Narutaka Ozawa, Yuhei Suzuki
We show the equivalence of several amenability type conditions for C*-dynamical systems. As an application, based on the Pimsner--Meyer construction, we give a new method to produce amenable actions on simple C*-algebras.
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For an amenable minimal topologically free dynamical system $α$ of a group on a compact metrizable space $Z$ and for a compact metrizable space $Y$ satisfying a mild condition, we construct a minimal skew product extension of $α$ on $Z\times Y$. This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of Rørdam and Sierakowski.
It is shown that the action of ${\rm SL}(3,\mathbb{Z})$ on the Stone-uCech boundary of ${\rm SL}(3,\mathbb{Z}) / {\rm SL}(2,\mathbb{Z}) $ is amenable. This confirms a prediction by Bekka and Kalantar.
Amenable wreath products with non almost finite actions of mean dimension zero
本文通过研究顺从圈积群的变构性，构造出平均维数为零但不具几乎有限性的顺从群极小拓扑自由作用，证明Kerr的Toms-Winter猜想动力学对应结论对这类作用不成立，并给出首批顺从变构群例子。
We prove that the crossed product Banach algebra $\ell^1(G,A;α)$ that is associated with a ${\mathrm C}^\ast$-dynamical system $(A,G,α)$ is amenable if $G$ is a discrete amenable group and $A$ is a strongly amenable ${\mathrm C}^\ast$-algebra. This is a consequence of the combination of a more general result with Paterson's characterisation of strongly amenable unital $\mathrm{C}^\ast$-algebras in terms of invariant means for their unitary groups.
McDuff factors from amenable actions and dynamical alternating groups
本文证明可均群在康托集上的拓扑自由作用产生的包含交错群的子群，其群冯诺依曼代数为McDuff因子，并推广到广义圈积的稳定性。
Examples of non-amenable, boundary-amenable dynamical systems
研究具有(AP)性质的离散可数群$\Gamma$的作用，当它以特定方式作用于可数集时，在$\partial_\beta \mathfrak{X}$上诱导作用拓扑可和，给出了在$\partial_\beta (\Gamma / \Lambda)$上作用的多种情形及相关应用。
We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…
By giving an interesting characterisation of amenable multiplicative unitaries in term of one dimensional representations, we show in a simple way that bicrossproducts of amenable locally compact groups is both amenable and coamenable.
We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra analogue is also obtained.
Amenable crossed product Banach algebras associated with a class of $\mathrm{C}^\ast$-dynamical systems
本文证明当离散群G顺从且C*-代数A为交换或有限维时，相应C*-动力系统的交叉积Banach代数ℓ^1(A,G,α)具有顺从性，并展望了后续研究方向。
Residually finite amenable groups that are not Hilbert-Schmidt stable
该研究构造了首批不具备Hilbert-Schmidt稳定性的剩余有限顺从群实例，涵盖多种群结构，还证明这些实例不具备算子HS稳定性，填补了相关领域的研究空白。
Every countable group admits amenable actions on stably finite simple C*-algebras
本文证明了每个可数群都能在稳定有限简单C*-代数上构造可数作用，通过正则表示的全福克空间和迹缩放自同构实现。
We prove that Wang's free orthogonal and free unitary quantum groups are weakly amenable and that their Cowling-Haagerup constant is equal to 1. This is achieved by estimating the completely bounded norm of the projections on the coefficients of irreducible representations of their compact duals. An argument of monoidal equivalence then allows us to extend this result to quantum automorphism groups of finite spaces and even yields some examples of weakly amenable non-unimodular discrete quantum groups with the Haagerup property.
We construct examples of principal groupoids that have weak containment but are not amenable, thus answering questions by Claire Anantharaman-Delaroche and Rufus Willett.
Amenable actions on ill-behaved simple C*-algebras
本文结合Rørdam与作者先前构造，首次给出非顺从群在既非稳定有限也非纯无限的简单可分核C*-代数上的顺从作用，并确保交叉积简单且含有限与无限投影。
A non-uniformly inner amenable group
本文构造了两个初等等价的可数 ICC 群，其中一个顺从而另一个非内顺从，并据此首次给出群冯·诺依曼代数不初等等价的例子，回答了领域内的一个公开问题。
In this last article of the series on outer actions of a countable dicrete amenable group on AFD factors, we analyze outer actions of a countable discrete free abelian group on an AFD factor of type $\text{III}_λ$, $0<λ< 1$, and compute outer conjugacy invariants. As a byproduct, we discover the asymmetrization technique for coboundary condition on a $\Bbb T$-valued cocycle of a torsion free abelian group, which might have been known by the group cohomologists. As the asymmetrization technique gives us a very handy criteria for coboundaries, we present it here in detail in the second section.
