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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 G be a locally compact group that acts on a C ∗ -algebra A through an action β. Generalizing an existing result for discrete group actions, we show that if there exist a tracial state ϕ on the full crossed product A ⋊ β G and a positive element x 0 in the center of A ⋊ β G with ϕ (x 0) ≠ 0 , then A ⋊ β G has strong property T if and only if G has property T and (A ⋊ β G , A) has strong property T. Furthermore, generalizing another existing result concerning property T for reduced group C ∗ -algebras, we show that if there exist a β-invariant tracial state τ on A and a continuous function f : G → A with compact support such that ∫ G τ (f 2 (s)) d s ≠ 0 , then the reduced crossed product A ⋊ β , r G has strong property T if and only if G has property T and (A ⋊ β , r G , A) has strong property T. [ABSTRACT FROM AUTHOR]