We will prove the following: Let
M be a finite von Neumann algebra with center Z and A a von Neumann
subalgebra of Z. Let Ω be the spectrum space of A and identify A with
C(Ω). Let 𝜖 be a σ-weakly continuous linear map of M onto A such that
𝜖(x∗x) = 𝜖(xx∗) ≧ 0 for every x ∈ M,𝜖(ax) = a𝜖(x) for every a ∈ A and
x ∈ M,𝜖(1) = 1 and 𝜖(x∗x)≠0 for every nonzero x ∈ M. For each ω ∈ Ω, let mω
denote the set of all x ∈ M with 𝜖(x∗x)(ω) = 0. Then mω is a closed ideal and the
quotient C∗-algebla M∕mω is a finite von Neumann algebra. Furthermore, if πω
denote the canonical homomorphism of M onto M∕mω, then πω(N) is a von
Neumann subalgebra of M∕mω for every von Neumann subalgebra N containing
A.
