arXiv subjects

Tomasz Kania

Publications and source records attributed to Tomasz Kania.

At least 37 records · Page 2Linked to original sources

arXiv2026

Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations

We isolate a coherent finite-separation strengthening of the Freese--Nation property and use it to characterise projective and weakly projective Boolean algebras. A Boolean algebra is projective if and only if it has a meet-closed decomposition base carrying such a coherent finite-separation map; it is weakly projective if and only if it has a meet-closed $π$-base carrying one. The global form of the same property is much stronger: the positive cone of a Boolean algebra carries a coherent finite-separation map if and only if the algebra is countable. Thus the projective and weakly projective characterisations are intrinsically local and cannot be strengthened by requiring the whole algebra to carry the coherent map except in the countable case.

math.LO↗

arXiv2026

Spectral constants for algebraic numerical ranges

We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If $Γ_n$ denotes the universal constant for algebraic elements of degree at most $n$, then \[ Γ_1=1, \qquad 2n-1\leqslantΓ_n<\infty\qquad(n\geqslant2). \] Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain \[ 3\leqslantΓ_2 \leqslant\sqrt{1+(2\mathrm e-1)^2}<4.55. \] For unital $C^*$-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang--Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.

math.OA↗

arXiv2026

Canonical reconstruction and forcing absoluteness of standard structures

We isolate a simple preservation principle governing when it is absolute, between transitive models of set theory, that a given algebraic or topological-algebraic structure has a standard form $F(X)$ indexed by a set $X$. The principle is: if the index $X$ (or a proxy for it) can be recovered from $F(X)$ by a uniform definable construction, then the class of structures isomorphic to some~$F(X)$ is downward absolute from forcing extensions. Answering a question raised by Noah Schweber, we deduce in particular that no group that fails to be a full symmetric group in the ground model can become one after forcing; the result holds already in ZF. The same mechanism applies to full transformation monoids, powerset Boolean algebras, full relation algebras, full clones, full partition lattices, products $R^X$ of finitely generated centrally indecomposable rings, the commutative $C^*$-algebras $\ell_\infty(X)$ and $c_0(X)$, full endomorphism rings, the operator algebras $\mathcal{B}(H)$ and $\mathcal{K}(H)$, and $\ell_1(X)$ as a real Banach lattice. In the motivating symmetric-group case, the same reconstruction gives more than descent: it yields a uniform $Π^1_1$ definition of fullness over transitive ZF-models. We then exhibit clean torsor obstructions, in the standard symmetric-model situation: finite covers $Y \times n$ already separate ZF-failure from ZFC-descent without any completeness caveat, and the finite-support normed space $c_{00}(I)$ provides the analogous Banach example. Bare-Banach-space isomorphism with $\ell_1(Γ)$ exhibits a genuine ZFC-descent. We conclude with the corresponding, relative, obstructions to $Π^1_1$-definability of standardness over transitive ZF-models.

math.LO↗

arXiv2026

Talagrand compacta, 2DCP, and pointwise quotients

We revisit Talagrand's CH compactum as a test object for the two-disjoint-copies property and for pointwise quotient questions. The two-disjoint-copies property, or 2DCP, is a topological sufficient condition for the existence of infinite-dimensional metrisable quotients of spaces $C_{\operatorname{p}}(X)$; recent work asks whether Talagrand's compactum has this property. Assuming $\diamondsuit(S)$ for a stationary co-stationary $S\subseteqω_1$, we carry out Talagrand's inverse-limit construction with additional diagonalisation. The resulting compactum $T$ keeps Talagrand's conclusions: $C(T)$ is Grothendieck, the weak-star compact ball $M_1(T)$ contains no copy of $βω$, and $T$ has no non-trivial convergent sequences. At the same time, no two disjoint non-metrisable closed subspaces of $T$ are homeomorphic; hence $T$ has no 2DCP and is not locally homogeneous. We also give a ZFC example of a perfect compact space with 2DCP which is not locally homogeneous and contains neither $βω$ nor $2^ω$. Finally, we isolate a general locally convex observation, in the spirit of the Banakh--Gabriyelyan theory of the Josefson--Nissenzweig property, showing that pointwise quotients onto $(\ell_p)_{\operatorname{p}}$, $1\leqslant p<\infty$, force the Josefson--Nissenzweig property. Consequently Talagrand compacta have no classical pointwise sequence quotients $(c_0)_{\operatorname{p}}$, $(\ell_p)_{\operatorname{p}}$, or $(\ell_\infty)_{\operatorname{p}}$. The full metrisable quotient problem for these $C_{\operatorname{p}}$-spaces remains open. Several open problems are included.

