arXiv subjects

Martino Lupini

Publications and source records attributed to Martino Lupini.

At least 19 recordsLinked to original sources

arXiv2026

Solution to Dixmier's Problem about spectra of C*-algebras

We solve Dixmier's 1967 problem about spectra of simple $C^*$-algebras. A function between spectra is called Borel-definable when it is induced by a Borel map between standard Borel spaces of unitary representations. Let $Γ=\mathrm{SL}_3(\mathbb{Z})$, let $K$ be its completion with respect to the congruence kernels modulo $2^n$, let $A$ be the canonical anticommutation relations (CAR) algebra, and let $B=C(K)\rtimes_rΓ$ be the reduced crossed product. The algebras $A$ and $B$ are simple, separable, unital, exact, and antiliminary; $A$ is nuclear, whereas $B$ is nonnuclear. There is no Borel-definable injection from the spectrum of $B$ to the spectrum of $A$. In fact, there is a probability measure on the pure-state space of $B$ such that every Borel lift of a Borel-definable function from the spectrum of $B$ to the spectrum of $A$ takes values in a single unitary-equivalence class almost everywhere. Answering a question of Simon Thomas, we also prove that, for every countable amenable group $H$, there is no Borel-definable injection from the unitary dual of the free group $F_\infty $ on infinitely many generators to the unitary dual of $H$. There is a fixed probability measure on a family of infinite-dimensional irreducible representations of $F_\infty$ such that every Borel lift of a Borel-definable function takes values in a single unitary-equivalence class almost everywhere. We also show that, in contrast, the spectra of any two separable nuclear non-type-I $C^*$-algebras admit a Borel-definable bijection.

math.OA↗

arXiv2026

The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and $\mathbb{G}_r(0)$ is generically monothetic

We prove various properties of the universal valued Abelian groups $\mathbb{G}_r(N)$ constructed by Niemiec, for $r\in\{1,\infty\}$ and $N\in\{0,2,3,\ldots\}$. First we show that the completion of $Γ_0=\bigoplus_{n\in\mathbb{N}}\mathbb{Q}/\mathbb{Z}$ (for $N=0$) or $Γ_N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}/N\mathbb{Z}$ (for $N\geq 2$) with respect to a generic invariant metric, bounded by $1$ when $r=1$, is isometrically group-isomorphic to $\mathbb{G}_r(N)$, recovering a result of Doucha in the case when $r=\infty $ and $N=0$. This confirms an expectation of Doucha for the group $Γ_N$. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of $\mathbb{G}_r(N)$. We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that $\mathbb{G}_r(0)$ is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.

math.GR↗

arXiv2026

Derivation length and automorphism length of unital C*-algebras

This paper is a contribution to the study of the ordinal-valued invariants of derivation and automorphism length. Akemann--Pedersen and Elliott proved that a separable unital C*-algebra has derivation length $0$ if and only if it has automorphism length at most $1/2$ if and only if it is a finite direct sum of homogeneous C*-algebras and simple C*-algebras. Kadison--Lance--Ringrose and Somerset proved that a separable unital C*-algebra has derivation length at most $1$ if and only if it has automorphism length at most $1$ if and only if its primitive spectrum has finite connecting order. In this paper, we prove a complete comparison between the two lengths: either \begin{equation*} \ell _{\mathrm{Aut}}\left( A\right) =\ell _{\mathrm{aut}}\left( A\right) \end{equation*} or, for some countable ordinal $α$ other than $1$ or a limit ordinal, \begin{equation*} \ell _{\mathrm{aut}}\left( A\right) =α\quad \text{and}\quad \ell _{\mathrm{Aut}}\left( A\right) =α+1/2\text{.} \end{equation*}

math.OA↗

arXiv2026

Applications of Borel-definable homological algebra to locally compact groups

We apply the description of the left heart of the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups in terms of groups with a Polish cover and Borel-definable group homomorphisms to study injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left hearts of its thick subcategories. In particular, we prove that the left hearts of the following categories have no nonzero injective objects: locally compact Polish abelian groups, compactly generated locally compact Polish abelian groups, locally compact Polish abelian groups of finite ranks, abelian Lie groups, totally disconnected locally compact Polish abelian groups, topological torsion locally compact Polish abelian groups, locally compact Polish abelian topological $p$-groups for any prime $p$.

