On factorial states of operator algebras

Results for the factorial state space of a C¹-algebra A which are analogous to results of 11., 12., 572–612),Tomiyama and Takesaki (Tohôku Math. J. (2) 13 (1961), 498–523) for the pure state space. It is shown that A is prime if and only if the (type I) factorial states are dense in the state space. It follows that every factorial state is a w¹-limit of type I factorial states. The factorial state space of a von Neumann algebra is determined, and it is shown that if A is unital and acts non-degenerately on a Hilbert space then the factorial state space of the generated von Neumann algebra restricts precisely to the factorial state space of A. It is shown that the set of factorial states is w¹-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.