Brent Nelson, Michigan State University
Uniqueness of almost periodic states on hyperfinite factors
Murray and von Neumann initiated the study of "rings of operators" in the 1930's. These rings, now known as von Neumann algebras, are unital *-algebras of operators acting on a Hilbert space that are closed under the topology of pointwise convergence. Elementary examples include square complex matrices and essentially bounded measurable functions, but the smallest honest examples come from infinite tensor products of matrix algebras. These latter examples are factors—they have trivial center—and are hyperfinite—they contain a dense union of finite dimensional subalgebras. Highly celebrated work of Alain Connes from 1976 and Uffe Haagerup from 1987 showed that these infinite tensor products are in fact the unique hyperfinite factors. Haagerup eventually provided several proofs of this uniqueness, and one from 1989 included as a corollary a uniqueness result for so-called periodic states. This result only holds for some infinite tensor products of matrix algebras and is known to fail for certain other examples, but in recent joint work with Mike Hartglass we show that it can be extended to the remaining examples when periodicity is generalized to almost periodicity. In this talk, I will discuss these results beginning with an introduction to von Neumann algebras that assumes no prior knowledge of the field.
MC 5501