math.GR↗

arXiv2026

The Borel complexity of non-Archimedean operator ranges

We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely $\boldsymbol{Π}_{1+λ+n+2}^{0}$ for a countable ordinal $λ$ that is either zero or a limit ordinal, and a finite ordinal $n$. Considering Fréchet spaces produces the additional complexity classes $\boldsymbol{Π}_{λ}^{0}$ for a countable limit ordinal $λ$.

math.FA↗

arXiv2026

The Exact Completion of the Category of Polish Groups

This article shows that the exact completion as a regular category of the homological category of Polish groups and continuous group homomorphisms is the category of groups with a Polish cover and Borel-definable group homomorphisms. We also obtain several equivalent characterizations for the morphisms in the exact completion. An analogous description is deduced for several important subcategories of the category of Polish groups.

math.CT↗

arXiv2026

A general Universal Coefficient Theorem, and applications

In this paper we isolate a general Universal Coefficient Theorem in the context of abelian categories. We then apply it in the left heart of the quasi-abelian category of abelian Polish groups to obtain short exact sequences relating Steenrod homology and Čech cohomology, regarded as functors to such a category. These are then used to (1) intrinsically characterize the phantom subgroups of Čech cohomology via a recursive purely algebraic formula, which can be see as a higher order generalization of the Milnor exact sequence; (2) describe phantom subgroups of Čech cohomology in terms of higher order phantom maps, in the context of the homotopical description of Čech cohomology; (3) classify up to homotopy (phantom) maps on higher order versions of solenoid complements, and measure from the viewpoint of Borel complexity theory the complexity of such a classification problem.

math.AT↗

arXiv2026

Polarities, voltages, and capacitors: a categorical approach to hulls, envelopes, and completions

This article provides a general framework in the context of category theory where one can recognize as particular instances of the same abstract construction several notions of completion, envelope, and hull, such as the Boolean algebra completion of a Boolean algebra, the Dedekind--MacNeille completion of an ordered set, the multiplier ring of a ring, the multiplier algebra and the von Neumann envelope of a C*-algebra. Towards our goal, we lay the foundations of \emph{polarized} category theory, which is a refinement of classical category theory where categories are endowed with two distinguished classes of \emph{positive} and \emph{negative} arrows. We define in this context the notion of \emph{polarity}, and \emph{voltage}. We explain how a voltage can be created through a \emph{capacitor}, which is essentially a polarized version of the notion of reflective subcategory. In particular, this produces a \emph{completion functor} (which in the classical case is just the reflector) which assigns to each object its completion or hull. These applies even when the completion is not (and cannot be) given by a functor on the whole category, as it is most often the case. In this framework, we obtain a general theorem ensuring the existence and uniqueness of a functorial completion functor. The corresponding completion of each object is characterized by its two universal properties with respect to positive and negative arrows.

math.CT↗

arXiv2026

A new characterization of (pre)liminary C*-algebras

Given an arbitrary countable ordinal $α$, we introduce the notion of type $I_{α}$ C*-algebra and $α$-subhomogeneous C*-algebra. When $α=0$, these recover the notions of Fell C*-algebra and of commutative C*-algebra, respectively. When $α=n<ω$, these recover the notions of type $I_{n}$ C*-algebra and of $n$-subhomogeneous C*-algebra, respectively. We prove that a separable C*-algebra is liminary if and only if it is type $I_{α}$ for some $α<ω_{1}$, and it is preliminary (i.e., has no infinite-dimensional irreducible representation) if and only if it is $α$-subhomogeneous for some $α<ω_{1}$. We also prove that for any countable ordinal $α$ there exists a separable C*-algebra that is type $I_{α}$ and not type $I_{β}$ for $β<α$, and a separable C*-algebra that is $α$-subhomogeneous and not $β$-subhomogeneous for any $β<α$.

