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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]

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]

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]

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: 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]

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]