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Asger Tornquist

Publications and source records attributed to Asger Tornquist.

17 recordsLinked to original sources
arXiv2025

The happy coexistence of mad families and Laver measurability

Let $x$ denote a Laver real over $L$. We prove that in $L[x]$ there is a $\Pi^1_1$ infinite mad family. Since $\Pi^1_1$ and $\Sigma^1_2$ sets are Laver measurable in $L[x]$, this shows that there are examples of well-behaved classical pointclasses $\Gamma$, namely $\Gamma=\Pi^1_1$ and $\Gamma=\Sigma^1_2$, where $\Gamma$-uniformization and ``all sets in $\Gamma$ are Laver measurable'' hold, but there is a mad family in $\Gamma$. This result stands in contrast to that for reasonable pointclasses, the $\Gamma$-Ramsey property together with uniformization implies that there are no mad families in $\Gamma$.

math.LO↗
arXiv2015

Definability and almost disjoint families

We show that there are no infinite maximal almost disjoint ("mad") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at $κ<2^{\aleph_0}$, then no $κ$-Souslin infinite almost disjoint family can be maximal. Finally we show that if $\aleph_1^{L[a]}<\aleph_1$, then there are no $Σ^1_2[a]$ infinite mad families.

math.LO↗
arXiv2012

The Borel complexity of von Neumann equivalence

We prove that for a countable discrete group $Γ$ containing a copy of the free group $\F_n$, for some $2\leq n\leq\infty$, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of the ergodic a.e. free actions of $Γ$ are analytic non-Borel equivalence relations in the Polish space of probability measure preserving $Γ$ actions. As a consequence we obtain that the isomorphism relation in the spaces of separably acting factors of type $\II_1$, $\II_\infty$ and $\III_λ$, $0\leqλ\leq 1$, are analytic and not Borel when these spaces are given the Effros Borel structure.

math.DS↗
arXiv2012

Turbulence, orbit equivalence, and the classification of nuclear C*-algebras

We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C*-algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra O_2 on its closed subsets. The same bounds are obtained for affine homeomorphism of metrizable Choquet simplexes. As a by-product we recover a result of Kechris and Solecki, namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish group action. These results depend intimately on the classification theory of nuclear simple C*-algebras by K-theory and traces. Both of necessity and in order to lay the groundwork for further study on the Borel complexity of C*-algebras, we prove that many standard C*-algebra constructions and relations are Borel, and we prove Borel versions of Kirchberg's O_2-stability and embedding theorems. We also find a C*-algebraic witness for a K_σhard equivalence relation.

math.OA↗
arXiv2011

Orbit Equivalence and actions of F_n

In this paper we show that there are "E_0 many" orbit inequivalent free actions of the free groups F_n, $2\leq n\leq\infty$, by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.

math.GR↗
arXiv2011

Projective maximal families of orthogonal measures with large continuum

We study maximal orthogonal families of Borel probability measures on $2^ω$ (abbreviated m.o. families) and show that there are generic extensions of the constructible universe $L$ in which each of the following holds: (1) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families and $\mathfrak{b}=\mathfrak{c}=ω_3$ (in fact any reasonable value of $\mathfrak{c}$ will do). (2) There is a $Δ^1_3$-definable well order of the reals, there is a $Π^1_2$-definable m.o. family, there are no $\mathbfΣ^1_2$-definable m.o. families, $\mathfrak{b}=ω_1$ and $\mathfrak{c}=ω_2$.

math.LO↗
arXiv2009

Turbulence and Araki-Woods factors

Using Baire category techniques we prove that Araki-Woods factors are not classifiable by countable structures. As a result, we obtain a far reaching strengthening as well as a new proof of the well-known theorem of Woods that the isomorphism problem for ITPFI factors is not smooth. We derive as a consequence that the odometer actions of Z that preserve the measure class of a finite non-atomic product measure are not classifiable up to orbit equivalence by countable structures.

math.OA↗
arXiv2009

On the pointwise implementation of near-actions

We show that the continuum hypothesis implies that every measure preserving near-action of a group on a standard Borel probability space $(X,μ)$ has a pointwise implementation by Borel measure preserving automorphisms.

math.LO↗
arXiv2009

Borel reducibility and classification of von Neumann algebras

We announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras.

math.LO↗
arXiv2009

A co-analytic maximal set of orthogonal measures

We prove that if $V=L$ then there is a $Π^1_1$ maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj that no analytic set of measures can be maximal orthogonal.

math.LO↗
arXiv2009

Definable Davies' Theorem

We prove the following analogue of a Theorem of R.O. Davies: Every $Σ^1_2$ function $f:\R\times\R\to\R$ can be represented as a sum of rectangular $Σ^1_2$ functions if and only if all reals are constructible.

math.LO↗
arXiv2009

The Effective Theory of Borel Equivalence Relations

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver and Harrington-Kechris-Louveau show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on $ω$ is above equality on ${\cal P}(ω)$, the power set of $ω$, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on ${\cal P}(ω)$. In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's $O$ as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal $α$ the existence of $Π^0_1$ singletons whose $α$-jumps are Turing incomparable.

math.LO↗
arXiv2009

The classification problem for von Neumann factors

We prove that it is not possible to classify separable von Neumann factors of types $\II_1$, $\II_\infty$ or $\III_λ$, $0\leq λ\leq1$, up to isomorphism by a Borel measurable assignment of "countable structures" as invariants. In particular the isomorphism relation of type $\II_1$ factors is not smooth. We also prove that the isomorphism relation for von Neumann $\II_1$ factors is analytic, but is not Borel.

math.OA↗
arXiv2008

Conjugacy, orbit equivalence and classification of measure preserving group actions

We prove that if $G$ is a countable discrete group with property (T) over an infinite subgroup $H<G$ which contains an infinite Abelian subgroup or is normal, then $G$ has continuum many orbit inequivalent measure preserving a.e. free ergodic actions on a standard Borel probability space. Further, we obtain that the measure preserving a.e. free ergodic actions of such a $G$ cannot be classified up to orbit equivalence be a reasonable assignment of countable structures as complete invariants. We also obtain a strengthening and a new proof of a non-classification result of Foreman and Weiss for conjugacy of measure preserving ergodic, a.e. free actions of discrete countable groups.

math.OA↗