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Factoriality of q-Araki-Woods von Neumann algebras via conjugate variables

q-Araki-Woods von Neumann algebras were introduced in 2003 by Fumio Hiai, who combined two deformations of (the Gaussian picture of) the free group factors, namely q-Gaussian von Neumann algebras of Marek Bożejko, Burkhard Kummerer and Roland Speicher on one hand and free Araki-Woods factors of Dima Shlyakhtenko on the other hand. From the very beginning it was expected that these von Neumann algebras are factors (i.e. have trivial center) as soon as they are noncommutative (so when the `initial dimension' is greater than 1), and over the years this was established in most cases by a combination of efforts of several authors.

In this talk we will outline the history of the problem. We will begin with the notion of factoriality for von Neumann algebras, and introduce the famous free group factor problem. Then we will recall the construction of q-Araki-Woods von Neumann algebras, due to Hiai, combining earlier deformations of free group factors due to Bożejko, Kümmerer and Speicher on one hand, and to Shlyakhtenko on the other, and explain how we can use conjugate variables to establish their factoriality, and thereby solve the factoriality question in full generality.