The Non-commutative Flow of Weights on a Von Neumann Algebra

The flow of weights of Connes and Takesaki is a canonical functor from the category of separable factors to the category of ergodic flows. The non-commutative flow of weights is another canonical functor from the category of separable factors to the category of covariant systems of semi-finite von Neumann algebras equipped with trace scaling one parameter automorphism groups with conjugations as morphisms. The constructions of these two functors are very similar. The flow of weights functor is obtained by looking at all semi-finite normal weights on a factor with the Murray–von Neumann equivalence relation. The non-commutative flow of weights functor is obtained by relating an arbitrary pair of faithful semi-finite normal weights by the Connes cocycle. Not only does this construction put a period to the search for a canonical construction of the core of a factor of type III, but it also allows us to put the characteristic square of a factor obtained by Katayama, Sutherland, and Takesaki in a new perspective. The power of this new approach is seen in an ultimate solution to a long standing question of extending the extended modular automorphism of a dominant weight to an arbitrary weight, which has been left open ever since the introduction of extended modular automorphisms by Connes and Takesaki over 20 years ago. The construction of the functor ties together the theory of Lp-spaces of Haagerup, Kosaki, Hilsum, Terp, and Izumi to the structure theory of a factor of type III. In fact, the non-commutative flow of weights is obtained by the analytic continuation of Lp-spaces to a pure imaginary value of p.