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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]

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]

Let A be an F -algebra and S (A *) be the metric semi-group of normal states of the W * -algebra A * ⁠. We show that S (A *) determines A , when A * is of type I. In particular, when A is either the Fourier algebra A (G) or the Fourier–Stieltjes algebra B (G) of a type I locally compact group G , the metric semi-group S (A *) determines G up to opposition. A remark for the general case will also be given. [ABSTRACT FROM AUTHOR]

We obtained a "decomposition scheme" of C *-algebras. We show that the classes of discrete C *-algebras (as defined by Peligard and Zsidó), type II C *-algebras and type III C *-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C *-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C *-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal A d,1, the largest discrete essentially infinite ideal A d,∞, the largest type II finite ideal A II,1, the largest type II essentially infinite ideal A II,∞, and the largest type III ideal A III of any C *-algebra A such that A d,1 + A d,∞ + A II,1 + A II,∞ + A III is an essential ideal of A. This "decomposition" extends the corresponding one for W *-algebras. We also give a closer look at C *-algebras with Hausdorff primitive ideal spaces, AW *-algebras as well as local multiplier algebras of C *-algebras. We find that these algebras can be decomposed into continuous fields of prime C *-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group. Let A be any one of the (complex) Banach algebras: L1(G), M(G), WAP(G) and LUC(G), consisting of integrable functions, regular Borel complex measures, weakly almost periodic functions and bounded left uniformly continuous functions, respectively, on G. We show that the metric semigroup A1+: = {f ∈ A: f ≥ 0 and ∥f∥ = 1} (the convex structure is not considered) is a complete invariant for G. [ABSTRACT FROM AUTHOR]

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

In this short paper, we obtained some equivalent formulations of property T for a general locally compact quantum group 픾, in terms of the full quantum group C*-algebras and the *-representation of associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if 픾 is of Kac type, we show that 픾 has property T if and only if every finite-dimensional irreducible *-representation of is an isolated point in the spectrum of (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property T discrete quantum groups using bicrossed products. [ABSTRACT FROM AUTHOR]

We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group Γ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra L(Γ). Moreover, a countable discrete group Γ has property (T) if and only if for any action α of Γ on a von Neumann algebra N, every α-invariant state on N is a weak-*-limit of a net of normal α-invariant states. [ABSTRACT FROM PUBLISHER]

We define a projective tensor product of a family of unital normed *-algebras. We show that ifis a family of groups, then the ‘reducing of scalar’ of the projective tensor product ofis isometrically *-isomorphic to. [ABSTRACT FROM PUBLISHER]

In this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case. [ABSTRACT FROM PUBLISHER]

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