Almost unimodular groups

The theory of von Neumann algebras was initiated by Murray and von Neumann in the 1930's and has since become a very active research area with many deep results and applications to other fields of mathematics (and beyond). Groups and their representations have always been a rich source of examples of von Neumann algebras, and in this talk I will focus on examples coming from locally compact groups. The measure theoretic properties of such groups give rise to a canonical linear functional on the von Neumann algebra known as the Plancherel weight, and often the structure of the group is equivalent to some feature of the von Neumann algebra or its Plancherel weight. For example, a group admits a measure that is invariant under both left and right translation (i.e. it is unimodular) if and only if the Plancherel weight is tracial in the sense that it is invariant under unitary conjugation—just like the trace on matrices or bounded operators on a Hilbert space. I will show how a more general property of the Plancherel weight can likewise be characterized in terms of the structure of the group. The property, which is due to Alain Connes, is called almost periodicity, and it allows one to simplify the study of a von Neumann algebra through the use of representations of compact abelian groups. This is based on joint work with Aldo Garcia Guinto.

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