Almost periodic states and factors of type III1

AbstractWe construct a factor of type III1 which has no almost-periodic state (or weight). We exhibit a factor N of type II∞ and two automorphisms θ1 θ2 of N which are not in the same conjugacy class in Out N = AutNInt N though τθ1 = λτ, τθ2 = λτ with λ ϵ ]0, 1[, τ = Trace on N. We introduce and study two invariants Sd and τ for factors of type III1. We relate the closedness of Int M in Aut M to the absence of central sequences in the von Neumann algebra M