Search

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 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]

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]