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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]

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: 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]

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]

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]