arXiv subjects

Tomasz Kania

Publications and source records attributed to Tomasz Kania.

At least 19 recordsLinked to original sources

arXiv2026

A consistent failure of separable quotients for pointwise function spaces

Assuming Jensen's diamond principle, we construct an infinite compact zero-dimensional space $K$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. The space $K$ is separable and crowded, has weight $\aleph_1$ and cardinality $2^{\aleph_1}$, and is an Efimov space. We construct $K$ as an inverse limit of compact metrisable spaces indexed by the countable ordinals. At each nontrivial successor step, the projection has two-point fibres over a chosen closed set and singleton fibres elsewhere; this changes the weak-star limit of a selected sequence of finitely supported measures. We also prove that, for compact $X$, the existence of an infinite-dimensional separable quotient of $C_p(X)$ is equivalent to the existence of an infinite-dimensional metrisable quotient.

math.FA↗

arXiv2026

Weakly LUR norms and the Schur property

A separable real or complex Banach space fails the Schur property if and only if it admits an equivalent Gâteaux smooth, weakly locally uniformly rotund norm which is not midpoint locally uniformly rotund. The construction enlarges an LUR unit ball by a weakly compact set and adds a weighted Hilbert-space term. The resulting norms can be chosen arbitrarily close to any prescribed equivalent LUR norm, and are Gâteaux smooth whenever the prescribed norm is Gâteaux smooth; Fréchet smoothness is preserved as well. Weakly locally uniformly rotund norms which are not midpoint locally uniformly rotund are dense among all equivalent norms on each separable non-Schur space. For arbitrary real or complex Banach spaces, such a renorming exists exactly when the space is LUR-renormable and fails the Schur property. We give a direct construction for this last assertion.

math.FA↗

arXiv2026

Twisting exponential spectra

Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of $ab$ and $ba$ differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain--Delbaen space $X$ obtained from Motakis' construction with Calkin algebra isomorphic as a Banach algebra to $C(S^4)$, we construct $S,T\in\B(X\oplus X)$ such that \[ \frac12\in\varepsilon_{\B(X\oplus X)}(ST) \quad\text{and}\quad \frac12\notin\varepsilon_{\B(X\oplus X)}(TS). \] Moreover, $\ltsr\B(X\oplus X)=\rtsr\B(X\oplus X)=2$, which is the least possible stable rank for such an example. More generally, if $K$ is compact metrisable and $X$ is any space arising from Motakis' construction for $C(K)$, then, for every $n\geqslant1$, \[ \ltsr\B(X^n)=\rtsr\B(X^n)= \begin{cases} \left\lceil \lfloor\dim K/2\rfloor/n\right\rceil+1,&\dim K<\infty, \infty,&\dim K=\infty. \end{cases} \] This also gives counterexamples of every finite stable rank at least two and of infinite stable rank. Finally, we relate every failure of exponential-spectral commutativity to the kernel of the first matrix stabilisation map on the index group.

math.FA↗

arXiv2026

Grothendieck and $\ell_\infty$-Grothendieck subspaces of $\ell_\infty$

Let $\mathfrak{c}=2^{\aleph_0}$. We prove that $\ell_\infty$ contains $2^{\mathfrak{c}}$ pairwise non-isomorphic non-reflexive Grothendieck subspaces. They may all be chosen to contain the canonical copy of $c_0$ and to have infinite-dimensional reflexive quotient. This is optimal and answers by the authors in 2021. We also show that the relative $\ell_\infty$-Grothendieck property depends on the specified copy of $c_0$ and is not an isomorphic invariant of the subspace. More generally, for every closed $E\subseteq\ell_\infty$ there is an $\ell_\infty$-Grothendieck subspace isomorphic to $\ell_\infty\oplus_\infty E$; taking $E=c_0$ answers a problem by González et al.

math.FA↗

arXiv2026

Halving the original Kalton--Roberts upper bound for nearly additive set functions

Let $K_{KR}$ be the least constant governing uniform approximation of approximately additive real-valued set functions by finitely additive signed measures. We prove that $$K_{KR}\le\frac{374167}{20000}=18.70835.$$ Starting from a norming functional for best uniform approximation, we form two families of almost extremal sets with complementary point frequencies. Intersections and biregular expanders then give two recombination schemes. The sharper estimate comes from using the schemes on different ranges and from cancelling the positive and negative errors with unequal weights. This improves the previously published bound $38.8$.

