OAI.RelativeGeneration.main_theorem

The theorem states that, for any complex Hilbert space H (complete, in a fixed universe), whenever small and large are unital star subalgebras of the bounded operators on H with small contained in large, both are type II₁ factors in the concrete sense used here, the relative commutant of small in large is trivial (every element of large commuting with all of small is a scalar multiple of the identity), and large has a separable predual, then there is a trace τ on large with the following properties. Here a type II₁ factor is an algebra in a space where 0 ≠ 1 that is closed in the weak operator topology, has scalar center, is finite (every isometry in it is unitary), and is diffuse (every nonzero projection in it has a nonzero proper subprojection in it). A separable predual means that large is conjugate-linearly isometrically isomorphic to the dual of some separable complex Banach space of the same universe level, with no separability required of H. The trace τ is a continuous linear functional on large that is normalized (τ(1)=1), positive, star-preserving, tracial (τ(xy)=τ(yx)), faithful (τ(xx)=0 forces x=0), and normal, meaning continuous for the ultraweak topology induced on large. The conclusion is that the set of unitaries u of large for which the weakly closed unital star algebra generated by small together with u equals large is both a Gδ set and dense in the unitary group of large, where the topology is the one generated by open balls for the trace two-distance d(u,v)=√τ((u−v)(u−v)), not the operator norm topology.