Amenable Representations Research Articles

Tame hereditary path algebras and amenability

AbstractIn this note we revisit the notion of amenable representation type introduced by Gábor Elek. We show that tame hereditary path algebras of quivers of extended Dynkin type over any field are of amenable type. This verifies a conjecture of Elek, which draws similarities to the tame‐wild dichotomy, for another class of tame algebras. We also show that path algebras of wild acyclic quivers over finite fields are not amenable.

Sugihara algebras and Sugihara monoids: Multisorted dualities

AbstractThe authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system R-mingle.This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications.

Infinite dimensional representations of finite dimensional algebras and amenability

We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

Return to the Poissonian city

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. Suppose further that connections are established using ‘near geodesics’, constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In Kendall (2011) it was shown that a scaled version of this flow has asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two ‘seminal curves’ defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to 0 with increasing computational effort.

Return to the Poissonian city

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. Suppose further that connections are established using ‘near geodesics’, constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In Kendall (2011) it was shown that a scaled version of this flow has asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two ‘seminal curves’ defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to anL1error which tends to 0 with increasing computational effort.

Strictly Amenable Representations of Reduced GroupC* -Algebras

We show that a locally compact group $G$ is amenable if and only if its reduced group $C^* $ -algebra $C^*_r(G)$ is nuclear and has a tracial state, which is also equivalent to $C^*_r(G)$ having a special kind of $^* $ -representation that we propose to call “strictly amenable representation”. Consequently, if $G$ is separable and connected, then $G$ is amenable if and only if $C^*_r(G)$ has a tracial state.

Hidden torsion, 3-manifolds, and homology cobordism

This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pairwise homology cobordant, yet remain indistinguishable by any prior known homology cobordism invariants. Additionally we give an answer to a question about transfinite lower central series of homology cobordant 3-manifold groups, asked by T. D. Cochran and M. H. Freedman.

Characterizations of amenable representations of compact groups

Let $G$ be a locally compact group and let $\pi $ be a unitary representation. We study amenability and H-amenability of $\pi $ in terms of the weak closure of $(\pi \otimes \pi )(G)$ and factorization properties of associated coefficient subspaces (or su

Modular actions and amenable representations

Consider a measure-preserving action Γ ↷ (X, μ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ ↷ X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action Γ ↷^α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free, measure-preserving, ergodic actions as well as two non-Borel bireducible ones.

Amenable representations and coefficient subspaces of Fourier-Stieltjes algebras

Amenable unitary representations of a locally compact group, $G$, are studied in terms of associated coefficient subspaces of the Fourier-Stieltjes algebra $B(G)$, and in terms of the existence of invariant and multiplicative states on associated von Neumann and $C^*$-algebras. We introduce Fourier algebras and reduced Fourier-Stieltjes algebras associated to arbitrary representations, and study amenable representations in relation to these algebras.

Amenable Representations and Reiter's Property for Kac Algebras

In this paper, we study amenable unitary corepresentations of Kac algebras. We also study some sorts of “noncommutative” Reiter's properties. As an application, we find some new equivalent conditions for the amenability of Kac algebras.

Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability property of a unitary representation \( \pi \) of a group G (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system \( ({\Bbb S}_{\pi}, G) \), where \( {\Bbb S}_{\cal H} \) is the unit sphere the Hilbert space of representation. We prove that \( \pi \) is amenable if and only if either \( \pi \) contains a finite-dimensional subrepresentation or the maximal uniform compactification of \( ({\Bbb S}_{\pi} \) has a G-fixed point. Equivalently, the latter means that the G-space \( ({\Bbb S}_{\pi}, G) \) has the concentration property: every finite cover of the sphere \( {\Bbb S}_{\pi} \) contains a set A such that for every \( \epsilon > 0 \) the \( \epsilon \)-neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of \( \pi \) is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \( {\Bbb S}_{\pi} \). As another corollary, a locally compact group G is amenable if and only if for every strongly continuous unitary representation of G in an infinite-dimensional Hilbert space \( {\cal H} \) the system \( ({\Bbb S}_{\cal H}, G) \) has the property of concentration.

Kazhdan's property (T) and amenable representations

Kazhdan's property (T) and amenable representations

Amenable unitary representations of locally compact groups

A notion of amenability for an arbitrary unitary group representation is introduced. This unifies and generalizes the notions of amenable homogeneous spaces and of inner-amenable groups. Amenable locally compact groups are characterized by the amenability of all their unitary representations. Amenable representations are characterized by several properties which are operator theoretic analogues of properties characterizing amenable groups. We give a generalization to arbitrary representations of Hulanicki-Reiter theorem. This is used in order to describe the amenable representations of the groups with Kazhdan property (T).