is a C*-algebra.  and some properties of C*-algebras. The GNS construction will be covered, which is known as the follows: every C*-algebras can be represented as a algebra of bounded linear operators on a Hilbert space. This construction gives us good starting point to study matrices over a C*-algebra. If time permits, I will introduce right Hilbert $A$-modules.

Free Additive Convolution and Subordination
Last week I reviewed the basic facts of scalar free independence. This week I will continue with a new convolution of probability measures called the free additive convolution. We will use a tool from complex analysis called subordination to get our convolution. It will then lead to a discussion of conditional expectation which will be the basis of operator valued freeness.

This fall the seminar will be a learning seminar with lectures by (willing) participants. The theme will be operator valued freeness. This is the non-commutative version of conditional independence, now we have independence over a subalgebra. In many cases the subalgebras are n x n matrices so this is quite a general situation. I will begin by reviewing the basic facts of scalar free independence. The seminar will follow a recent book by D. Kaliuzhnyi-Verbovetskyi and V. Vinnikov and some lecture notes of D. Jekel.