Subject / 22 records

Banach spaces and Banach lattices

Banach geometry, Banach lattices, bases, free constructions, and norm control.

Classification: Second Brain

22 records

• 10.5281/zenodo.236824122026

  The noncommutative weak extension principle under OCA

  Martino Lupini · C*-algebras, Descriptive set theory

  We prove that the Open Colouring Axiom implies the noncommutative weak extension principle. Every *-homomorphism between coronas of separable nonunital C*-algebras therefore splits, by a commuting projection, into a part with Borel graph and a part with nonmeagre kernel. This removes Martin's Axiom from Vignati and Yilmaz's theorem and answers their proposed weakening for Theorem A. In particular, all such homomorphisms with stable source are Borel under the Open Colouring Axiom alone. We also obtain Borel product lifts for bounded linear maps between reduced products of separable Banach spaces and separable C*-algebras which respect the canonical coordinate restrictions.

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• 10.5281/zenodo.235206832026

  Semiprojectivity and norm-controlled lifting in Banach lattices

  Martino Lupini · Orderability

  We characterize semiprojectivity of a real Banach lattice by almost contractive lifting through ideal quotients of separable Banach lattices, with no separability assumption on the domain. Semiprojectivity is equivalent to ℓ-openness, implies ℓ-closedness, and is unchanged if the finite ideal stage is allowed to depend on the norm tolerance. For separable domains it coincides with 1⁺-projectivity. We prove that arbitrary ℓ₁-sums preserve semiprojectivity without an order-unit hypothesis on the summands. In contrast, ℓ₁(ℕ) ⊕∞ ℝ is not semiprojective, although both summands are 1⁺-projective. Thus semiprojectivity need not pass to finite ℓ∞-sums or split extensions. The example is ∞-projective with infimum of uniform projectivity constants equal to 2, admits an equivalent 1⁺-projective lattice norm, and is not ℓ-closed.

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• 10.5281/zenodo.234699552026

  Order, norming faces, and symmetries of Tursi's Banach lattice

  Martino Lupini · Orderability, Groups

  Let L be Tursi’s separable universal homogeneous real Banach lattice. The set of decreasing positive sequences with infimum zero is coanalytic complete, even for sequences of components of one fixed quasi-interior vector, providing a fixed-lattice coanalytic-complete example for the decreasing-order-complexity question raised in Tursi’s 2024 survey. This gives a negative answer to the 2024 question of Avilés, Rosendal, Taylor, and Tradacete whether every separable Banach lattice belongs to their countable Fatou hierarchy. Unrestricted sequential order convergence has the exact complete complexity coA(Pi^1_1), the dual of one Souslin operation on coanalytic sets. We classify quasi-interior positive vectors by their norming Choquet simplices and answer Tursi’s documented 2021 automorphism-group universality question affirmatively. We prove universality of embedding orbits for every nonzero separable domain and of automorphism conjugacy, and establish concentration, lifting, and representation results for pointwise face kernels. Further results describe generic embedding orbits, ideals, and finite-dimensional approximate disposition.

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• 10.5281/zenodo.234405222026

  A countable norm construction for free Banach lattices under arbitrary convexity implications

  Martino Lupini · Orderability

  We give an explicit countable construction of the norm of the free Banach lattice of an arbitrary prescribed convexity type over a pre-ordered Banach space. Starting with the ordinary free Banach lattice, each step takes the gauge of a closed solid absolutely convex hull determined by the original convexity implications. If $p_k$ is the seminorm obtained after $k$ steps, then the free norm is $p_\omega=\inf_{k<\omega}p_k$. No continuity is required of the increasing numerical functions in the implications, and every antecedent is retained in full. The proof uses disjoint cutoffs that repair the antecedents while preserving the value of one positive functional. The same estimate gives quotient closure and an exact restriction of each successor gauge to finite-dimensional vector sublattices. Applying classical compactness to the resulting class yields a local norm formula over finite-dimensional linear input spaces and finite sets of complete implications. Consequently every strict decrease of the free norm is witnessed at a finite input dimension, a finite set of implications, and a finite stage. For each fixed separable input, a countable subfamily of complete conditions already determines the same free norm. For one presentation of the AM-space type, no finite stage computes the norm on all lattice expressions. The formula also gives an intrinsic isometry criterion, without providing the convenient geometric test asked for by de Jeu and Jiang.

