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Repositories

Latest version of each Zenodo concept by Logica Artificialis. A row shows only the files on that record, and a GitHub address only when the manuscript names one.

8 repositories

• 10.5281/zenodo.238333062026

  Finite configurations and nuclear quotients of Thompson's group F

  Martino Lupini · Thompson groups, C*-algebras, Operator algebras

  Let F be Thompson’s group, let P = F′, and let Ug denote the canonical prefix-table unitaries in the Cuntz algebra O₂. We prove that the finite ordinary tensor images Bn = C*(Uf⊗n : f ∈ F) and Cn = C*(Up⊗n : p ∈ P) are nuclear. A representation has finite separated derived capacity when products of sufficiently many displacement operators with separated interior supports vanish. Capacity at most d gives an actual marked quotient of B2d+2, or of C2d for a nontrivial P-representation. Infinite capacity is equivalent to admitting the regular representation as an actual quotient. Consequently, if F is nonamenable, every nonzero nuclear quotient of its full group C*-algebra, and likewise of C*(P), is a quotient of one of these finite canonical images. The finite nuclearity and norm estimates are unconditional; nonamenability enters only in passing from nuclearity to finite capacity. The result gives a countable cofinal family of nuclear quotients, with a degree depending on the quotient.

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• 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 $W^{1,1}(-1,1)$.

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• 10.5281/zenodo.234404832026

  Peripheral eigenspace dimensions for positive operators and semigroups

  Martino Lupini · Orderability, Operator algebras

  Let $T$ be a positive contraction on a monotonically complete complex Banach lattice with the Fatou property. We prove $$\dim\ker(\lambda I-T)\leq\dim\ker(\lambda^k I-T) \qquad (|\lambda|=1,\ k\in\mathbb Z),$$ where all infinite dimensions are identified with infinity. This answers an open problem posed by Jochen Glück in 2021 and extends the previously established root-of-unity case. The bounded bilateral trajectories of $T$ form an intrinsic Banach lattice by the fixed-space theorem. On this lattice, the shift is a lattice automorphism, and evaluation at zero identifies its unimodular eigenspaces with those of $T$. We also prove the corresponding multiplicity inequality for imaginary generator eigenspaces of positive contractive $C_0$-semigroups, and for positive power-bounded operators that are order continuous.

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• 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.

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• 10.5281/zenodo.234077612026

  Lipschitz embeddings of symmetric operator-ideal balls into Hilbert space

  Martino Lupini · Operator algebras

  No abstract on this deposit.

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• 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.

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• 10.5281/zenodo.228473142026

  Uniform cohomological gaps and operator-local repair

  Martino Lupini · Operator algebras

  No abstract on this deposit.

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• 10.5281/zenodo.226833362026

  On the Borel complexity of operator images

  Martino Lupini · Operator algebras, Descriptive set theory

  No abstract on this deposit.

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