math.OA↗

arXiv2025

Projective length, phantom extensions, and the structure of torsion modules

The notion of phantom extension of order a given ordinal $α$ has been introduced in collaboration with Casarosa, as an algebraic analogue of the order of a phantom map in topology, to study the structure of flat modules. In this companion paper we characterize phantom extension of \emph{torsion} modules over a countable Dedekind domain $R$. After localizing, one can assume that $R$ is a discrete valuation domain with maximal ideal generated by $p\in R$. In this case, the phantom extensions of order $α$ of a countable torsion module are precisely the $p^{ω\left( 1+α\right) }$-pure extensions introduced by Nunke in the 1960s. A module has projective length at most $α$ if and only if it is a projective object with respect to the exact structure defined by phantom extensions of order $α$. We prove that a countable torsion module has projective length at most $α$ if and only if it is reduced and has Ulm length at most $1+α$, if and only if it is the colimit of a presheaf of finite torsion modules over a countable well-founded forest of rank at most $1+α$.

math.GR↗

arXiv2025

Projective length, phantom extensions, and the structure of flat modules

We consider the natural generalization of the notion of the order of a phantom map from the topological setting to triangulated categories. When applied to the derived category of the category of countable flat modules over a countable Dedekind domain, this yields a notion of\emph{\ phantom extension} of order $α<ω_{1}$. We provide a complexity-theoretic characterization of the module $\mathrm{Ph}% ^{α}\mathrm{Ext}\left( C,A\right) $ of phantom extensions of order $α$ with respect to the structure of \emph{phantom Polish module} on $\mathrm{Ext}\left( C,A\right) $ obtained by considering it as an object of the left heart of the quasi-abelian category of Polish modules. We use this characterization to prove the following Dichotomy Theorem: either all the extensions of a countable flat module $A$ are trivial (which happens precisely when $A$ is divisible) or $A$ has phantom extensions of arbitrarily high order. By producing canonical phantom projective resolutions of order $α$, we prove that phantom extensions of order $α$ define on the category of countable flat modules an exact structure $\mathcal{E}_{α}$ that is hereditary with enough projectives, and the functor $\mathrm{Ph}^{α}% \mathrm{Ext}$ is the derived functor of $\mathrm{Hom}$ with respect to $\mathcal{E}_{α}$. We prove a Structure Theorem characterizing the objects of the class $\mathcal{P}_{α}$ of countable flat modules that have \emph{projective length at most }$α$ (i.e., are $\mathcal{E}_{α}$-projective) as the direct summands of colimits of presheaves of finite flat modules over well-founded forests of rank $% 1+α$ regarded as ordered sets. This can be seen as the first analogue in the flat case of the classical Ulm Classification Theorem for torsion modules.

math.LO↗

arXiv2025

The classification problem for extensions of torsion-free abelian groups, I

Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free $R$-module for some subring $R\subseteq \mathbb{Q}$. Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation $\mathcal{R}_{\mathbf{Ext}\left( C,A\right) }$ of isomorphism of extensions of $C$ by $A$. This complements a previous result by the same author, settling the case when $C$ is torsion and $A$ is arbitrary. We establish the main result within the framework of Borel-definable homological algebra, recently introduced in collaboration with Bergfalk and Panagiotopoulos. As a consequence of our main results, we will obtain that if $C$ is torsion-free and $A$ is either a free $R$-module or a torsion group with bounded components, then an extension of $C$ by $A$ splits if and only if it splits on all finite-rank subgroups of $C$. This is a purely algebraic statements obtained with methods from Borel-definable homological algebra.

math.AC↗

arXiv2025

Homological algebra of pro-Lie Polish abelian groups

In this paper, we initiate the study of pro-Lie Polish abelian groups from the perspective of homological algebra. We extend to this context the type-decomposition of locally compact Polish abelian groups of Hoffmann and Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish abelian groups is a thick subcategory of the category of Polish abelian groups. We completely characterize injective and projective objects in $\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough projectives but not enough injectives and homological dimension $1$. We also completely characterize injective and projective objects in the category of non-Archimedean Polish abelian groups, concluding that it has enough injectives and projectives and homological dimension $1$. Injective objects are also characterized for the categories of topological torsion Polish abelian groups and for Polish abelian topological $p$-groups, showing that these categories have enough injectives and homological dimension $1$.

math.AC↗

arXiv2024

The definable content of homological invariants I: $\mathrm{Ext}$ & $\mathrm{lim}^1$