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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.234404152026

  Lipschitz positive decompositions of von Neumann preduals

  Martino Lupini · Mathematics

  We prove that the selfadjoint predual of a complex von Neumann algebra admits a Lipschitz positive decomposition if and only if the algebra is subhomogeneous. In particular, infinite-dimensional selfadjoint trace class gives a negative answer to the Lipschitz decomposition problem, confirming Messerschmidt's 2018 conjecture. For a nonzero subhomogeneous algebra of maximal matrix degree $d$, the least Lipschitz constant of a positive majorant $q(x)\geq\pm x$ is attained and equals the corresponding constant $c_d$ for $d\times d$ trace class. Among nonzero von Neumann algebras, the optimal constant equals one exactly for the abelian algebras. The finite-dimensional constants satisfy $$c\log\log N\leq c_N\leq C\log N$$ for all sufficiently large $N$, with absolute constants $c,C>0$. The lower bound applies to arbitrary positive majorants; the upper bound comes from the classical absolute-value estimate.

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• 10.5281/zenodo.234403022026

  Isometric canonical maps into free convex Banach lattices

  Martino Lupini · Mathematics

  For every $1<p<\infty$, we construct a three-dimensional real Banach lattice $X$ which is not $p$-convex with constant one, although its canonical positive map into the free $p$-convex Banach lattice with constant one is isometric. Dimension three is minimal. In every dimension $n\geq3$, such examples can be arbitrarily close to $\ell_p^n$ under the coordinate lattice isomorphism. An elementary example answers the necessity question raised by Marcel de Jeu and Xingni Jiang in 2025 for general convexity types; the three-dimensional construction gives a negative answer even for ordinary constant-one $p$-convexity. We characterize isometry by an exact factorization of dual functionals through two positive coordinates. For a Banach lattice, the criterion says that every sublattice of $X^*$ generated by two disjoint positive elements is $p'$-concave with constant one. The counterexample separates this two-generator condition from full concavity. At $p=\infty$, in contrast, isometry forces the original norm to be an AM-space norm.

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• 10.5281/zenodo.234399852026

  Critical ideals and covering widths of ideal Schauder bases

  Martino Lupini · Descriptive set theory

  An ideal Schauder basis of a Banach space expands every vector as a series whose partial sums Sₙx converge along an ideal of subsets of ℕ, rather than along the cofinite sets. Its critical ideal is the least ideal with this property: it is generated by the finite sets and the error sets {n: ‖x − Sₙx‖ ≥ ε}. We determine the possible descriptive complexities of critical ideals and study when they are P-ideals. If the coordinate functionals are continuous, the critical ideal is complete for exactly one of the classes Fσ, Fσδ and Fσδσ, and all three cases occur; it is Fσ if and only if it is a P⁺-ideal, and it is always Fσ in a reflexive space. Since analyticity of a critical ideal forces continuity of the coordinates, an analytic critical ideal is Borel. A critical ideal need not be a P-ideal, even for a basis of a separable Hilbert space; the P-property is equivalent to a uniform bound on the number of vectors whose error sets are needed to cover a member of the ideal. Every proper analytic P-ideal is isomorphic to the critical ideal of a basis of ℓ₁ ⊕ c₀, but there is a pathological Fσ P-ideal on ℕ that is not the critical ideal of any basis of any Banach space. These results answer Questions 11.3 and 11.4 of Kwela and Swaczyna (2025) negatively, and their Question 11.2 negatively as stated and positively up to isomorphism. Elementary coordinatewise examples also illustrate a negative answer to their Question 11.1; the dense-span ℓ₁ existence phenomenon is already implicit in classical non-strong Markushevich basis constructions. We also show that every proper nonpathological Fσ P-ideal I is the critical ideal of a basis of a reflexive space and satisfies I = Fin(ψ) = Exh(ψ) for a nonpathological lower semicontinuous submeasure ψ; this answers a question of Lopez-Abad, Olmos-Prieto and Uzcátegui-Aylwin (2025).

