We prove an inverse theorem for the Gowers $U^2$-norm for maps $G\to\mathcal M$ from an countable, discrete, amenable group $G$ into a von Neumann algebra $\mathcal M$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert\cdot\Vert$. We use this result to prove a stability result for unitary-valued $\varepsilon$-representations $G\to\mathcal U(\mathcal M)$ with respect to $\Vert\cdot \Vert$.
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.
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.
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…
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…
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 argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $ϕ$ of $S$ onto a group $G$, we show, under an assumption on $\ker(ϕ)$, that $S$ has weak containment if and only if $G$ is amenable and $\ker(ϕ)$ has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show that all graph inverse semigroups have weak containment and that Nica's inverse semigroup $\mcT_{G,P}$ of a quasi-lattice ordered group $(G,P)$ has weak containment if and only if $(G,P)$ …
We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a n…
We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Γ$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Γ$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Lück rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups…
We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on X, it is shown that the entropy of the a…
We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rk_F for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rk_F depends on rk continuously. When an ame…
Recently Bingbing Liang and Hanfeng Li computed the mean dimension and metric mean dimension for algebraic actions of amenable groups. We show how to extend their computation of metric mean dimension to the case of sofic groups, provided that the dual module is finitely generated. Additionally, we show that when the dual module is finitely presented that the mean dimension is the von Neumann rank. The proof also goes through introducing \ell^{p}-analogues of metric mean dimension, which may be seen as an obstruction to the equality of mean dimension and metric mean dimension.
We give algebraic characterizations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of their Boolean inverse semigroups of compact open bisections. It yields in turn algebraic characterizations of both properties for inverse semigroups in terms of subquotients of their booleanizations.
Let $G$ be a locally compact group and $(Φ, Ψ)$ be a complementary pair of $N$-functions. In this paper, using the powerful tool of porosity, it is proved that when $G$ is an amenable group, then the Figà-Talamanca-Herz-Orlicz algebra ${\A}_Φ(G)$ is a Banach algebra under convolution product if and only if $G$ is compact. Then it is shown that ${\A}_Φ(G)$ is a Segal algebra, and as a consequence, the amenability of ${\A}_Φ(G)$ and the existence of a bounded approximate identity for ${\A}_Φ(G)$ under the convolution product is discussed. Furthermore, it is shown that for a compact abelian group…
We recast the Foelner condition in an operator algebraic setting and prove that it implies a certain dimension flatness property. Furthermore, it is proven that the Foelner condition generalizes the existing notions of amenability and that the enveloping von Neumann algebra arising from a Foelner algebra is automatically injective. As an application we show how our techniques unify the previously known results concerning vanishing of L^2-Betti numbers for amenable groups, groupoids and quantum groups and moreover provides a large class of new examples of algebras with vanishing L^2-Betti numbe…
The notion of sofic equivalence relation was introduced by Gabor Elek and Gabor Lippner. Their technics employ some graph theory. Here we define this notion in a more operator algebraic context, starting from Connes' embedding problem, and prove the equivalence of this two definitions. We introduce a notion of sofic action for an arbitrary group and prove that amalgamated product of sofic actions over amenable groups is again sofic. We also prove that amalgamated product of sofic groups over an amenable subgroup is again sofic.
We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.
We relate Fuglede-Kadison determinants to entropy of algebraic actions of sofic groups in essentially complete generality. This generalizes recent results of Hanfeng Li and Andreas Thom from the amenable case to the sofic case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the residually finite case. Moreover, the proof given is the first calculation of entropy of algebraic actions to avoid a nontrivial determinant approximation. We apply our results to a problem of Christopher Deninger, generalizing results of Hanfeng Li and David Kerr. Finally, we show that in many cases, …
The analytic von Neumann regular closure $R(Γ)$ of a complex group algebra $\CΓ$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(Γ)$ containing $\CΓ$. We prove that all the algebraic von Neumann regular closures corresponding to sofic representations of an amenable group are isomorphic to $R(Γ)$. This result can be viewed as a structural generalization of Lück's Approximation Theorem. \noindent The main tool of the proof which might be of independent interest is that an amenable group algebra $KΓ$ over any field $K$ …
We show that for any amenable group Γand any ZΓ-module M of type FL with vanishing Euler characteristic, the entropy of the natural Γ-action on the Pontryagin dual of M is equal to the L2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module ZΓ/ZΓf is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in ZΓ. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szegő-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra …
Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not pro…
Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearly shows that people have hardly been looking at the last chapter dealing with the so-called "inverse systems", unless in very particular situations. The purpose of this paper is to provide for the first time the full explanation of this chapter within the framework of the formal …
In this work we present algebraic conditions on an inverse semigroup S (with zero) which imply that its associated tight groupoid G_{tight)(S) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.
Cartan inverse semigroups and twisted Steinberg algebras
University of Sydney（悉尼大学）；Victoria University of Wellington（惠灵顿维多利亚大学）；Queen’s University Belfast（贝尔法斯特女王大学）；MSU-Iligan Institute of Technology（棉兰老国立理工大学）
