Abstract (Elzinga): In the early nineties, Bozejko and Speicher were the first to construct a family of type II1 factors, called q-Gaussians, which could be seen as q-deformations of the free group factors [1]. This family was well studied in the subsequent years (eg [2]), but the isomorphism-class as a function of q remained undetermined. In 2014, a partial resolution was achieved by Guionnet and Shlyakhtenko [3], through the introduction of ideas from optimal transport to free probability.
In this talk, we will review the construction of the q-Gaussian von Neumann algebras and explain the techniques introduced in [3] and how to apply them, picking up the necessary tools from free probability along the way.
- Bozejko & Speicher, CMP 137, 3, 519-531 (1991)
- Ricard, CMP 257, 3, 659-665 (2005)
- Guionnet & Shlyakhtenko, Inventiones Mathematicae 197, 3, 613-661