Amenable actions of compact and discrete quantum groups on von Neumann algebras
本文利用非交换 Lp-空间技术证明紧量子群动力系统中强等变顺从性与等变顺从性等价，并通过对偶推广到离散量子群，同时首次给出顺从离散量子群非顺从作用于冯·诺依曼代数的显式例子。
We study the existence of minimal dynamical systems, their orbit and minimal orbit-breaking equivalence relations, and their applications to C*-algebras and K-theory. We show that given any finite CW-complex there exists a space with the same K-theory and cohomology that admits a minimal homeomorphism. The proof relies on the existence of homeomorphisms on point-like spaces constructed by the authors in previous work, together with existence results for skew product systems due to Glasner and Weiss. To any minimal dynamical system one can associate minimal equivalence relations by breaking orb…
Almost finiteness and groups of dynamical origin
本文提出良好子群性质，证明其对顺从群作用蕴含比较性，从而确立多类动力起源群在有限维空间上的自由作用几乎有限，并得到可分类交叉积。
We establish several new characterizations of amenable $W^*$- and $C^*$-dynamical systems over arbitrary locally compact groups. In the $W^*$-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of $(M,G,α)$ converging point weak* to the identity of $G\bar{\ltimes}M$. In the $C^*$-setting, we prove that amenability of $(A,G,α)$ is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$, as well as a particular case of the pos…
We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related…
Simplices of maximally amenable extensions in II$_1$ factors
本文构造了可分 II$_1$ 因子及其极大交换子代数，使极大可扩空间与 $n$ 维实单纯形仿射同构，首次实现恰好具有 $n$ 个极大可扩因子扩张的 masa 实例。
Strongly solid group factors which are not interpolated free group factors
本文构造了具有 Haagerup 性质且弱顺从常数为1的非顺从 ICC 群，证明其生成的 II_1 因子具有强刚性，且不同构于任何插值自由群因子。
Let $F\subseteq H\subseteq G$ be closed subgroups of a locally compact group. In response to a 1972 question by Eymard, we construct an example where the homogeneous factor-space $G/F$ is amenable in the sense of Eymard-Greenleaf, while $H/F$ is not. (In our example, $G$ is discrete.) As a corollary which answers a 1990 question by Bekka, the induced representation $\ind_H^G(ρ)$ can be amenable in the sense of Bekka even if $ρ$ is not amenable. The second example, answering another question by Bekka, shows that $\ind_H^G(ρ)$ need not be amenable even if both the representation $ρ$ and the cose…
In a previous paper the authors developed an operator-algebraic approach to Lewis Bowen's sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable.
An example of an infinite amenable group with the ISR property
本文证明正整数集上的有限支撑置换群具有不变 von Neumann 子代数刚性（ISR）性质，并完全刻画了其所有不变子代数，首次给出具有 ISR 性质的无限顺从群例子。
We generalize a theorem of Chifan and Ioana by proving that for any, possibly type III, amenable von Neumann algebra $A_0$, any faithful normal state $φ_0$ and any discrete group $Γ$, the associated Bernoulli crossed product von Neumann algebra $M=(A_0,φ_0)^{\mathbin{\bar{\otimes}}Γ}\rtimes Γ$ is solid relatively to $\mathcal{L}(Γ)$. In particular, if $\mathcal{L}(Γ)$ is solid then $M$ is solid and if $Γ$ is non-amenable and $A_0 \neq \mathbb{C}$ then $M$ is a full prime factor. This gives many new examples of solid or prime type $\mathrm{III}$ factors. Following Chifan and Ioana, we also obta…
Non-amenable C$^*$-superrigid groups that are not W$^*$-superrigid
本文构造了非amenable且无法从群von Neumann代数重建的C$^*$-超刚群，并展示了同时满足C$^*$-和W$^*$-超刚性质的自由积群例子。
We establish that a second countable locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable. We give some examples and applications.
