Normal states are determined by their facial distances

Let M be a semi‐finite von Neumann algebra with normal state space S ( M ) . For any ϕ ∈ S ( M ) , letM ϕ : = { x ∈ M : x ϕ = ϕ x }be the centralizer of ϕ with center Z ( M ϕ ) . We show that for ϕ , ψ ∈ S ( M ) , the following are equivalent.ϕ = ψ .Z ( M ψ ) ⊆ Z ( M ϕ )andϕ | Z ( M ϕ )= ψ | Z ( M ϕ ).ϕ , ψ have the same distances to all the closed faces of S ( M ) . As an application, we give an alternative proof of the fact that metric preserving surjections between normal state spaces of semi‐finite von Neumann algebras are induced by Jordan∗ ‐isomorphisms between the underlying algebras. We then use it to verify some facts concerning F ‐algebras and Fourier algebras of locally compact quantum groups.