math.CO↗

arXiv2026

Maximal right ideals of the Banach algebra of bounded operators on a Banach space

We study finitely generated maximal right ideals of the Banach algebra $\mathcal{B}(E)$ of bounded operators on a complex Banach space $E$. Every maximal right ideal is either fixed by a non-zero functional or contains the ideal of finite-rank operators; when $E$ is infinite-dimensional, each non-fixed maximal right ideal in fact contains the ideal of inessential operators. Using the elementary representation of finitely generated right ideals as lifting ideals $\operatorname{Lift}(T)=\{TU:U\in\mathcal{B}(E,E^n)\}$, where $T\in\mathcal{B}(E^n,E)$ for some $n\in\mathbb{N}$, we identify the exact operator-theoretic obstruction. The ideal $\operatorname{Lift}(T)$ contains the finite-rank operators precisely when $T$ is surjective, and it equals $\mathcal{B}(E)$ precisely when $T$ is right invertible. If $T$ is surjective but not right invertible, then $\operatorname{Lift}(T)$ is maximal exactly when the row operator $[T\ S]$ is right invertible for every $S\in\mathcal{B}(E)\setminus\operatorname{Lift}(T)$. We apply this framework, together with duality, pullback, lattice-theoretic and cardinality arguments, to obtain maximal right ideals which are not finitely generated for large classes of Banach spaces. These include the following infinite-dimensional spaces: reflexive spaces, separable spaces with an unconditional Schauder decomposition into a countably infinite sequence of non-zero subspaces, spaces containing a complemented copy of $\ell_1$, KB-spaces, Lebesgue spaces $L_p(μ)$ for $1\leqslant p<\infty$, full Orlicz spaces with order-continuous norm, and scalar-plus-compact spaces. We obtain the stronger conclusion that every finitely generated maximal right ideal is fixed for Hilbert spaces, $\ell_1(Γ)$-spaces, reflexive spaces with the bounded approximation property, and the mixed spaces $\ell_1(Γ)\oplus H$ with $H$ a separable Hilbert space.

math.FA↗

arXiv2026

There is no universal separable Banach algebra

We prove that no separable Banach algebra is universal for homomorphic embeddings of all separable Banach algebras, whether embeddings are merely bounded or required to be contractive. The same holds in the commutative category. The proof is short and direct. For each closed subspace $X$ of $c_0$ we build a separable commutative Banach algebra $A_X$ whose multiplication records the canonical pairing between $X^*$ and $X$. Any injective homomorphism of $A_X$ into a Banach algebra $B$ forces the identity operator on $X^*$ to factor through $B$. Thus a universal separable Banach algebra would be complementably universal for the family of duals of subspaces of $c_0$, contrary to a theorem of Johnson and Szankowski.

math.FA↗

arXiv2026

Pointwise function spaces over compacta are not weak Banach spaces

Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.

math.FA↗

arXiv2026

Banach's Isometric Conjecture over the Complex Field

We complete Banach's isometric conjecture over the complex field. More precisely, if $X$ is a complex normed space and, for some $2\leqslant n<\dim_{\C}X$, all its $n$-dimensional complex subspaces are isometric as metric spaces, then the norm is induced by a Hermitian inner product. We also prove the quaternionic counterpart. The central geometric argument first treats real star bodies without convexity or central symmetry; applied to circled complex or quaternionic bodies, it shows that mutually real-linearly equivalent hyperplane sections force the ambient body to be a Hermitian ellipsoid. The proof adapts the bundle-degree mechanism introduced by Lu and Yang for the real case. Finally, we obtain extensions to absolutely homogeneous functions, graded Fréchet spaces, metrisable locally convex spaces, and compatible translation-invariant metrics.

