Reiji Tomatsu
Z.-J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper, we extend this work to the case of discrete quantum groups. That is, we show that a discrete quantum group, where we do not assume its unimodularity, has an invariant mean if and only if it is strongly Voiculescu amenable.
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We classify outer actions (or $\mathscr{G}$-kernels) of discrete amenable groupoids on injective factors. Our method based on unified approach for classification of discrete amenable groups actions, and cohomology reduction theorem of discrete amenable equivalence relations. We do not use Katayama-Takesaki type resolution group approach.
We apply the Evans-Kishimoto type argument for centrally free actions of discrete amenable groups on McDuff factors, and classify them. Especially, we present a different proof that the Connes-Takesaki modules are complete cocycle conjugacy invariants for centrally free actions of discrete amenable groups on injective factors.
We present a simple unified proof of classification of discrete amenable group actions on injective factors. Our argument does not depend on types of factors. We also show the second cohomology vanishing theorem for arbitrary cocycle crossed actions of discrete amenable groups on the injective factor of type II$_1$.
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.
The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some usefull properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups.
We introduce the notion of Zimmer amenability for actions of discrete quantum groups on von Neumann algebras. We prove generalizations of several fundamental results of the theory in the noncommutative case. In particular, we give a characterization of Zimmer amenability of an action $α:\Bbb G\curvearrowright N$ in terms of $\hat{\Bbb{G}}$-injectivity of the von Neumann algebra crossed product $N\ltimes_α\Bbb G$. As an application we show that the actions of any discrete quantum group on its Poisson boundaries are always amenable.
In this paper, we introduce actions of fusion algebras on unital $C^*$-algebras, and define amenability for fusion algebraic actions. Motivated by S.\ Neshveyev et al.'s work, considering the co-representation ring of a compact quantum group as a fusion algebra, we define the canonical fusion algebraic (for short, CFA) form of a discrete quantum group action on a compact quantum space. Furthermore, through the CFA form, we define FA-amenability of discrete quantum group actions, and present some basic connections between FA-amenable actions and amenable discrete quantum groups. As an applicati…
Using the continuous decomposition, we classify strongly free actions of discrete amenable groups on strongly amenable subfactors of type III$_λ, 0 < λ< 1$. Winsløw's fundamental homomorphism is a complete invariant. This removes the extra assumptions in the classification theorems of Loi and Winsløw and gives a complete classification up to cocycle conjugacy.
Let $G$ be a countable discrete amenable group, ${\cal M}$ a McDuff factor von Neumann algebra, and $A$ a separable nuclear weakly dense C$^*$-subalgebra of ${\cal M}$. We show that if two centrally free actions of $G$ on ${\cal M}$ differ up to approximately inner automorphisms then they are outer conjugate by an approximately inner automorphism, in the operator norm topology, which makes $A$ invariant. In addition, when $A$ is unital, simple, and with a unique tracial state and $α$ is an automorphism of $A$ we also show that the aperiodicity of $α$ on the von Neumann algebra is equivalent to…
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.
A Nica-amenable submonoid of a group which fails to embed in an amenable group
构造了一个群的弱拟格序子幺半群，其半群C*-代数是核的，但该幺半群不能嵌入任何可amenable群，从而表明Nica amenable性和核性不刻画可嵌入性。
We construct the HNN extension of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is K-amenable.
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.
Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations
本文利用Evans-Kishimoto交织论证对离散顺从群到遍历变换全群正规化子的作用进行分类，且证明不依赖遍历变换的类型。
We prove a weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting.
Groups with rapid decay and trivial amenable radical are C*-simple
本文证明具有快速衰减性和平凡可解根的有限生成群必为C*-单群，建立了这两类性质间的联系。
Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras s…
In 1990, Lind, Schmidt and Ward gave a formula for the entropy of certain $\mathbb{Z}^n$-dynamical systems attached to Laurent polynomials $P$, in terms of the (logarithmic) Mahler measure of $P$. We extend the expansive case of their result to the noncommutative setting where $\mathbb{Z}^n$ gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede-Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.
