Related papers: A property (T) for C*-algebras

Let $1 \in A \subset B$ be an inclusion of unital C*-algebras of index-finite type and depth 2. Suppose that $A$ is infinite dimensional simple with $tsr(A) = 1$ and SP-property. Then $tsr(B) \leq 2$. As a corollary when $A$ is a simple…

Let $G$ be a subgroup of a discrete (countable) group $\Gamma$. We introduce a notion of relative inner amenability of $G$ in $\Gamma$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss…

We give two characterizations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for…

For a number of properties of C*-algebras, including real rank zero, stable rank one, pure infiniteness, residual hereditary infiniteness, the combination of pure infiniteness and the ideal property, the property of being an AT algebra with…

Let $n$ be a positive integer. We introduce a concept, which we call the $n$-filling property, for an action of a group on a separable unital $C^*$-algebra $A$. If $A=C(\Omega)$ is a commutative unital $C^*$-algebra and the action is…

We characterise Geometric Property (T) by the existence of a certain projection in the maximal uniform Roe algebra $C_{u,\max}^*(X)$, extending the notion of Kazhdan projection for groups to the realm of metric spaces. We also describe this…

We study the problem of determining when the reduced twisted group C*-algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a…

We prove that the crossed product $C^*$-algebra $C^*_r(\Gamma,\partial\Gamma)$ of a free group $\Gamma$ with its boundary $\partial\Gamma$ naturally sits between the reduced group $C^*$-algebra $C^*_r\Gamma$ and its injective envelope…

We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$ to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a manner reminiscent of several tracial approximation properties. We are particularly…

A discrete group $\Gamma$ is C*-simple if the C*-algebra $C_\lambda^*(\Gamma)$ generated by the range of the left regular representation $\lambda$ on $\ell^2(\Gamma)$ is simple. In this case, $\Gamma$ acts faithfully on the Furstenberg…

We define a basis property that an inclusion of C*-algebras $\mathcal O_\infty\subset A$ may have, and give various conditions for the property to hold. Some applications are considered. We also give a characterization of open projections…

Let $\Omega$ be a class of unital $\rm C^{*}$-algebras. The class of ${\rm C^*}$-algebras which are asymptotical tracially in $\Omega$, denoted by ${\rm AT}\Omega$. In this paper, we will show that the following class of ${\rm…

We consider three notions of divisibility in the Cuntz semigroup of a C*-algebra, and show how they reflect properties of the C*-algebra. We develop methods to construct (simple and non-simple) C*-algebras with specific divisibility…

In this paper, let $A$ be a unital separable simple infinite dimensional C*-algebra which has uniform property $\Gamma$. Let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a finite group which has the weak tracial Rokhlin property.…

We give a simple definition of property T for discrete quantum groups. We prove the basic expected properties: discrete quantum groups with property T are finitely generated and unimodular. Moreover we show that, for "I.C.C." discrete…

The reduced C*-algebra of a countable linear group G is shown to be simple if and only if G has no nontrivial normal amenable subgroups. Moreover, these conditions are shown to be equivalent to the uniqueness of tracial state on the…

We introduce the Property (C) for a unital commutative sub-C*-algebra $D$ of a unital C*-algebra $A$, a version of the relative comparison property using almost normalizers. Under the assumption of this property, the $\mathcal Z$-absorption…

D. A. Kahzdan first put forth property (T) in relation to the study of discrete subgroups of Lie groups of finite co-volume. Through a combinatorial approach, we define an analogue of property (T) for regular graphs. We then prove the basic…

In this paper, we discuss some equivalent definitions of Property $\Gamma$ for a type II$_1$ von Neumann algebra. Using these equivalent definitions, we prove that the Pisier's similarity degree of a type II$_1$ von Neumann algebra with…

We show that for a large class of actions $\Gamma \curvearrowright \mathcal{A}$ of $C^*$-simple groups $\Gamma$ on unital $C^*$-algebras $\mathcal{A}$, including any non-faithful action of a hyperbolic group with trivial amenable radical,…