Normal states of type III factors

Abstract

Let M be a factor of type III with separable predual and with normal states φ1,...,φk ω with ω faithful. Let A be a finite-dimensional C*-subalgebra of M. Then it is shown that there is a unitary operator u ε M such that φi o Ad u=ω on A for i = 1,..., k. This follows from an embedding result of a finite-dimensional C*-algebra with a faithful state into M with finitely many given states. We also give similar embedding results of C*-algebras and von Neumann algebras with faithful states into M. Another similar result for a factor of type II1 instead of type III holds.