Summary: Let \(M\) be a von Neumann algebra with separable predual. For a normal semi-finite weight \(\varphi\) on \(M\), denote by \(M_\varphi\) the von Neumann subalgebra generated by \[ \lbrace u\in M:u\text{ is a unitary satisfying } u\varphi u^* = \varphi\rbrace. \] Let \(\mathcal{Z}(M_\varphi)\) be the center of \(M_\varphi\), and \(\mathfrak{W}(M)\) be the set of normal semi-finite weights on \(M\). When \(M\) has no type \(\text{III}_1\) part (but could have a non-trivial type \(\text{III}\) part), for every faithful weights \(\varphi\), \(\psi \in \mathfrak{W}(M)\) with \(\varphi\) being strictly semi-finite, if \(M_\varphi \subseteq M_\psi\), then there is a positive self-adjoint operator \(h\) affiliated with \(\mathcal{Z}(M_\varphi)\) such that \(\psi = \varphi_h\). This does not hold for the hyper-finite type \(\text{III}_1\) factor. When \(M\) has no type \(\text{III}_1\) part, we verify that for strictly semi-finite weights \(\varphi\), \(\psi \in \mathfrak{W}(M)\) with \(M_\varphi \subseteq M_\psi\) and \(\varphi |_{M_\psi^+} = \psi |_{M_\psi^+}\), one has \(\varphi = \psi\). This is not true for the hyper-finite type \(\text{III}_1\) factor. Denote by \(\mathfrak{W}_\mathbf{z}(M)\) the subset of \(\mathfrak{W}(M)\) consisting of weights \(\varphi\) with \(\varphi |_{\mathcal{Z}(M_\varphi)^+}\) being semi-finite. When \(M\) is the direct sum of a semi-finite algebra and a type \(\text{III}_0\) algebra, we show that for \(\varphi ,\psi \in \mathfrak{W}_\mathbf{z}(M)\), if \(\mathcal{Z}(M_\varphi)\subseteq \mathcal{Z}(M_\psi)\) and \(\varphi |_{\mathcal{Z}(M_\psi)^+} = \psi |_{\mathcal{Z}(M_\psi)^+}\), then \(\varphi = \psi\). This fails for any type \(\text{III}_\lambda\) factor when \(\lambda \in (0,1)\). Using the above, we establish that when \(M\) has no type \(\text{III}_1\) part, the distances of a normal state on \(M\) to closed faces of the normal state space of \(M\) uniquely determine this state.
For Part I, see [A. T. M. Lau et al., ibid. 52, No. 3, 505–514 (2020; Zbl 1480.46075)].
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