| Résume | To every (countable, discrete) group G, one can construct its group von Neumann algebra L(G), which is a certain completion of the group ring C[G]. It is natural to wonder whether or not there is any connection between elementary equivalence of groups G and H and their group von Neumann algebras L(G) and L(H) (viewed as structures in continuous logic). We begin by showing that there is no implication in general in either direction. We then discuss recent work with Matthew Harrison-Trainor, where we show that back-and-forth equivalence (in the sense of computability theory) between the groups implies back-and-forth equivalence of the group von Neumann algebras. Finally, we comment on some partial results, joint with Jennifer Pi, concerning elementary equivalence for group von Neumann algebras associated to free groups. No prior knowledge of von Neumann algebra theory will be assumed. |