This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category, introduced herein, of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $Λ$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $Λ$ with isomorphic classical invariants $\textrm{Ext}(Λ,\mathbb{Z}) $. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(Λ)\curvearrowright\mathrm{Ext}(Λ,\mathbb{Z}))$ of classifying all group extensions of $Λ$ by $\mathbb{Z}$ up to base-free isomorphism, when $Λ=\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $ d\geq 1$.

math.LO↗

arXiv2024

The definable content of homological invariants II: Čech cohomology and homotopy classification

This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from $\mathsf{GPC}$ to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of $d$-spheres or $d$-tori for any $d\geq 1$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors $\check{\mathrm{H}}^n_{\mathrm{def}}$ to show that a seminal problem in the development of algebraic topology, namely Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement $S^3\backslashΣ$ to the $2$-sphere, is essentially hyperfinite but not smooth. In the course of this work, we record Borel definable versions of a number of classical results bearing on both the combinatorial and homotopical formulations of Čech cohomology; in aggregate, this work may be regarded as laying foundations for the descriptive set theoretic study of the homotopy relation on the space of maps from a locally compact Polish space to a polyhedron, a relation which embodies a substantial variety of classification problems arising throughout mathematics.

math.LO↗

arXiv2024

Definable (co)homology, pro-torus rigidity, and (co)homological classification

We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable spaces up to homotopy. In particular, we show that pro-tori are completely classified up to homeomorphism by their definable homology. This is in contrast with the fact that, for example, there exist uncountably many pairwise non-homeomorphic solenoids with the same Steenrod homology groups. We similarly develop a definable cohomology theory which strengthens Čech cohomology and we show that it completely classifies complements of pro-tori up to homeomorphism. We also apply definable cohomology theory to the study of the space $\left[ X,S^{2}\right] $ of homotopy classes of continuous functions from a solenoid complement $X$ to the $2$-sphere, which was initiated by Borsuk and Eilenberg in 1936. It was proved by Eilenberg and Steenrod in 1940 that the space $\left[ X,S^{2}\right] $ is uncountable. We will strengthen this result, by showing that each orbit of the canonical action $\mathrm{Homeo}% \left( X\right) \curvearrowright \left[ X,S^{2}\right] $ is countable, and hence that such an action has uncountably many orbits. This can be seen as a rigidity result, and will be deduced from a rigidity result for definable automorphisms of the Čech cohomology of $X$. We will also show that these results still hold if one replaces solenoids with pro-tori. We conclude by applying the machinery developed herein to bound the Borel complexity of several well-studied classification problems in mathematics, such as that of automorphisms of continuous-trace $C^{*}$-algebras up to unitary equivalence, or that of Hermitian line bundles, up to isomorphism, over a locally compact second countable space.

math.AT↗

arXiv2023

(Looking For) The Heart of Abelian Polish Groups

We prove that the category $\mathcal{M}$ of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category $\mathcal{A}$ of abelian Polish groups in the sense of Beilinson--Bernstein--Deligne and Schneiders. Thus, $\mathcal{M}$ is an abelian category containing $\mathcal{A}$ as a full subcategory such that the inclusion functor $\mathcal{A}\rightarrow \mathcal{M}$ is exact and finitely continuous. Furthermore, $\mathcal{M}$ is uniquely characterized up to equivalence by the following universal property: for every abelian category $\mathcal{B}$, a functor $\mathcal{A}\rightarrow \mathcal{B}$ is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor $\mathcal{M}\rightarrow \mathcal{B}$. In particular, this provides a description of the left heart of $\mathcal{A}$ as a concrete category. We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish $R$-modules, for a given Polish group or Polish ring $R$; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

math.LO↗

arXiv2022

Complexity classes of Polishable subgroups

In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah--Solecki. In particular, we obtain a characterization of such canonical approximations in terms of their Borel complexity class. As an application we provide a complete list of all the possible Borel complexity classes of Polishable subgroups of Polish groups or, equivalently, of the ranges of continuous group homomorphisms between Polish groups. We also provide a complete list of all the possible Borel complexity classes of the ranges of: continuous group homomorphisms between non-Archimedean Polish groups; continuous linear maps between separable Fréchet spaces; continuous linear maps between separable Banach spaces.

math.LO↗