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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]

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 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]

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]

In this paper, we will prove that if A is a C*-algebra with an effective coaction by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras. [ABSTRACT FROM AUTHOR]