Subject / 33 records

Topological dynamics and amenability

Minimal flows, compactifications, concentration, amenability, and symmetry groups.

Classification: Second Brain

Connected through shared records

Operator algebras7 Banach spaces and Banach lattices3 Homological algebra and derived limits3 Polish groups and descriptive complexity3 Measure and ergodic theory2 Model theory and definability2 Geometric group theory1 Groupoids and Cartan pairs1 Ordered groups1

Reading routes

Symmetry, concentration, and boundedness →

Search within Topological dynamics and amenability

33 records

10.5281/zenodo.238320582026
Charmenability of Thompson's group F
Martino Lupini · Thompson groups, C*-algebras, Amenability
We prove that Thompson's group F and its commutator subgroup F′ are charmenable. The main assertion is that every nonempty compact convex conjugation-invariant family of normalized positive-definite functions on either group contains a trace. Its proof uses commuting groups supported on disjoint intervals and a stationary averaging argument on their joint von Neumann algebra. It is independent of nonamenability of F. The character condition in charmenability follows from the classification of finite factor representations by Dudko and Medynets. We also prove that F is not charfinite and that F′ is charfinite precisely when F is nonamenable. Every nonzero simple quotient of either full group C*-algebra is scalar or is the corresponding reduced group algebra, and every nonzero nuclear quotient admits a multiplicative character.https://github.com/eulogica/charmenability-of-thompsons-group-f
No files · GitHub named
10.5281/zenodo.234405102026
Geometric criteria for the sequential Dini property
Martino Lupini · Operator algebras
Let $C$ be a closed pointed convex cone in a real Banach space $X$ , and set $q_C(x)=\max\{\operatorname{dist}(x,C),\operatorname{dist}(x,-C)\}.$ We prove that every increasing weakly convergent sequence converges in norm if and only if every weakly null sequence $(x_n)$ satisfying $q_C(x_n)\to0$ is norm null. Equivalent conditions are a sandwich principle on relatively weakly compact sets and uniform equivalence of the $q_C$ and norm metrics on each weakly compact subset, with a modulus depending on the weakly compact set. These give intrinsic geometric criteria for the property in a characterization question posed by Jochen Glück in 2024. They distinguish the sequential property from normality, which requires a uniform lower bound for $q_C$ on the entire unit sphere. Rosenthal's theorem yields an equivalent criterion through $\ell^1$ basic sequences. In particular, on a space containing no copy of $\ell^1$ , the property is equivalent to normality. A closed nonnormal cone with the property exists on a Banach space exactly when it contains $\ell^1$ , with no generating requirement on the cone. The criteria also show failure of the property for the pointwise cones of $C^k[-1,1]$ , $k\geq1$ , and $W^{1,1}(-1,1)$ .
PDF · LaTeX · GitHub named
10.5281/zenodo.234388512026
Universal minimal flows from pointwise completions and atomic decomposition
Martino Lupini · Orderability, Operator algebras, Descriptive set theory
We organize explicit universal minimal flows of mapping and transformation groups around two standard reductions: passage to a compact group completion and removal of a closed extremely amenable normal subgroup. Finite interpolation identifies the pointwise completions of path, loop, gauge, and section groups; for disconnected compact Lie targets the completion also records the clopen information in the source. For measurable maps into a compact metrizable group K, the established atomless fixed-point theorem gives M(L0(Y, μ; K)) = KAμ, where Aμ is the set of atoms. Combining these reductions with known structure and compact-extension theorems yields atomic formulas for measure-space transformations, Lp-isometries, and automorphism groups of operator algebras. The general Banach-valued statement uses complex fibres; real Hilbert fibres are treated separately. We also give the formula M(FΩ ⋊ Sym(Ω)) = M(F)Ω × LO(Ω) for a Polish group F with metrizable universal minimal flow and a countable set Ω, with the natural skew action. Its ordered relational special cases have earlier Ramsey-theoretic treatments. The results are presented as explicit consequences and extensions of the cited completion, extreme-amenability, and structural results.
PDF · LaTeX · GitHub named
10.5281/zenodo.233835602026
Proximally irreducible affine extensions: composition and maximal covers
Martino Lupini · Groups
We prove that the composition of proximally irreducible affine extensions is proximally irreducible, answering a question of Bal for arbitrary topological groups and compact Hausdorff affine flows. The proof combines localization, Baire category, and compactness for finite convex witnesses, without requiring closed extreme-point sets. Composition, factorization, inverse limits, and uniform density bounds yield maximal proximally irreducible covers, unique up to a unique isomorphism over the base and characterized by a universal property. Extensions of a fixed base form a complete lattice, realized by closed invariant spaces of affine functions on its maximal cover. Passing to probability flows recovers maximal proximal covers of minimal flows and gives canonical refinements, pullbacks, and functorial lifts. For the trivial group, Gleason covers describe maximal domains and classify polytope extension lattices. For separable groups, extensions of metrizable bases admit metrizable approximations that detect nontriviality. Examples delimit the affine pullback and metrizability statements.
PDF · LaTeX · GitHub named
10.5281/zenodo.233625992026
Exactness and amenability at infinity for locally compact groupoids
Martino Lupini · Groupoids, Amenability, Groups
For a second-countable locally compact Hausdorff étale groupoid G, we prove the equivalence of exactness of Cr*(G), exactness of reduced crossed products, and strong amenability at infinity. These conditions are also equivalent to exactness of any one twisted reduced algebra. The amenable action can be chosen with a proper open surjective anchor admitting a continuous section. This answers Anantharaman–Delaroche’s 2016 exactness questions and the étale case of LaLonde’s 2017 question about exact transformation groupoids. Consequently, exactness together with canonical weak containment implies amenability; for compact unit space, exactness is equivalent to amenability of the Furstenberg boundary action. We also prove permanence of strong amenability at infinity under continuous strongly surjective homomorphisms between second-countable groupoids with Haar systems, when the kernel and target have this property. No Haar system on the kernel is required. Finally, we construct an inner-exact étale group bundle with exact fibers and nonexact reduced algebra, supplying the example asked for by Anantharaman–Delaroche in 2016 and answering LaLonde’s 2017 group-bundle question negatively. Its fibers are hyperbolic groups converging strongly to a free group, while their Cayley graphs fail uniform property A.
PDF · LaTeX · GitHub named
10.5281/zenodo.233421812026
Bounded geometry in Polish groups: locally compact models, amenability, and simple examples
Martino Lupini · Orderability, Descriptive set theory, Amenability
