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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]

We will extend the main result of [1] to the non-unital case with a totally different proof. More precisely, we give an abstract characterization of an arbitrary self-adjoint weak⁎-closed subspace of L (H) (equipped with the induced matrix norm, the induced matrix cone and the induced weak⁎-topology). In order to do this, we give a matrix analogue of a result of Bonsall for ⁎-operator spaces equipped with closed matrix cones. [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]

For an inductive system of metric spaces that are coarsely embeddable into Hilbert spaces, its directed limit in the category of metric spaces may not admit a coarse embedding into a Hilbert space. In this article, we will show that the corresponding directed limit in the category of coarse spaces (with morphisms being bornologous maps) is always coarsely embeddable into a Hilbert space. [ABSTRACT FROM AUTHOR]

As a non-unital analogue of the main result in [6] , we show in this article that if A is a separable quasi-central C ⁎ -algebra with property T and is nuclear, then there is a sequence of positive integers { n k } k ∈ N such that the c 0 -direct sum of the family { M n k } k ∈ N of matrix algebras is an ideal J A of A and that A / J A has no tracial state. In particular, a separable C ⁎ -algebra A is a c 0 -direct sum of matrix algebras if and only if A is quasi-central, is nuclear, has property T and every unital simple quotient of A has a tracial state. We also show that a separable property T C ⁎ -algebra A is residually finite dimensional if and only if A admits an amenable faithful tracial state and every ideal of A has non-zero center. [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 introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure E on a set X is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set P 0 (X) of non-empty subsets of X and show that it induces the Hausdorff coarse structure on P 0 (X). On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure U on a set X is induced by a map d from X × X to a partially ordered set (with no requirement on d) if and only if U admits a base B such that B ∪ { ⋂ U } is closed under arbitrary intersections. In this case, U is actually defined by a pseudo uniform metric. We also show that a uniform structures U comes from a pseudo uniform metric that takes values in a totally ordered set if and only if U admits a totally ordered base. Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set. [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]

Abstract Extending the notion of the spectrum Σ f of an element f in an ultrametric Banach algebra over a complete valued field (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum σ R , S Ber (a) of an element a in a unital Banach algebra R over a commutative unital Banach ring S (in particular, when R is a Banach ring). This spectrum is a compact subset of the affine analytic line A S 1 over S , and the latter can be identified with the suitable equivalence classes of all elements in all complete valued fields that are Banach S -algebras. If R is generated by a as a unital Banach S -algebra, then σ R , S Ber (a) coincides with the spectrum M (R) of R. If R is a unital complex Banach algebra but we regard it as a Banach ring only, then σ R , Z 1 Ber (a) is the quotient of the ordinary spectrum σ R C (a) by complex conjugation. For a non-Archimedean complete valued field k and an infinite dimensional ultrametric Banach k -space E with an orthogonal basis, if u ∈ L (E) is a completely continuous operator, we show that several different ways to define the spectrum of u give the same compact set σ L (E) , k Ber (u). As an application, we give a relation between σ L (E) , k Ber (u) and zeros of the Fredholm determinant det ⁡ (1 − t u) (as defined by Serre) in complete valued field extensions of k. [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]

Let 1 ≤ p ≤ + ∞ . We show that the positive part of the closed unit ball of a non-commutative L p -space, as a metric space, is a complete Jordan ⁎ -invariant for the underlying von Neumann algebra. [ABSTRACT FROM AUTHOR]

In this article, we define and study property T for continuous homomorphisms between topological groups. If G is a locally compact group, we show that λ G : G → U ( L 2 ( G ) ) has property T if and only if either G is compact or G is non-amenable. Moreover, the abelianization G ab : = G / [ G , G ] ‾ is compact if and only if every continuous homomorphism from G to any abelian topological group has property T . Moreover, we show that G has property ( T , FD ) if and only if any continuous homomorphism from G to any compact group has property T . In the case when G is almost connected, the above is also equivalent to the canonical map from G to its Bohr compactification being a quotient map. We also give some new equivalent forms of the strong property T of a locally compact group. As a consequence, if G is a second countable and has strong property T and H is a closed subgroup of G , there exist at most one G -invariant mean on G / H . [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]

Abstract: Let be a Hausdorff topological vector space with the underlying vector space being a Hilbert space such that is coarser than the norm topology. A densely defined - -continuous operator on is called -maximal if it has no non-trivial - -continuous extension, and it is said to be -adjointable if its adjoint is also - -continuous. We show that if is locally convex, the collection of all densely defined -maximal -adjointable operators is a -algebra under the multiplication given by the -maximal extension of the composition and the involution given by the -maximal extension of the adjoint. Examples include rigged Hilbert spaces and -algebras. In the general (not necessarily locally convex) case, we associate with a -algebra which is a -subalgebra of when is locally convex. If is the measure topology on corresponding to a tracial von Neumann algebra , then the image of the representation of the measurable operator algebra on the completion of with respect to , can be regarded as a -subalgebra of . In the case when is normable, it is shown that is a Banach -algebra. Examples of such Banach -algebras include (under a suitable norm) as well as , where and are the spaces of Hilbert–Schmidt operators and of trace-class operators respectively, on . [Copyright &y& Elsevier]

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]