math.GN↗

arXiv2026

Pure infiniteness and primary factorisation

We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP). We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}_E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$. For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar. We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E_{\operatorname{R}}$ does not have the uniform PFP.

math.FA↗

arXiv2026

Amenability constants for unconditional sums of Banach algebras

We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family $(A_i)_{i\in I}$ of Banach algebras and a Banach sequence lattice $E$ on~$I$, the $E$-sum $\bigl(\bigoplus_{i\in I} A_i\bigr)_{\!E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C_E := \sup\{\|χ_F\|_E: F\subseteq I\text{ finite}\}<\infty$, we prove that this $E$-sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two-sided estimate \[ \sup_{i\in I}\text{AM}(A_i) \;\le\; \text{AM}\Bigl(\bigl(\textstyle\bigoplus_{i\in I} A_i\bigr)_{\!E}\Bigr) \;\le\; C_E^2\,\sup_{i\in I}\text{AM}(A_i). \] We show that the factor $C_E^2$ is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of $C_E$ is necessary whenever infinitely many summands are non-zero and the sum admits a bounded approximate identity. As applications, we recover the classical formula $\text{AM}\bigl(c_0\text{-}\bigoplus_{i\in I} A_i\bigr) = \sup_{i\in I}\text{AM}(A_i)$ for arbitrary (possibly uncountable) index sets, extend it to weighted $c_0$-spaces, and characterise amenability for Orlicz sequence algebra sums. We also record how these unconditional criteria give obstructions within the conditional framework of James-type $J$-sums. Finally, we investigate weak amenability of $E$-sums. We prove that weak amenability passes to summands, that $E$-sums of commutative weakly amenable algebras are weakly amenable, and--contrasting sharply with the Johnson amenability picture--that for $1 < p < \infty$, the $\ell_p$-sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the $c_0$-type regime ($C_E < \infty$), we establish two-sided estimates for weak amenability constants with constants depending only on $C_E$.

math.FA↗

arXiv2026

Uniform Property (S)

We introduce and investigate a quantitative version of Steinhaus' property$(S)$ for Banach spaces, called the \emph{uniform property$(S)$}. A Banach space$X$ is said to have uniform$(S)$ if for every pair of distinct unit vectors $x,y\in X$ and every$a>0$, the difference of the perturbed norms $$ \sup_{\|z\|\le a}\big|\|x+z\|-\|y+z\|\big| $$ is bounded below by a positive function of$a$ and$\|x-y\|$. We compute this modulus exactly for the spaces $L_1(μ)$ with atomless measure$μ$, $$ U_{L_1(μ)}(d;a)=\Big(\tfrac{4a}{2+d}\wedge 1\Big)d. $$ The class of spaces with uniform$(S)$ is stable under ultrapowers, Bochner-$L_1$ constructions, and contains all Gurari\uı spaces as well as Banach lattices of almost universal disposition. In particular, every Banach space embeds isometrically into a non-strictly convex Banach space of the same density having uniform$(S)$. We further exhibit an explicit equivalent renorming of$\ell_1(Γ)$, $$ \|x\|_S=\big(\|x\|_1^2+\|x\|_2^2\big)^{1/2}, $$ which endows$\ell_1(Γ)$ and all its ultrapowers with uniform$(S)$. These results settle, inZFC, several open questions about the quantitative geometry of property$(S)$ posed by Kochanek and the second-named author.