We develop locally compact models and applications for Polish groups of bounded geometry. Building on Hrushovski's quasi-model theorem and the ultrafilter framework of Krupiński–Pillay, we give an explicit coarse-density calculation for a model with compact conjugation-invariant multiplication defect and exact control of the intrinsic coarsely bounded bornology. Applications yield the Haagerup property under topological amenability and two-sided entropy estimates under metric amenability. Packing and model obstructions treat the stated classes of groups. For McCleary's classical simple periodic coloured-order group, we compute the pointwise Polish topology and intrinsic coarse geometry, obtaining a non-locally compact quasi-line.
PDF · LaTeX · GitHub named
10.5281/zenodo.233387882026
Automatic equivariant Jiang–Su stability in the nonunital tracial setting
Martino Lupini · C*-algebras, Amenability, Groups
Let A be a separable simple nuclear C*-algebra that absorbs the Jiang–Su algebra Z and admits a nonzero densely defined lower semicontinuous trace. Using the Gaussian construction of the October 2026 OpenAI preprint, we prove that every action α of a countable discrete amenable group on A is cocycle conjugate to α ⊗ idZ. This answers the general tracial part of Szabó’s 2019 conjecture on automatic equivariant Jiang–Su stability, extending the October 2026 unital theorem to algebras with possibly unbounded traces and to trace-scaling actions. The extension applies that construction in a tracial quotient of Kirchberg’s central sequence algebra and uses the nonunital absorption results of Szabó and Wouters. The transfer is expressed as a local equivariant Gamma criterion for nonseparable unital algebras admitting relative copies of Z. Its uniform trace-splitting estimates allow the original action and its full cone of traces to be retained. Combined with the established purely infinite case, the result gives automatic equivariant Jiang–Su stability for all separable simple nuclear Z-stable algebras.
PDF · LaTeX · GitHub named
10.5281/zenodo.231718172026
Tame functions and Asplund representations of additive Banach groups
Martino Lupini · Operator algebras, Groups
We prove that tame functions on the additive group of c0 do not determine its norm topology, answering questions of Megrelishvili and Glasner–Megrelishvili. The proof combines Gowers’ positive-block stabilization with Megrelishvili’s independent-translates configuration. The obstruction extends by homomorphism pullback to Banach spaces containing c0. Full-sphere stabilization gives arbitrarily large subspace balls of approximate constancy and shows that every point of a closed norm ball lies in the tame closure of its boundary sphere. In contrast, a dyadic phase map and the coordinate-rotation construction of Antunes, Ferenczi, Grivaux and Rosendal give a strong-operator embedding of c0 into the isometry group of an Asplund space isomorphic to c0, with almost periodic scalar coefficients. Thus c0 is Asplund but not reflexively representable, settling the Asplund-versus-reflexive problem; even its tame algebra fails to determine the topology. Applications give local copies of βℕ in every proper semigroup compactification and exclude c0 from compact-open automorphism groups of finite-rank compact topological median algebras. For locally compact abelian H, tame functions determine the uniform topology of c0(H) exactly when H is totally disconnected. For second-countable H, these groups are always Asplund representable.
PDF · LaTeX · Data
10.5281/zenodo.231332632026
Universal minimal flows of automorphism groups of locally finite trees
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228906182026
Rosenthal representations of homogeneous spaces: rank-one classifications and higher-rank criteria
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228863602026
Low-complexity factors of compact affine semigroup actions
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228833532026
Isomorphism of finitely generated bi-orderable elementary amenable groups is universal
Martino Lupini · Orderability, Amenability, Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228526312026
Ordered Compactifications and Circular Ellis Semigroups
Martino Lupini · Orderability
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228452362026
Skew-amenability in finite-regularity topologies for path, loop, and gauge groups
Martino Lupini · Amenability, Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228399152026
Charmenability of Leavitt unit groups and Clifford-lamp groups
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228146892026
$\mathbf{\Sigma^0_2}$ boundaries in $\ell^1$-homology from amenable convergence actions
Martino Lupini · Amenability
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228137522026
A matrix convexity characterization of the noncommutative Poulsen simplex
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228133152026
Malliavin--Malliavin and whirly amenability: Borel observations, translation displacement, and total variation
Martino Lupini · Descriptive set theory, Amenability
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.228097652026
Generic monothetic subgroups of the measure automorphism group
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227764742026
Universal minimal flows of homeomorphism groups of solenoids and Knaster continua
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227646152026
Universal minimal flows of automorphism groups\\of stable Kirchberg algebras
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227616302026
Universal minimal flows for unital Kirchberg and AF algebras
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227582302026
Borel invariant means on countable inverse limits of amenable groups
Martino Lupini · Descriptive set theory, Amenability, Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227493622026
Whirly monothetic groups that are not L\'evy
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227436042026
Amenability and skew-amenability of C1 path, loop, and one-dimensional gauge groups
Martino Lupini · Amenability, Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227378542026
Concentrating measures and whirly amenability of Doucha's universal metric group
Martino Lupini · Amenability, Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227378282026
L\'evy stabilizers of the Poulsen simplex, the Gurarij space, and their noncommutative analogues
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227375692026
Universal minimal flows of inductive limits of Lie groups
Alessandro Codenotti, Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227224532026
Extreme amenability of the automorphism group of the Jiang-Su algebra
Martino Lupini · Amenability, Groups
No abstract on this deposit.
PDF · LaTeX · Data
10.5281/zenodo.227221902026
Extreme amenability of the automorphism group of the Razak-Jacelon algebra
Martino Lupini · Amenability, Groups
No abstract on this deposit.
LaTeX · Data
10.5281/zenodo.227189882026
Universal minimal flows of Cuntz automorphism and unitary groups
Martino Lupini · Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.227135582026
Extreme amenability of the additive group of $L^p$ for $0<p<1$
Martino Lupini · Amenability, Groups
No abstract on this deposit.
PDF · LaTeX
10.5281/zenodo.226918402026
Homogeneity of complete M-faces of the noncommutative Poulsen simplex
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX · Data

