|
Date Time |
Location | Speaker |
Title – click for abstract |
|
08/30 4:00pm |
BLOC 306 |
Daniel Perales Texas A&M University |
S-transform in finite free probability
We show a simple way to obtain the limiting spectral distribution of a sequence of polynomials (with increasing degree) directly using their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients to Voiculescu's S-transform of the limiting measure. In the framework of finite free probability, this ratios of coefficients can be understood as a new notion of finite S-transform, which satisfies several analogous properties to those of the S-transform in free probability, including multiplicativity and monotonicity. We will mention some of the main ingredients of the proof that include topics of independent interest such as a partial order in the set of polynomials, and a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials. Then we will go over some applications. Joint work with Octavio Arizmendi, Katsunori Fujie and Yuki Ueda (arXiv:2408.09337). |
|
09/06 4:00pm |
BLOC 306 |
Carl Pearcy Texas A&M University |
On restrictions of operators on Hilbert space to a half space |
|
09/20 4:00pm |
BLOC 306 |
Ken Dykema Texas A&M University |
On operator-valued R-diagonal and Haar unitary elements (Joint work with John Griffin)
R-diagonal elements are naturally defined by conditions on the free cumulants of the pair consisting of the element and its adjoint. In the tracial, scalar-valued context, it is known (due to pioneering work of Nica and Speicher) that being R-diagonal is equivalent to having the same *-distribution as an element with a polar decomposition z=u|z|, where u and |z| are *-free and where u is a Haar unitary. In the operator-valued context (namely, B-valued where B is an operator algebra), this is no longer the case. Freeness need not occur, and even notions of Haar unitary are more complicated in the operator-valued setting. We will (1) examine different notions of operator-valued Haar unitary (2) introduce the notion of a free bipolar decomposition and (3) discuss a specific result about free bipolar decompositions of B-valued circular elements (which are a very special case of B-valued R-diagonal elements) when B is two-dimensional. |
|
10/11 4:00pm |
BLOC 306 |
Carl Pearcy Texas A&M University |
A structure theorem for essentially quasinilpotent operators
In this talk it will be shown that every operator in a large class of essentially quasinilpotent operators, up to similarity, has a 3 x 3 operator matrix with particularly nice properties. |
|
10/18 4:00pm |
BLOC 306 |
Tao Mei Baylor University |
Hilbert transform, Cotlar’s identity, and Hyperbolic groups
The classical Hilbert transform is a cornerstone of analysis, known for its fundamental role in both analysis and probability. A key approach to establishing its Lp-boundedness is through Cotlar's identity, a powerful tool that not only yields optimal constants for the Lp bounds of the Hilbert transform but also generalizes to broader settings where the notion of "analytic functions" is meaningful. In this talk, I will revisit Cotlar’s identity and explore how modified versions extend to free groups and hyperbolic groups |
|
10/25 4:00pm |
BLOC 306 |
Zhiyuan Yang Texas A&M University |
A dual of positive maps between von Neumann algebras with weights
We discuss a basic duality of positive maps between von Neumann algebras with faithful normal states introduced by L. Accardi and C. Cecchini in 1982. This duality was later generalized by Dénes Petz in 1984 to the cases of weights. We will prove this duality following Petz's argument. And as a direct application, we show that for any weight decreasing positive map (between von Neumann algebras with n.s.f. weight), there is a normal weight decreasing positive map such that these two positive maps coincide on the domain of the weight. In particular, this covers the well-known fact that any state decreasing map is automatically normal. |
|
11/01 4:00pm |
BLOC 306 |
Carl Pearcy Texas A&M University |
A structure theorem for a class of essentially quasinilpotent operators on Hilbert space
It is widely thought that quasinilpotent operators are the most difficult to understand. In this talk I will obtain a structure theorem for a class of such operators that may allow some progress in their understanding. |
|
11/08 4:00pm |
BLOC 306 |
Carl Pearcy Texas A&M University |
On transitive subspaces of operators on finite dimensional Hilbert spaces
This topic has been around for awhile, but there are only a few papers that address it. I will discuss what I have learned about it and a few results that I have obtained. |
|
11/22 4:00pm |
BLOC 306 |
Junchen Zhao Texas A&M University |
Free products and rescalings involving non-separable von Neumann algebras
For a non-separable self-symmetric abelian von Neumann algebra A, we study rescalings of the free product of n copies of A with LF_r to define a new mutually non-isomorphic continuous family of non-separable interpolated free products that has a rescaling formula and a free product addition formula. Explicit computations will be given to demonstrate well-definedness of this family, their free product, and free products with finite-dimensional or hyperfinite von Neumann algebras. Using this family, we can also show the Murray-von Neumann fundamental group of the infinite free product of A is all of (0, \infty). This is joint work with Ken Dykema. |