Department of Mathematics

Mathematical Physics and Operator Algebras

  •  Brent Nelson, Michigan State University
  •  Tomita–Takesaki theory for von Neumann algebras
  •  09/28/2021
  •  11:10 AM - 12:00 PM
  •  C304 Wells Hall
  •  Brent Nelson (banelson@msu.edu)

Tomita–Takesaki theory is a powerful (and often necessary) tool for studying von Neumann algebras that lack tracial states. It shows that any state on a von Neumann algebra  $M$ automatically induces an action $\mathbb{R}\curvearrowright M$ (i.e. a non-commutative dynamical system) that then allows one to construct a crossed product von Neumann algebra $M\rtimes \mathbb{R}$. As the notation suggests, this crossed product is an analogue of semidirect products for groups, and it both contains $M$ and encodes the action via unitary elements. It turns out this crossed product always admits a trace (albeit an infinite one) and a dual action $\mathbb{R}\curvearrowright (M\rtimes \mathbb{R})$, and this structure was used by Alain Connes in 1973 to give a classification of the so-called type III von Neumann algebras. In this expository talk, I will provide an introduction to these ideas that does not require any previous experience with von Neumann algebras aside from some functional analysis (e.g. Hilbert spaces, bounded operators, and dual spaces).

 

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Michigan State University
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