Basile Morando (ENS Lyon)
To any locally compact group G, one can associate a von Neumann algebra L(G), generated by the image of G under its left regular representation. This algebra reflects decomposition properties of the representation: L(G) is a factor — i.e., has trivial center — if and only if the regular representation does not split as a direct sum of two disjoint subrepresentations.
In the discrete case, Murray and von Neumann showed in 1943 that L(G) is a factor if and only if all non-trivial conjugacy classes are infinite. By contrast, for non-discrete groups, determining factoriality becomes more subtle.
In this talk, we present a new sufficient criterion for factoriality of L(G), when G is a totally disconnected group. This criterion, based on a growth condition for the conjugation orbits of cosets, allows us to prove that the von Neumann algebra of the Neretin group is a factor — providing the first known example of a simple, non-discrete group with this property.
If time permits, we will also discuss implications of this criterion for determining the type of L(G), and for understanding factoriality of crossed product associated to G-actions on von Neumann algebras.