math.FA↗

arXiv2026

Primariness and the Primary Factorisation Property

We study the relation between primariness of Banach spaces and the stronger operator-theoretic notions of the primary factorisation property (PFP) and the uniform primary factorisation property (UPFP). We revisit several classical primariness arguments and isolate the additional information needed to factor the identity through arbitrary operators. In the separable setting, this recovers quantitative factorisation versions of the Casazza--Kottman--Lin method for spaces with symmetric bases and treats the exceptional cases of $\ell_1$ and $\ell_\infty$. We then develop support-reduction and free-selection tools for uncountable direct sums, allowing one to transfer primariness and the PFP/UPFP from countable building blocks to non-separable $\ell_p$-, $c_0$- and more general symmetric sums. As applications, we obtain, among others, the primariness of $C[0,1]^*$ under the negation of the Continuum Hypothesis and UPFP results for uncountable sums of ordinal $C(α)$-spaces. Finally, using the finite-block representation of $\mathcal B(\ell_p)$, we prove a uniform primary factorisation theorem for the Banach space $\mathcal B(\ell_p)$, $1<p<\infty$, and end with open problems concerning the gap between primariness and factorisation.

math.FA↗

arXiv2026

Semiprojective Banach lattices

We introduce a norm-controlled notion of semiprojectivity for Banach lattices, requiring liftability of contractive lattice homomorphisms through inductive limits of closed ideals with arbitrarily small loss of norm control. Our main result establishes that, for a compact metric space $X$, the Banach lattice $C(X)$ is semiprojective if and only if $X$ is an absolute neighbourhood retract. Notably, this characterisation is strictly more permissive than its $C^*$-algebraic counterpart: by a theorem of S\orensen and Thiel, $C(X)$ is semiprojective in the category of $C^*$-algebras and $*$-homomorphisms if and only if $X$ is an ANR of dimension at most one. The dimensional obstruction disappears in the Banach-lattice setting because lattice homomorphisms between $C(K)$-spaces are automatically weighted composition operators, and therefore no commutation relations need to be lifted. We also show that uncountable $\ell_1$-sums of $1^+$-projective Banach lattices with topological order units are semiprojective but need not be $1^+$-projective, establishing that the two notions are genuinely distinct. On the negative side, we prove that $\ell_p$ and $L_p([0,1])$ for $p \in (1,\infty)$ as well as Orlicz spaces are not semiprojective.

math.FA↗

arXiv2026

Finite-codimensional subspaces of Daugavet spaces: projection constants and minimal projections

Over the real or complex field, we establish a duality formula for projection constants of finite-codimensional subspaces of Banach spaces with the Daugavet property. If \[ Y=\bigcap_{j=1}^n \ker f_j \subset X, \qquad W=\operatorname{span}\{f_1,\dots,f_n\} \subset X^*, \] then \[ λ(Y,X)=1+λ(W,X^*), \] and minimal projections onto $Y$ correspond exactly to weak$^*$-continuous minimal projections onto $W$. This yields, in particular, a complete description of the hyperplane case: every hyperplane has projection constant $2$, and $\ker f$ admits a minimal projection if and only if $f$ attains its norm. We then specialise to the real space $X=C[0,1]$. Our second ingredient is a transfer principle from duplication-stable finite-dimensional subspaces of $\ell_1^N$ to piecewise-constant subspaces of $L_1[0,1]\subset M[0,1]=C[0,1]^*$. For the regular symmetric spaces constructed by Chalmers and the second-named author and the second named author and Prophet, respectively, the transferred subspaces retain their projection constants but admit no weak$^*$-continuous minimal projections. Passing to annihilators yields finite-codimensional subspaces of the real space $C[0,1]$ for which the infimum defining the projection constant is not attained. As a consequence, for every $Λ\in[2,\infty)$ there exists a finite-codimensional subspace $Y$ of the real space $C[0,1]$ such that \[ λ(Y,C[0,1])=Λ, \] and the infimum defining $λ(Y,C[0,1])$ is not attained. For each even codimension $n$ we moreover realise every value in the interval $(2,1+β_n]$, where \[ β_n = \mathsf E_{{\mathsf P}_n}\Bigl|\sum_{j=1}^n \varepsilon_j\Bigr| = n2^{-n}\binom{n}{n/2} \sim \sqrt{\frac{2n}π}, \] $(\varepsilon_j)$ is a Rademacher family on $Ω_n=\{-1,1\}^n$, and $\mathsf{P}_n$ is the uniform probability measure.

