Title: Projective representations of almost unimodular groups

Abstract: Given a locally compact group G with a 2-cocycle ω: G × G → T , Colin Sutherland showed that any left Haar measure uniquely determines a faithful normal semifinite weight on the associated twisted group von Neumann algebra. This weight, which we call the twisted Plancherel weight, is tracial if and only if G is unimodular, and for countable discrete groups it is the usual tracial state. In the setting of non-unimodular groups, the modular automorphism group of the necessarily non-tracial twisted Plancherel weight is explicitly determined by the so-called modular function of G. The twisted group von Neumann algebra is generated by the left regular ω-projective representation of G. In 1958, George Mackey showed that ω-projective representations of G are connected to the representations of the central extension of G by T, when G is second countable. In this talk, we will introduce the class of "almost unimodular groups" for which the twisted Plancherel weight is almost periodic, in the sense of Connes from 1972. We will also give some examples of such groups admitting a 2-cocycle such that the group von Neumann algebras are purely infinite and not factors, but the twisted group von Neumann algebras are semifinite factors.