We introduce a notion of entropy for automorphisms of discrete groups which admit amenable actions on a compact space. This entropy is dual to classical topological entropy in the sense that if G is discrete and abelian then our notion of entropy agrees with the topological entropy of the induced automorphism on the (compact) dual group of G. We prove a number of basic properties of this dual entropy and give a few calculations. In particular, we are able to give precise calculations for arbitrary automorphisms of crystallographic groups.
We study uniform perturbations of crossed product C$^*$-algebras by amenable groups. Given a unital inclusion of C$^*$-algebras $C\subseteq D$ and sufficiently close separable intermediate C$^*$-subalgebras $A$, $B$ for this inclusion with a conditional expectation from $D$ onto $B$, if $A=C\rtimes G$ with $G$ discrete amenable, then $A$ and $B$ are isomorphic. Furthermore, if $C\subseteq D$ is irreducible, then $A=B$.
We study actions of countable discrete amenable groups on unital separable simple nuclear Z-absorbing C*-algebras. Under a certain assumption on tracial states, which is automatically satisfied in the case of a unique tracial state, the crossed product is shown to absorb the Jiang-Su algebra Z tensorially.
We prove that the unitary equivalence classes of extensions of C*_r(G) by any sigma-unital stable C"-algebra, taken modulo extensions which split via an asymptotic homomorphism, form a group which can be calculated from the universal coefficient theorem of KK-theory when G is a free product of a countable collection of amenable groups.
We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.
In this paper, we give examples of maximal amenable subalgebras of the free group factor of two generators. More precisely, we consider two copies of the hyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take a Haar unitary $u_i$ which generates a Cartan subalgebra of it. We show that the von Neumann subalgebra generated by the self-adjoint operator $u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. This provides infinitely many non-unitary conjugate maximal amenable MASAs.
The Cuntz algebra O_n is presented as a partial crossed product in which an amenable group partially acts on an abelian C*-algebra. The partial action is related to the Cuntz groupoid for O_n and connections are made with non-self-adjoint subalgebras of O_n, particularly the Volterra nest subalgebra. These ideas are also extended to the M_k(O_n) context.
Amenable absorption in von Neumann algebras of hyperbolic groups
证明双曲群相关的冯·诺依曼代数满足可吸收性质：任何无限极大可分子群H的冯·诺依曼子代数若与L(H)有扩散交集则必包含于L(H)。
A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the…
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 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…
We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group.
We associate a cohomological invariant to each outer action of a group on a factor, and classify them by the invariant in the case that the group is a countable discrete amenable group and the factor is appoximately finite dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called the intrinsic modular obstruction. The invariant for an outer action alpha is given as the pull back of the intrinsic modular obstruction, which is called the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first part of the theory and presents general theory. In the case tha…
By classifying $S$-maximal amenable subgroups of algebraic groups over a global field of characteristic zero, we obtain a complete classification of maximal amenable subgroups up to commensurability in the respective arithmetic groups. Futhermore, we prove that these commensurably maximal amenable subgroups are singular and therefore give rise to maximal amenable von Neumann subalgebras.
We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index…
Strongly 1-bounded inner amenable groups
本文研究了有限生成icc内可 amenable群，证明其生成强1有界的II_1因子。
We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.
We investigate the notion of relatively amenable topological action and show that the action of Thompson's group $T$ on $S^1$ is relatively amenable with respect to Thompson's group $F$. We use this to conclude that $F$ is exact if and only if $T$ is exact. Moreover, we prove that the groupoid of germs of the action of $T$ on $S^1$ is Borel amenable.
We describe conditions that characterize amenability for groups in terms of positive definite functions valued in a von Neumann algebra.
We prove that for any infinite, maximal amenable subgroup $H$ in a hyperbolic group $G$, the von Neumann subalgebra $LH$ is maximal amenable inside $LG$. It provides many new, explicit examples of maximal amenable subalgebras in II$_1$ factors. We also prove similar maximal amenability results for direct products of relatively hyperbolic groups and orbit equivalence relations arising from measure-preserving actions of such groups.