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• 10.5281/zenodo.234249602026

  Robust fixed points and finite classifying spaces: CAT(0) actions, Banach targets and group fillings

  Martino Lupini · Groups

  For a finitely presented group, we characterize the fixed-point property for all isometric actions on complete CAT(0) spaces by finite systems of polynomial distance inequalities. An infeasible system gives a uniform tolerance for simultaneous errors in the generating maps, defining relations and midpoint geometry. For finite presentations, this yields an algorithm that halts exactly when the property holds. On genuine CAT(0) spaces and prescribed Banach classes, continuous iterations give locally Hölder retractions onto common fixed sets and continuous choices in parameter families. For one arithmetic presentation we evaluate a nonlinear generator tolerance of 10−232 on genuine CAT(0) spaces.

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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.234055082026

  A universal Banach–Mazur stability radius for von Neumann algebras

  Martino Lupini · Mathematics

  We prove cohomological criteria for ordinary complex Banach–Mazur stability of von Neumann algebras, without separability, type, or density restrictions. A fixed algebra has a positive stability radius if its second bounded Hochschild cohomology vanishes and its third coboundary space is norm closed. For any class closed under countable bounded direct products, vanishing in degrees two and three gives one radius for the entire class. Assuming the degree-two and degree-three vanishing theorem in the September 2026 preprint of OpenAI, this yields one $\varepsilon>0$ such that, for all complex von Neumann algebras $M,N$,

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• 10.5281/zenodo.234048042026

  Polynomial estimates for K-convexity from Walsh decay and cotype

  Martino Lupini · Mathematics

  We obtain polynomial bounds for the norm K(X) of the Rademacher projection on a Banach space X. If the probability-normalized Walsh transform on the s-dimensional cube has norm at most 1/2 on L2(X), then K(X) ≤ 9000s log(6000s2). Consequently, for each 1 < p ≤ 2, the type constant Tp(X) satisfies K(X) ≲p Tp(X)p/(p−1)(1 + log Tp(X)). This improves the exponential upper bound in Naor’s 2022 quantitative question, without determining optimal dependence.

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• 10.5281/zenodo.233421992026

  Efficient contractions of the Urysohn isometry group

  Martino Lupini · Descriptive set theory, Groups

  We construct a contraction of the Urysohn isometry group with uniform and coarse control. Pointwise stabilizers of nonempty finite sets admit contractions uniformly continuous for the left uniformity, and the global contraction preserves bounded-displacement sets. Ordinary contractibility is due to Melleray, while the quantitative realization and embedding inputs are due to Lešnik. The construction here adds guided surjective back-and-forth, inverse evaluation control, and uniform endpoint estimates. A preliminary bounded-chain argument shows that the bounded-image clause of efficient contractibility follows from the stated uniform-continuity hypothesis.

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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.231424712026

  The linear isometry group of the Holmes space

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.229298252026

  Continuous Lipschitz spaces on topometric spaces and free-space dynamics

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.229297642026

  The countable cube with the supremum metric is not Lipschitz universal

  Martino Lupini · Mathematics

  The countable cube with the supremum metric is not Lipschitz universal

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• 10.5281/zenodo.228340732026

  Contractibility and Banach-space representations of the Carderi--Thom groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228131892026

  Distortion of Banach-space unit spheres under their linear isometry groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 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.

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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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