We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.
We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $\mathcal{O}_\infty$ and $\mathcal{O}_2$. This generalizes results known for finite groups and poly-$\mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a prec…
The dynamical Kirchberg-Phillips theorem
本文建立任意局部紧群在 Kirchberg 代数上一类可和作用的余圈共轭分类，证明其由等变 Kasparov 理论决定，并解决及恢复相关猜想与结果。
In this paper, we construct a semigroup associated to an action of countable discrete group on a compact Hausdorff space, that can be regarded as a higher dimensional generalization of the type semigroup. Using this generalized type semigroup we obtain a new characterization of dynamical comparison. This answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither necessarily the acting group is amenable nor the action is minimal.
On operator Connes-amenability of the Fourier-Stieltjes algebra
本文研究了傅里叶-斯蒂尔杰斯代数B(G)的算子康尼斯-可 amenability性质，通过构造反例推翻了关于B(G)与群G可 amenability关系的猜想。
We show that certain amenable subgroups inside $\tilde{A}_2$-groups are singular in the sense of Boutonnet and Carderi. This gives a new family of examples of singular group von Neumann subalgebras. We also give a geometric proof that if $G$ is an acylindrically hyperbolic group, $H$ is an infinite amenable subgroup containing a loxodromic element, then $H<G$ is singular. Finally, we present (counter)examples to show both situations happen concerning maximal amenability of $LH$ inside $LG$ if $H$ does not contain loxodromic elements.
We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representat ions.
Tracially amenable actions and purely infinite crossed products
本文引入迹顺从作用概念并给出等价刻画，证明含自由群F2的群在简单、单位、Z-稳定C*-代数上的外迹顺从作用具有纯无限交叉积，并提供自由群在AF-代数上的具体例子。
We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels' celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong …
We give an example of a non-compact, locally compact group $G$ such that its Fourier-Stieltjes algebra $B(G)$ is operator amenable. Furthermore, we characterize those $G$ for which $A^*(G)$ - the spine of $B(G)$ as introduced by M. Ilie and the second named author - is operator amenable and show that $A^*(G)$ is operator weakly amenable for each $G$.
We prove that the $L^1$-algebra of any non-Kac type compact quantum group does not satisfy operator biflatness. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counter example to the conjecture that $L^1(\Gb)$ is operator amenable if and only if $\Gb$ is amenable and co-amenable for any locally compact quantum group $\Gb$. The result also implies that the $L^1$-algebra of a locally compact quantum group is operator biprojective if and only if $\Gb$ is compact and of Kac type.
In this article, we give explicit examples of maximal amenable subalgebras of the $q$-Gaussian algebras, namely, the generator subalgebra is maximal amenable inside the $q$-Gaussian algebras for real numbers $q$ with its absolute value sufficiently small. To achieve this, we construct a Riesz basis in the spirit of Rădulescu and develop a structural theorem for the $q$-Gaussian algebras.
We study stability properties of amenable locally compact quantum groups under the bicrossed product construction. We obtain as our main result an equivalence between amenability of the bicrossed product and amenability of the matched quantum groups used as building ingredients of the bicrossed product. Finally, we give examples of non-amenable locally compact quantum groups obtained by a bicrossed product construction.
A subgroup of an amenable group is amenable. The $C^*$-algebra version of this fact is false. This was first proved by M.-D. Choi who proved that the non-nuclear $C^*$-algebra $C^*_r(\ZZ_2*\ZZ_3)$ is a subalgebra of the nuclear Cuntz algebra ${\cal O}_2$. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same. In fact he proved that certain of the $C^*$-algebras studied by J. Cuntz and W. Krieger [10] can be constructed naturally as crossed product algebras. For example if the group $Γ$ a…
We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is non-amenable, yet is boundedly approximately contractible. The example embeds into a homogeneous von Neumann algebra. We also observe that there are singly generated, biflat subalgebras of finite Type I von Neumann algebras, which are not amenable (and hence are not isomorphic to C*-algebras). Such an example can be used to show that a certain extension property for commutative operator algebras, which is shown in arXiv:1012.4259 to follow from amenability, does not necessarily imply amenability.
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