We introduce an Ulam-type stability condition for positive definite maps defined on a countable group and prove that this condition characterizes amenability.
We construct inner amenable groups G with infinite conjugacy classes and such that the associated II_1 factor does not have property Gamma of Murray and von Neumann. This solves a problem posed by Effros in 1975.
Vanishing of $\ell^2$-Betti numbers for inner amenable groups
该研究解决了内顺从群领域的公开问题，证明可数内顺从群所有次数的$\u2113^2$-贝蒂数均消失。
RFD property for groupoid C*-algebras of amenable groupoids and for crossed products by amenable actions
本文研究了可算群作用下C*-代数的交叉积及可算群体C*-代数的RFD性质，通过动作特征化RFD属性，并推广Bekka定理至动力学方向。
We construct inner amenable and ICC groups having no ergodic, free, probability-measure-preserving and stable action. This solves a problem posed by Jones-Schmidt in 1987.
On inner-amenability and boundary actions
研究离散可数群\(\Gamma\)，证明若\(\Gamma\)是 ICC 内顺从非顺从的则不满足(AO)性质，推广该现象；还证明若\(\Gamma\)是特定离散子群则不可能内顺从，推广了相关已知事实。
When amenable groups have real rank zero $C^*$-algebras
该研究探讨离散顺从群具有实秩零$C^*$-代数的条件，在局部有限群对应实秩零代数的已知结论基础上，证明了实秩零条件下初等顺从且有限Hirsch长度的正规子群必为局部有限群，推进了相关开放问题的研究。
We show that a Fell bundle B = {B_t}_{t \in F}, over an arbitrary free group F, is amenable, whenever it is orthogonal (in the sense that B_x^* B_y = 0, if x and y are distinct generators of F) and semi-saturated (in the sense that B_{ts} coincides with the closed linear span of B_t B_s, when the multiplication ``ts'' involves no cancelation).
We show that, if $G$ is an amenable group and $H$ is a hyperbolic group, then the free product $G\ast H$ is weakly amenable. A key ingredient in the proof is the fact that $G\ast H$ is orbit equivalent to $\mathbb{Z}\ast H$.
We prove that a wreath product of an abelian group and a non-amenable group is not strongly Ulam stable. Previously this was known for groups containing free subgroups, due to work of Burger, Ozawa and Thom, and for some surface groups, due to work by Kazhdan. We also show that these wreath products are not deformation rigid.
Non-Commutative Maximal Inequalities for State-Preserving Actions of amenable groups
本文研究了可算群在非可换$ L^1 $空间上保持态作用的极大不等式及遍历定理，并通过Neveu分解得到随机遍历定理。
A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, we show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group.
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。
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.
We introduce inner amenability for discrete p.m.p. groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations which either are stable or have a non-trivial central sequence in their full group.
We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-sta…
Kida and Tucker-Drob recently extended the notion of inner amenability from countable groups to discrete p.m.p. groupoids. In this article, we show that inner amenable groupoids have "fixed priced 1" in the sense that every principal extension of an inner amenable groupoid has cost 1. This simultaneously generalizes and unifies two well known results on cost from the literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p. equivalence relation admitting a nontrivial asymptotically central sequence in its full group has cost 1, and (2) a theorem of Tucker-Drob stating that…
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。
On sofic approximations of non amenable groups
本文对每个初始次顺从的非顺从群构造了两个在任何万有sofic群自同构下均不共轭的sofic逼近，回答了Paunescu的问题并推广了Elek-Szabo唯一性定理。
We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup $S$, the action of $S$ on its spectrum is amenable if and only if every action of $S$ is amenable.
On the amenable subalgebras of group von Neumann algebras
本文从动力学角度研究可数群的群冯·诺依曼代数的子代数，证明极大不变顺从子代数的存在性，并引入不变概率测度概念，证明顺从测度支撑于该极大子代数。
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