Topological dynamics and amenability

Connected through shared records

Reading routes

Charmenability of Thompson's group F

Geometric criteria for the sequential Dini property

Universal minimal flows from pointwise completions and atomic decomposition

Proximally irreducible affine extensions: composition and maximal covers

Exactness and amenability at infinity for locally compact groupoids

Bounded geometry in Polish groups: locally compact models, amenability, and simple examples

Automatic equivariant Jiang–Su stability in the nonunital tracial setting

Tame functions and Asplund representations of additive Banach groups

Universal minimal flows of automorphism groups of locally finite trees

Rosenthal representations of homogeneous spaces: rank-one classifications and higher-rank criteria

Low-complexity factors of compact affine semigroup actions

Isomorphism of finitely generated bi-orderable elementary amenable groups is universal

Ordered Compactifications and Circular Ellis Semigroups

Skew-amenability in finite-regularity topologies for path, loop, and gauge groups

Charmenability of Leavitt unit groups and Clifford-lamp groups

$\mathbf{\Sigma^0_2}$ boundaries in $\ell^1$-homology from amenable convergence actions

A matrix convexity characterization of the noncommutative Poulsen simplex

Malliavin--Malliavin and whirly amenability: Borel observations, translation displacement, and total variation

Generic monothetic subgroups of the measure automorphism group

Universal minimal flows of homeomorphism groups of solenoids and Knaster continua

Universal minimal flows of automorphism groups\\of stable Kirchberg algebras

Universal minimal flows for unital Kirchberg and AF algebras

Borel invariant means on countable inverse limits of amenable groups

Whirly monothetic groups that are not L\'evy

Amenability and skew-amenability of C1 path, loop, and one-dimensional gauge groups

Concentrating measures and whirly amenability of Doucha's universal metric group

L\'evy stabilizers of the Poulsen simplex, the Gurarij space, and their noncommutative analogues

Universal minimal flows of inductive limits of Lie groups

Extreme amenability of the automorphism group of the Jiang-Su algebra

Extreme amenability of the automorphism group of the Razak-Jacelon algebra

Universal minimal flows of Cuntz automorphism and unitary groups

Extreme amenability of the additive group of $L^p$ for $0<p<1$

Homogeneity of complete M-faces of the noncommutative Poulsen simplex