Abstract: In this short article, we study tensor products of not necessarily unital operator systems (for short, NUOS). We will define canonical functorial NUOS tensor products (in a similar fashion to Kavruk et al. (2011) ) as well as a subclass of them consisting of reduced functorial NUOS tensor products (that are defined through a unitalization process). We show that if a NUOS is -nuclear (in the sense that there is only one NUOS tensor product of with any NUOS ), then is trivial. However, if is a unital operator system, then is -nuclear (in the sense that there is only one reduced NUOS tensor product of with any NUOS ) if and only if is -nuclear in the sense of Han and Paulsen (2011)  (i.e. there is only one unital operator system tensor product of with any unital operator system ). On the other hand, a -algebra is -nuclear if and only if is a nuclear -algebra. [Copyright &y& Elsevier]

In this short note, we give a brief study of the *-algebra R ⊗∞ which is given by certain “inductive limit process” of the algebraic tensor product R ⊗n of the C*-algebra R representing quantum mechanics with one particle. In particular, we will study “feasible pure states” of R ⊗∞ and their GNS constructions. [Copyright &y& Elsevier]

Abstract: In this note, we study geometric unitaries of JB-algebras (in particular, self-adjoint parts of -algebras). By using order theoretical arguments, we show that the geometric unitaries of a JB-algebra are precisely the central algebraic unitaries (or central symmetries). [Copyright &y& Elsevier]

Abstract: In this paper we will give a thorough study of the notion of property (T) for -algebras (as introduced by M.B. Bekka) as well as a slightly stronger version of it, called “strong property (T)” (which is also an analogue of the corresponding concept in the case of discrete groups and type II1-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property (T) and strong property (T). We will also relate them to certain (T)-type properties of the unitary group of the underlying -algebra. [Copyright &y& Elsevier]

Abstract: Let be a -correspondence with a continuous action by an amenable locally compact group G. We will show that if is the induced -correspondence and β is the induced action on , then . As applications, we obtain [B. Abadie, Takai duality for crossed products by Hilbert -modules, preprint arXiv:0709.1122v1, Proposition 4.5] and [A. Kumjian, D. Pask, -algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19 (6) (1999) 1503–1519, Corollary 2.5]. [Copyright &y& Elsevier]

Abstract: Inspired by the recent work of Bekka, we study two reasonable analogues of property for not necessarily unital -algebras. The stronger one of the two is called “property ” and the weaker one is called “property .” It is shown that all non-unital -algebras do not have property (neither do their unitalizations). Moreover, all non-unital σ-unital -algebras do not have property . [Copyright &y& Elsevier]

Abstract: In this article, we define a natural Banach ∗-algebra for a -dynamical system which is slightly bigger than (they are the same if A is finite-dimensional). We will show that this algebra is ∗-regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and ∗-regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press]. [Copyright &y& Elsevier]

In this article, we will give the definition of the Ext-functor for the category ℳS of (completely bounded) counital right S-comodules (where S is a counital Hopf C*-algebra that is either unital or nuclear) using the derived functor approach. Moreover, we will give a brief discussion on the injective objects in ℳS. [ABSTRACT FROM AUTHOR]

We will study some permanence properties of C*-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C*-unique group. Moreover, any amenable discrete group is a retract of a discrete C*-unique group. [Copyright &y& Elsevier]

In this paper, we study the duality theory of Hopf C[sup*]-algebras in a general “Hilbert-space-free” framework. Our particular interests are the “full duality” and the “reduced duality”. In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf C[sup*]-algebra. This study of reduced duality and Fourier algebra is found to be useful in the study of other aspects of Hopf C[sup*]-algebras (see e.g. [12–14]). [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]

Let (Ω , A , μ) and (Γ , B , ν) be two arbitrary measure spaces, and p ∈ [ 1 , ∞ ]. Set S (L p (μ)) + : = { f ∈ L p (μ) : ‖ f ‖ p = 1 ; f ≥ 0 μ -a.e. } , that is, the positive part of the unit sphere of L p (μ). We show that every surjective isometry Φ : S (L p (μ)) + → S (L p (ν)) + can be extended (necessarily uniquely) to an isometric order isomorphism from L p (μ) onto L p (ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ , B , ν) is localizable (in particular, when it is σ -finite). On the other hand, we show that for compact Hausdorff spaces X and Y , if Φ is a surjective isometry from the positive part of the unit sphere of C (X) to that of C (Y) , then there is a homeomorphism τ : Y → X satisfying Φ (f) (y) = f (τ (y)) for f ∈ S (C (X)) + and y ∈ Y. [ABSTRACT FROM AUTHOR]

Let Γ be a discrete group that acts on a unital C ⁎ -algebra A through an action β. We study property T as well as its stronger variant of the full and the reduced crossed products. We show that if A has a β -invariant tracial state, then the reduced crossed product A ⋊ β , r Γ has strong property T if and only if Γ has property T and (A ⋊ β , r Γ , A) has strong property T. If Γ has the Haagerup property, the strong property T of (A ⋊ β , r Γ , A) in the above statement can be replaced by that of A. Furthermore, the strong property T of the full crossed product is shown to coincide with that of the reduced one. In the case of property T , we obtain the corresponding result when the group is either ICC or abelian. As an application, a non-compact locally compact group G is amenable if and only if C b (G) ⋊ lt , r G d does not have strong property T , where G d is the group G equipped with the discrete topology and lt is the left translation action. [ABSTRACT FROM AUTHOR]