Contiguity and Entire Separability of States on von Neumann Algebras

© 2017 Springer Science+Business Media New YorkWe introduce the notions of the contiguity and entirely separability for two sequences of states on von Neumann algebras. The ultraproducts technique allows us to reduce the study of the contiguity to investigation of the equivalence for two states. Here we apply the Ocneanu ultraproduct and the Groh–Raynaud ultraproduct (see Ocneanu (1985), Groh (J. Operator Theory, 11, 2, 395–404 1984), Raynaud (J. Operator Theory, 48, 1, 41–68, 2002), Ando and Haagerup (J. Funct. Anal., 266, 12, 6842–6913, 2014)), as well as the technique developed in Mushtari and Haliullin (Lobachevskii J. Math., 35, 2, 138–146, 2014)