math.FA↗

arXiv2026

Bourgain-uo sequential completeness in vector lattices

We revisit Bourgain's 1981 counterexample to the sequential completeness of the `pointwise plus domination' convergence on $\ell_1$ from the perspective of vector lattices. In this setting, we show that for sequences the associated notion of Bourgain--uo convergence coincides with ordinary order convergence. Motivated by Bourgain's construction, we introduce a strengthened, subsequence-invariant notion of Cauchy sequence: a sequence $(x_n)$ in a vector lattice $E$ is called Buo-Cauchy if for every strictly increasing sequence $(n_k)$ the differences $x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$. We first show that sequential Buo-completeness forces $σ$-order completeness. Thus every non-$σ$-order complete vector lattice fails sequential \Buo-completeness. In particular, free Banach lattices $\mathrm{FBL}(E)$ are not sequentially Buo-complete whenever $\dim E>1$. On the positive side, we prove that the classical sequence lattices $c_0$ and $\ell_\infty$ are sequentially Buo-complete: every Buo-Cauchy sequence converges in order, and hence in the Buo sense. Finally, we obtain a sharp metric characterisation for bounded Lipschitz function lattices: the vector lattice $\mathrm{Lip}_b(X)$ of bounded Lipschitz functions on a metric space $(X,d)$ is sequentially Buo-complete if and only if $X$ is uniformly discrete.

math.FA↗

arXiv2026

$(λ^+)$-injective Banach spaces

In a companion paper (Studia Math., 2023), we proved for every $λ\in(1,2]$ the existence of a $(λ^+)$-injective renorming of $\ell_\infty$ that is not $λ$-injective, thereby establishing a~forgotten theorem of Pełczyński in that range. The complementary range $λ\in(2,\infty)$ was left open. In the present paper, we resolve this remaining case: for every $λ>2$ we construct a Banach space that is $(λ^+)$-injective but not $λ$-injective, completing Pełczyński's theorem for all $λ>1$. The construction uses a single device: the `zero-sum' subspace $Σ_N(Y)\subset Z_\infty^N$, which multiplies the relative projection constant by $μ_N=2-2/N$ while preserving non-attainment. Iterating this operation reduces the problem to the range $(1,2]$ already covered by the companion paper. Since the ambient spaces arising in the iteration are finite $\ell_\infty$-sums of $\ell_\infty$, the resulting examples may be realised as subspaces of~$\ell_\infty$. We also prove that if two Banach spaces are each isometrically isomorphic to their own square and each is isometric to a $1$-complemented subspace of the other, then their Banach--Mazur distance is at most $9+6\sqrt{3}$. Consequently, we obtain the estimate $\operatorname{dist}(L_\infty[0,1],\ell_\infty)\le 9+6\sqrt{3}$, thereby improving a recent result of Korpalski and Plebanek.

math.FA↗

arXiv2026

Raja's covering index of $L_p$ spaces

We study Raja's covering index $Θ_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving \[ Θ_H(n)=n^{-1/2}\qquad(n\in\mathbb N); \] in particular $Θ_H(2)=1/\sqrt2$, thus answering a question of Raja about the precise two-piece covering index of $\elltwo$. For scalar-valued Lebesgue spaces $L_p(μ)$, $1\le p<\infty$, we construct an explicit block decomposition of the unit ball yielding the upper bound $Θ_{L_p(μ)}(n)\le n^{-1/p}$ for all $n\in\mathbb{N}$; in particular $Θ_{\ell_p}(n)\le n^{-1/p}$. For $1<p<\infty$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja's general lower bound to give the sharp asymptotic estimate $Θ_{L_p(μ)}(n)\asymp n^{-1/p}$. We also obtain uniform upper bounds $Θ_{L_p(μ;E)}(n)\le n^{-1/p}$ for Bochner spaces $L_p(μ;E)$ over non-atomic $σ$-finite measure spaces, with constants independent of the Banach space $E$; this shows that, at the level of power-type upper estimates, the covering index decays at the same rate regardless of the asymptotic geometry of~$E$ and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds $Θ_{L_p(M,τ)}(n)\gtrsim n^{-1/r}$ for non-commutative $L_p(M,τ)$ spaces associated with semifinite von Neumann algebras, where $r=\min\{p,2\}$. We do not attempt to optimise the exponent or constants in the non-commutative setting.

math.FA↗

arXiv2026

Non-standard analysis for coherent risk estimation: hyperfinite representations, discrete Kusuoka formulae, and plug-in asymptotics

