Learning Seminar on the Basics of Brown Measure

Ping Zhong, University of Wyoming

The Brown measure was introduced by L.G. Brown in 1983. It is a generalized notion of spectral measure which applies to non-normal operators living in a suitable non-commutative probability framework -- the framework of a so-called W*-probability space. The purpose of this learning seminar is to provide an accessible entry point to the notion of Brown measure, with an eye towards becoming able to do calculations of Brown measures in examples which come from free probability.

The meetings of the seminar will be on Zoom, typically on Tuesdays 2:30-4 pm; but the first meeting is on Monday, June 21, at 2:30 pm, with Meeting ID 968 4191 7390 and passcode 939234.

The main sources used for the presentations will be the papers [1], [2], and also the Chapter 11 of the monograph [3].

References:

[1] U. Haagerup, F. Larsen. Brown's spectral distribution measure for $R$-diagonal elements in finite von Neumann algebras, Journal of Functional Analysis 176 (2000), 331-367.

[2] U. Haagerup, H. Schultz. Brown measures of unbounded operators affiliated with a von Neumann algebra, Mathematica Scandinavica 100 (2007), 209-263. Also available as arXiv:math/0605251.

[3] J.A. Mingo, R. Speicher. Free probability and random matrices, volume 35 in the series of Fields Institute Monographs, Springer Verlag 2017.