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

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]

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]

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]