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Let M$M$ be a von Neumann algebra with separable predual. For a normal semi‐finite weight φ$\varphi$ on M$M$, denote by Mφ$M_\varphi$ the von Neumann subalgebra generated by {u∈M:uis a unitary satisfyinguφu∗=φ}.$$\begin{equation*} \hspace*{20pt}\lbrace u\in M:u\text{ is a unitary satisfying } u\varphi u^* = \varphi\rbrace.\hspace*{-20pt} \end{equation*}$$Let Z(Mφ)$\mathcal {Z}(M_\varphi)$ be the center of Mφ$M_\varphi$, and W(M)$\mathfrak {W}(M)$ be the set of normal semi‐finite weights on M$M$. When M$M$ has no type III1$\mathrm{III}_1$ part (but could have a non‐trivial type III$\mathrm{III}$ part), for every faithful weights φ,ψ∈W(M)$\varphi , \psi \in \mathfrak {W}(M)$ with φ$\varphi$ being strictly semi‐finite, if Mφ⊆Mψ$M_\varphi \subseteq M_\psi$, then there is a positive self‐adjoint operator h$h$ affiliated with Z(Mφ)$\mathcal {Z}(M_\varphi)$ such that ψ=φh$\psi = \varphi _h$. This does not hold for the hyper‐finite type III1$\mathrm{III}_1$ factor. When M$M$ has no type III1$\mathrm{III}_1$ part, we verify that for strictly semi‐finite weights φ,ψ∈W(M)$\varphi ,\psi \in \mathfrak {W}(M)$ with Mφ⊆Mψ$M_\varphi \subseteq M_\psi$ and φ|Mψ+=ψ|Mψ+$\varphi |_{M_\psi ^+} = \psi |_{M_\psi ^+}$, one has φ=ψ$\varphi = \psi$. This is not true for the hyper‐finite type III1$\mathrm{III}_1$ factor. Denote by Wz(M)$\mathfrak {W}_\mathbf {z}(M)$ the subset of W(M)$\mathfrak {W}(M)$ consisting of weights φ$\varphi$ with φ|Z(Mφ)+$\varphi |_{\mathcal {Z}(M_\varphi)^+}$ being semi‐finite. When M$M$ is the direct sum of a semi‐finite algebra and a type III0$\mathrm{III}_0$ algebra, we show that for φ,ψ∈Wz(M)$\varphi ,\psi \in \mathfrak {W}_\mathbf {z}(M)$, if Z(Mφ)⊆Z(Mψ)$\mathcal {Z}(M_\varphi)\subseteq \mathcal {Z}(M_\psi)$ and φ|Z(Mψ)+=ψ|Z(Mψ)+$\varphi |_{\mathcal {Z}(M_\psi)^+} = \psi |_{\mathcal {Z}(M_\psi)^+}$, then φ=ψ$\varphi = \psi$. This fails for any type IIIλ$\mathrm{III}_\lambda$ factor when λ∈(0,1)$\lambda \in (0,1)$. Using the above, we establish that when M$M$ has no type III1$\mathrm{III}_1$ part, the distances of a normal state on M$M$ to closed faces of the normal state space of M$M$ uniquely determine this state. [ABSTRACT FROM AUTHOR]

Let M be a semi‐finite von Neumann algebra with normal state space S(M). For any ϕ∈S(M), let Mϕ:={x∈M:xϕ=ϕx} be the centralizer of ϕ with center Z(Mϕ). We show that for ϕ,ψ∈S(M), the following are equivalent. ϕ=ψ.Z(Mψ)⊆Z(Mϕ) and ϕ|Z(Mϕ)=ψ|Z(Mϕ).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. [ABSTRACT FROM AUTHOR]

We prove that amenability of a unitary co-representation [ABSTRACT FROM AUTHOR]

We define two canonical cohomology theories for Hopf C*-algebras and for Hopf von Neumann algebras (with coefficients in their comodules). We then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups are considered in more detail. E-mail: c.k.ng@qub.ac.uk 2000 Mathematical Subject Classification: primary 46L05, 46L55; secondary 43A07, 22D25. [ABSTRACT FROM AUTHOR]