Searcharxiv⌕ Search

arXiv subjects

Ben De Bondt

Publications and source records attributed to Ben De Bondt.

9 recordsLinked to original sources
arXiv2026

Trace-norm rigidity for reduced products of unitary groups and matrix algebras

We study homomorphisms, with a focus on isomorphisms, between the tracial metric reduced products of finite dimensional unitary groups and of matrix algebras. A variant of Ulam stability for unitary groups and a classification of the almost surjective continuous homomorphisms between finite dimensional unitary groups are proved and then used to show that all isomorphisms of product form of these tracial reduced products are induced by almost permutations of the coordinates and coordinatewise application of automorphisms. We prove coordinate recognition for these reduced products and obtain under set theoretic assumptions rigidity and classification results for their full automorphism groups. For tracial reduced matrix algebras we obtain such rigidity result in the more general context of center-preserving $*$-homomorphisms.

math.OA↗
arXiv2025

Adding cofinal countable sequences through multiple regular cardinals by ssp forcing

We present a direct construction of stationary set preserving forcings that make $ω$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $κ> ω_1$. In addition, it is made possible to ensure that no other uncountable regular cardinals from the ground model acquire countable cofinality in the forcing extension. Our method is elementary, being based on a combinatorial argument by Foreman and Magidor together with generalizations of typical side-condition arguments and needs no assumptions beyond $\mathsf{ZFC}$.

math.LO↗
arXiv2024

A metric lifting theorem

In a recent article by Farah and the authors, a strong lifting theorem was proved for a class of coordinate-respecting maps between reduced products of discrete structures, hereby working under mild Forcing Axioms. We generalise this lifting theorem to the metric setting.

math.LO↗
arXiv2024

Trivial Isomorphisms between Reduced Products

We introduce a general method for showing under weak forcing axioms that reduced products of countable models of a theory $T$ have as few automorphisms as possible. We show that such forcing axioms imply that reduced products of countably infinite or finite fields, linear orders, trees, or random graphs have only trivial automorphisms. We also show that Todorčević's Open Colouring Axiom, $\mathsf{OCA}_{\mathrm{T}}$, implies that all automorphisms of $\mathcal{P}(\mathbb{N})/{\mathrm{Fin}}$ are trivial.

math.LO↗
arXiv2024

Increasing the second uniform indiscernible by strongly ssp forcing

We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}(λ,μ)$ that (under $\mathsf{NS}_{ω_1}$ precipitous + existence of $H_θ^#$ for a sufficiently large regular $θ$) increases the second uniform indiscernible $\mathbf{u}_2$ beyond some given ordinal $λ$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $\mathbf{G}_M^{cap}(X)$ and the catching-capturing games $\mathbf{G}_M^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_θ$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way.

math.LO↗
arXiv2024

Saturation of reduced products

We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fréchet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not $\aleph_2$-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that $M$ is isomorphic to an ultrapower (associated with an ultrafilter on $\mathbb N$) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product $M$ is isomorphic to an ultrapower if and only if the theory of $M$ is stable. All of these conclusions apply for reduced products associated with $F_σ$ ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal $\mathcal Z_0$, or any other analytic P-ideal that is not $F_σ$, is not even $\aleph_1$-saturated if its theory is unstable.

math.LO↗
arXiv2022

Games on AF-algebras

We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_α\}_{α< ω_1}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_α\equiv_αA_β$ for every $α< β$. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal{K} \equiv_{ω+ 2 \cdot α+2} B \otimes \mathcal{K}$ implies $K_0(A) \equiv_αK_0(B)$, for every unital $\mathrm{C}^\ast$-algebras $A$ and $B$.

math.LO↗
arXiv2020

Filter-dependent versions of the Uniform Boundedness Principle

For every filter $\mathcal F$ on $\mathbb N$, we introduce and study corresponding uniform $\mathcal F$-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness principles for sequences of continuous linear maps by coinciding with these principles when the filter $\mathcal F$ equals the Fréchet filter of cofinite subsets of $\mathbb N$. We determine combinatorial properties for the filter $\mathcal F$ which ensure that these uniform $\mathcal F$-boundedness principles hold for every Fréchet space. Furthermore, for several types of Fréchet spaces, we also isolate properties of $\mathcal F$ that are necessary for the validity of these uniform $\mathcal F$-boundedness principles. For every infinite-dimensional Banach space $X$, we obtain in this way exact combinatorial characterisations of those filters $\mathcal F$ for which the corresponding uniform $\mathcal F$-boundedness principles hold true for $X$.

math.FA↗