Examples of Prime von Neumann Algebras

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Let {(Mi,φi) : i = 1, 2, . . .} be a family of injective von Neumann algebras on separable Hilbert spaces with a faithful normal state φi on each Mi and M be the reduced free product von Neumann algebra of (Mi,φi), i ∈ N. If there is a normal conditional expectation from M onto a non-injective von Neumann subalgebra N of M, then N is prime, i.e., N = N1⊗N2 implies that either N1 or N2 is a von Neumann algebra of type I. This result provides many examples of prime von Neumann algebras. These examples of prime von Neumann algebras include prime factors given by Ge (type II1) and by Shlyakhtenko (Type III). In our proof we combine Ozawa’s new techniques for solid von Neumann algebras with Shlyakhtenko’s “matrix model” techniques for the free ArakiWoods factors.