We develop a non-standard analysis framework for coherent risk measures and their finite-sample analogues, coherent risk estimators, building on recent work of Aichele, Cialenco, Jelito, and Pitera. Coherent risk measures on $L^\infty$ are realised as standard parts of internal support functionals on Loeb probability spaces, and coherent risk estimators arise as finite-grid restrictions. Our main results are: (i) a hyperfinite robust representation theorem that yields, as finite shadows, the robust representation results for coherent risk estimators; (ii) a discrete Kusuoka representation for law-invariant coherent risk estimators as suprema of mixtures of discrete expected shortfalls on $\{k/n:k=1,\ldots,n\}$; (iii) uniform almost sure consistency (with an explicit rate) for canonical spectral plug-in estimators over Lipschitz spectral classes; (iv) a Kusuoka-type plug-in consistency theorem under tightness and uniform estimation assumptions; (v) bootstrap validity for spectral plug-in estimators via an NSA reformulation of the functional delta method (under standard smoothness assumptions on $F_X$); and (vi) asymptotic normality obtained through a hyperfinite central limit theorem. The hyperfinite viewpoint provides a transparent probability-to-statistics dictionary: applying a risk measure to a law corresponds to evaluating an internal functional on a hyperfinite empirical measure and taking the standard part. We include a standardd self-contained introduction to the required non-standard tools.

q-fin.RM↗

arXiv2025

Dimension-dependent bounds for the SDIEP via phase optimisation and Paley-type constructions

We refine the cycle-walk (Fourier) template of Gnacik and the author to quantify when a~$δ$-Sule\uımanova spectrum $(1,λ_2,\dots,λ_n)$ (with $λ_j\le 0$) is realised by a symmetric doubly stochastic matrix. For the canonical cycle basis we compute the \emph{exact} size-dependent threshold \[ δ_n \;=\; 1-\frac{1}{2\cos^2\!\Big(\fracπ{4n}ρ(n)\Big)}, \quad ρ(n)\in\{0,1,2,4\}\ \text{determined by } n\bmod 8, \] which improves $1/2$ if and only if $8\nmid n$; we also prove sharpness for that template. We then introduce an \emph{optimally phase-aligned} cycle basis which removes the `$8\mid n$' artefact and yields better sufficient bound \[ δ_n^{\rm (ph)} \;=\; \begin{cases} \displaystyle 1-\dfrac{1}{2\cos^2(π/n)}, & n\equiv 0\pmod{4},\\[2mm] \displaystyle 1-\dfrac{1}{2\cos^2(π/2n)}, & n\equiv 2\pmod{4},\\[2mm] \displaystyle 1-\dfrac{1}{2\cos^2(π/4n)}, & n\ \text{odd}, \end{cases} \] so that $δ_n^{\rm (ph)}<\tfrac12$ for \emph{every} $n\ge3$ and $δ_n^{\rm (ph)}=δ_n$ unless $8\mid n$. Next, on abelian $2$-groups, the Walsh--Hadamard basis has coherence $M=1$ and hence suffices for \emph{all} Sule\uımanova lists ($δ=0$); the same conclusion holds in every Hadamard order (\emph{e.g.}, Paley families).

math.SP↗

arXiv2025

The $π$-property of a Banach space along a filter

We examine the analyticity of the class of separable Banach spaces possessing the $π$-property, defined in terms of convergence along a filter. Our results establish that this class is $Σ^1_3$ whenever the underlying filter is analytic (as a subset of the Cantor set $Δ$). Furthermore, we demonstrate that if the filter is countably generated, the class of such spaces is $Σ^1_2$ with respect to any admissible Polish topology on the family of closed subspaces of $C(Δ)$.

math.FA↗

arXiv2024

Differential embeddings into algebras of topological stable rank 1

We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterise in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.

math.OA↗

arXiv2024

Discrete subgroups of normed spaces are free

Ancel, Dobrowolski, and Grabowski (Studia Math., 1994) proved that every countable discrete subgroup of the additive group of a normed space is free Abelian, hence isomorphic to the direct sum of a certain number of copies of the additive group of the integers. In the present paper, we take a set-theoretic approach based on the theory of elementary submodels and the Singular Compactness Theorem to remove the cardinality constraint from their result and prove that indeed every discrete subgroup of the additive group of a normed space is free Abelian.

math.FA↗