Title: Almost unimodular groups

Abstract: Just like discrete groups, locally compact groups provide a rich source of von Neumann algebras through their left regular representations. However, by moving beyond the discrete setting one loses the canonical tracial state on these algebras and must instead work with an infinite weight called the Plancherel weight. This weight can even have non-trivial modular theory (i.e. be non-tracial), but this is fortunately controlled by the so-called modular function of the group: a continuous homomorphism of the group into the positive reals that serves as the Radon–Nikodym derivative between left and right Haar measures. One example of this control is that the Plancherel weight is tracial if and only if the group is unimodular in the sense that the modular function is identically one. In this talk, I will discuss the class of locally compact groups one obtains when traciality of the Plancherel weight is loosened to almost periodicity. The resulting class of course contains all unimodular groups (e.g. all discrete or compact groups), but also all totally disconnected groups. Moreover, the von Neumann algebras associated to the groups in this class are easier to study because the almost periodicity of the Plancherel weight ensures there is a large tracial subalgebra that determines much of the structure. This talk is based on joint work with Aldo Garcia Guinto and will not assume any prior knowledge of locally compact groups beyond the existence of a left Haar measure.

Professor Nelson is giving a colloquium talk, Uniqueness of almost periodic states on hyperfinite factors, at 3:30pm-4:20pm the same of this talk and you are welcome to attend.