A factor M is of type III1 if and only if the action of its unitary group on its state space by inner automorphisms is topologically transitive in the norm topology. © 1978.
Connes, A., & Størmer, E. (1978). Homogeneity of the state space of factors of type III1. Journal of Functional Analysis, 28(2), 187–196. https://doi.org/10.1016/0022-1236(78)90085-X