math.FA↗

arXiv2026

Law equivalence for Ornstein--Uhlenbeck dynamics driven by Lévy noise

For stochastic partial differential equations driven by Lévy noise, understanding when changes in the drift operator preserve the law of the solution is fundamental to filtering, control, and simulation. We extend law-comparison results for Ornstein--Uhlenbeck processes from bounded drift operators to generators of $C_0$-semigroups on a separable Hilbert space. The argument separates the problem into a Gaussian channel and a jump--drift channel. The Gaussian channel is governed by an inverse-covariance Hilbert--Schmidt perturbation condition. The jump--drift channel is handled by a directional Cameron--Martin condition, formulated conditionally on the jump path; no unconditional Novikov estimate is needed for this step. We prove that these hypotheses give absolute continuity of path laws on the Skorohod space, and equivalence whenever the Gaussian channels are equivalent. For purely jump noise we prove a rigidity phenomenon: absolute continuity forces the two solutions to coincide. For analytic semigroups and compound Poisson jumps we give verifiable sufficient conditions in terms of fractional smoothing and the boundedness of $Q^{-1/2}(\widetilde A-A)A^{-β}$. Diagonal examples show the sharp role of the Cameron--Martin and inverse-covariance requirements.

math.PR↗

arXiv2026

Approximate homomorphisms on orthomodular lattices

The stability programme initiated by Ulam asks when approximate solutions to algebraic identities must lie near exact ones. For lattices, this leads to the question of when a map that nearly preserves joins and meets can be approximated by a genuine lattice homomorphism. Badora--Kochanek--Przebieracz developed a neighbourhood-based framework for distributive lattices, centred on a separation (sandwich) lemma that constructs an exact join homomorphism between a join-subhomomorphism and a join-superhomomorphism via an order envelope. We revisit this mechanism and identify the single step at which distributivity is used: a decomposition identity for elements lying below a join. Without distributivity, separation can fail already in the modular lattice $M_3$ and in a small finite orthomodular lattice. On the positive side, we show that separation holds in arbitrary lattices whenever the lower bounding map is isotone. For orthomodular lattices---algebraic models of quantum logic---we develop a blockwise stability theory on Boolean blocks. Approximate identities on compatible pairs yield exact homomorphic selections on each block (for joins, for meets, and for both operations under bi-admissibility). We present several gluing criteria for assembling blockwise selections, and we give a concrete finite example showing that gluing can fail when block overlaps are non-trivial. Finally, in the spirit of Kalton--Roberts, we obtain blockwise approximation results for nearly additive functions on orthomodular lattices by finitely additive measures, with an illustration on finite-dimensional projection lattices.

math.LO↗

arXiv2026

Higher traces as boundary averages on finite-dimensional normed spaces

Let $X$ be an $N$-dimensional real normed space, let $1\leqslant k\leqslant N$, and set $V=Λ^kX$ and $m=\binom Nk$. We characterise the probability measures $η$ on the unit sphere of $V$ for which \[ \operatorname{tr}(Λ^kA)=m\int w^\sharp\big((Λ^kA)w\big)dη(w) \] holds for every $A\in\operatorname{End}(X)$: this is equivalent to $m\int w\otimes w^\sharp dη(w)=\operatorname{Id}_V$. The cone probability measure always satisfies this condition, giving a canonical higher-trace formula for every norm. Normalised Euclidean hypersurface measure also does so under a scalar-commutant symmetry hypothesis, including spaces with a $1$-symmetric basis. We further obtain atomic and polyhedral formulae and show that, within a natural power-weighted family, cone measure is the unique universally isotropic member; for hypersurface measure the first-order obstruction is precisely the degree-$2$ spherical harmonic component of the support function.

math.FA↗

arXiv2026

Hamana's injective envelope as a maximal rigid multiplier cover

Let $A$ be a unital $C^*$-algebra. We call an $A$-multiplier cover a pair $(E,ι)$ consisting of a $C^*$-algebra $E$ and a faithful non-degenerate $*$-homomorphism $ι\colon A\to M(E)$. Ordering such covers by $A$-preserving unital completely positive maps between multiplier algebras, we study those covers for which the inclusion $A\subseteq M(E)$ is rigid in Hamana's sense. We prove that Hamana's injective envelope $I(A)$ is a maximal rigid $A$-multiplier cover and that, conversely, a rigid cover is maximal if and only if its multiplier algebra is canonically $*$-isomorphic to $I(A)$ over $A$. Thus maximal rigid multiplier covers provide an order-theoretic characterisation of the injective envelope. In the commutative case $A=C(X)$, this recovers the familiar realisation $I(C(X))\cong C(G(X))\cong M(C_0(U))$ for a dense cozero set $U$ in the Gleason cover $G(X)$, in a form inspired by Błaszczyk's concise construction of the Gleason cover.

math.OA↗

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↗