Jacek Brodzki, Graham A. Niblo, Nick Wright
In this paper we show that if a discrete group $G$ acts properly isometrically on a discrete space $X$ for which the uniform Roe algebra $C_u^*(X)$ is exact then $G$ is an exact group. As a corollary, we note that if the action is cocompact then the following are equivalent: The space $X$ has Yu's property A; $C^*_u(X)$ is exact; $C_u^*(X)$ is nuclear.
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It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Cech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups.
We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.
$\mathrm{C}^*$-exactness and property A for group actions
本文证明了离散群作用下的Schreier图具有性质A当且仅当其置换表示生成精确的$\mathrm{C}^*$-代数，从而推广了Cayley图的经典等价性和Sako关于一致Roe代数的定理。
In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.
We provide a reference for basic categorial properties of the categories of (possibly non-unital) $\mathbb{C}$-linear $*$-categories or $C^{*}$-categories, and (not necessarily unit-preserving) functors. Generalizing the classical case of algebras with $G$-action, we extend the construction of crossed products to categories with $G$-action. We will show that the crossed product functor preserves exact sequences and excisive squares and sends weak equivalences to equivalences.
We extend a theorem of Haagerup and Kraus in the C*-algebra context: for a locally compact group with the approximation property (AP), the reduced C*-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group C*-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.
We study the C*-algebras and von Neumann algebras associated with the universal discrete quantum groups. They give rise to full prime factors and simple exact C*-algebras. The main tool in our work is the study of an amenable boundary action, yielding the Akemann-Ostrand property. Finally, this boundary can be identified with the Martin or the Poisson boundary of a quantum random walk.
Given a $C^*$-dynamical system $(A,G,α)$, with $G$ a discrete group, Schur $A$-multipliers and Herz--Schur $(A,G,α)$-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of $A \rtimes_{α, r} G$. The resulting characterisations of exactness and SOAP of $A \rtimes_{α, r} G$ generalise the corresponding statements for the reduced group $C^*$-algebra.
Exactness and Fell bundles with the approximation property over inverse semigroups
本文研究了具有逼近性质的Fell包 over逆半群的减少横截代数的精确性，证明其等价于Fell包的单位纤维的精确性，并扩展了作者关于局部紧豪斯多夫第二可数群oid作用于可分C*-代数的结果。
We define Ozawa's notion of bi-exactness to discrete quantum groups, and then prove some structural properties of associated von Neumann algebras. In particular, we prove that any non amenable subfactor of free quantum group von Neumann algebras, which is an image of a faithful normal conditional expectation, has no Cartan subalgebras.
We generalize Kirchberg's weak exactness to inclusions of C*-algebras in von Neumann algebras and study some characterizations and permanence properties which are similar to those of exact groups. We then consider a similar condition to Ozawa's condition (AO) with our weak exactness and generalize Ozawa's theorem for bi-exact groups. As a corollary, we give new examples of prime factors.
A locally compact groupoid is said to be exact if its associated reduced crossed product functor is exact. In this paper, we establish some permanence properties of exactness, including generalizations of some known results for exact groups. Our primary goal is to show that exactness descends to certain types of closed subgroupoids, which in turn gives conditions under which the isotropy groups of an exact groupoid are guaranteed to be exact. As an initial step toward these results, we establish the exactness of any transformation groupoid associated to an action of an exact groupoid on a loca…
Weak exactness and amalgamated free product of von Neumann algebras
本文证明弱正合冯·诺依曼代数的融合自由积仍弱正合，方法利用Toeplitz-Pimsner代数的泛性质及双模上的局部凸拓扑刻画弱正合性。
Generalizing Block and Weinberger's characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of $G$ on its Stone-\vCech compactification we obtain a homological characterization of exactness of the group, answering a question of Nigel Higson.
Given a countable residually finite group, we construct a compact group K and two elements w and u of K with the following properties: The group generated by w and the cube of u is amenable, the group generated by w and u contains a copy of the given group, and these two groups are dense in K. By combining it with a construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda and formation of diagonal products, we construct an example for which the latter dense group is non-exact. Our proof employs approximations in the space of marked groups of LEF ("Locally Embeddable into…
We study exactness of groups and establish a characterization of exact groups in terms of the existence of a continuous linear operator, called an invariant expectation, whose properties make it a weak counterpart of an invariant mean on a group. We apply this operator to show that exactness of a finitely generated group $G$ implies the vanishing of the bounded cohomology of $G$ with coefficients in a new class of modules, which are defined using the Hopf algebra structure of $\ell_1(G)$.
We construct explicit approximating nets for crossed products of C*-algebras by actions of discrete quantum groups. This implies that C*-algebraic approximation properties such as nuclearity, exactness or completely bounded approximation property are preserved by taking crossed products by actions of amenable discrete quantum groups. We also show that the noncommutative topological entropy of a transformation commuting with the quantum group action does not change when we pass to the canonical extension to the crossed product. Both these results are extended to twisted crossed products via a s…
Groupoid exactness and the weak containment problem
研究局部紧群体中精确性的类比，证明六个精确性概念的等价性，并探讨群体可接受性与C*-代数关系。
We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf $G$-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf $G$-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on $\ell_{\infty}$-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.
It is shown by Houdayer-Isono that a group measure space von Neumann algebra is a full factor if the group is countable discrete and bi-exact, and the action is strongly ergodic, essentially free and non-singular. Recently, bi-exactness for locally compact groups was introduced by Brothier-Deprez-Vaes. In this paper, we will show that Houdayer-Isono type result holds for bi-exact locally compact groups.
Bi-exact Wreath-like Product Groups
本文证明了顺从群与双精确群的花环型积群具有双精确性，进而获得群冯诺依曼代数的坚实性及麦克达夫超刚性的新例子。
Actions of quantum groups on dual operator spaces and their crossed products
该研究探讨量子群在对偶算子空间及其交叉积上的作用，通过函子的等价性刻画量子群的逼近性质，并将Fubini交叉积函子的正合性与量子群的可和性质关联起来。
In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long s…
Let $Γ$ be a discrete group. A $C^*$-algebra $A$ is an exotic $C^*$-algebra (associated to $Γ$) if there exist proper surjective $C^*$-quotients $C^*(Γ)\to A\to C^*_r(Γ)$. In this paper, we show that a large class of exotic $C^*$-algebras have poor local properties. More precisely, we demonstrate the failure of local reflexitity, exactness, and local lifting property. Additionally, $A$ does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when $A$ is from the class of exotic $C^*$-algebras defined by Brown and Guentner. In order to achieve the main results…
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