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Martino Lupini

Publications and source records attributed to Martino Lupini.

At least 37 records · Page 2Linked to original sources
arXiv2021

The Ramsey property for Operator spaces and noncommutative Choquet simplices

The noncommutative Gurarij space $\mathbb{\mathbb{\mathbb{NG}}}$, initially defined by Oikhberg, is a canonical object in the theory of operator spaces. As the Fra\"ıssé limit of the class of finite-dimensional nuclear operator spaces, it can be seen as the noncommutative analogue of the classical Gurarij Banach space. In this paper, we prove that the automorphism group of $\mathbb{\mathbb{NG}}$ is extremely amenable, i.e.\ any of its actions on compact spaces has a fixed point. The proof relies on the Dual Ramsey Theorem, and a version of the Kechris--Pestov--Todorcevic correspondence in the setting of operator spaces. Recent work of Davidson and Kennedy, building on previous work of Arveson, Effros, Farenick, Webster, and Winkler, among others, shows that nuclear operator systems can be seen as the noncommutative analogue of Choquet simplices. The analogue of the Poulsen simplex in this context is the matrix state space $\mathbb{NP}$ of the Fra\"ıssé limit $A(\mathbb{NP})$ of the class of finite-dimensional nuclear operator systems. We show that the canonical action of the automorphism group of $\mathbb{NP}$ on the compact set $\mathbb{NP}_1$ of unital linear functionals on $A(\mathbb{NP})$ is minimal and it factors onto any minimal action, whence providing a description of the universal minimal flow of \textrm{Aut}$\left( \mathbb{NP}% \right) $.

math.OA↗
arXiv2021

Computable topological abelian groups

We study the algorithmic content of Pontryagin - van Kampen duality. We prove that the dualization is computable in the important cases of compact and locally compact totally disconnected Polish abelian groups. The applications of our main results include solutions to questions of Kihara and Ng about presentations of connected Polish spaces, and an unexpected arithmetical characterisation of direct products of solenoid groups among all Polish groups.

math.LO↗
arXiv2021

On Rado conditions for nonlinear Diophantine equations

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture hold for the equations $x^{2}-xy+ax+by+cz=0$ and $x^{2}-y^{2}+ax+by+cz=0$ for $a,b,c\in \mathbb{Z}$ such that $abc=0$ or $% a+b+c=0$. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in $\mathbb{Z}$ such as $\left\{ x,x+y,xy+x+y\right\} $, building on the recent result on the partition regularity of $\left\{ x,x+y,xy\right\} $.

math.CO↗
arXiv2020

Definable $\mathrm{K}$-homology of separable C*-algebras

In this paper we show that the $\mathrm{K}$-homology groups of a separable C*-algebra can be enriched with additional descriptive set-theoretic information, and regarded as definable groups. Using a definable version of the Universal Coefficient Theorem, we prove that the corresponding definable $\mathrm{K}$-homology is a finer invariant than the purely algebraic one, even when restricted to the class of UHF C*-algebras, or to the class of unital commutative C*-algebras whose spectrum is a $1$-dimensional connected subspace of $\mathbb{R}^{3}$.

math.OA↗
arXiv2020

Dilation theory in finite dimensions and matrix convexity

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most $1$, and of the Putinar-Sandberg numerical range dilation theorem. As a key tool, we prove versions of Carathéodory's and of Minkowski's theorem for matrix convex sets.

math.FA↗
arXiv2020

The Ramsey property for Banach spaces and Choquet simplices

We show that the Gurarij space $\mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $\mathbb{P}$ and we prove that it consists of the canonical action on $\mathbb{P}$ itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable. These results are obtained via several Kechris-Pestov-Todorcevic correspondences, by establishing the approximate Ramsey property for several classes of finite-dimensional Banach spaces and function systems and their versions with distinguished contractions. This is the first direct application of the Kechris-Pestov-Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.

math.DS↗
arXiv2019

The Ramsey properties for Grassmannians over $\mathbb R$, $\mathbb C$

In this note we study and obtain factorization theorems for colorings of matrices and Grassmannians over $\mathbb{R}$ and ${\mathbb{C}}$, which can be considered metric versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.

math.CO↗
arXiv2019

Uniqueness, universality, and homogeneity of the noncommutative Gurarij space

We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg as the Fra\"ıssé limit of the class of finite-dimensional $1$-exact operator spaces. As a consequence we deduce that the concommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable $1$-exact operator spaces. We also prove that $\mathbb{NG}$ is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal TRO to the triple envelope is an isomorphism. We deduce from this fact that $\mathbb{NG}$ does not embed completely isometrically into an exact C*-algebra, and it is not completely isometrically isomorphic to a C*-algebra or to a TRO. We also provide a canonical construction of $\mathbb{NG}$, which shows that the group of surjective complete isometries of $\mathbb{NG}$ is universal among Polish groups. Analog results are proved in the commutative setting and, more generally, for $M_{n}$-spaces. In particular, we provide a new characterization and canonical construction of the Gurarij Banach space.

math.OA↗
arXiv2018

Games orbits play and obstructions to Borel reducibility

We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.

math.LO↗
arXiv2018

Applications of model theory to C*-dynamics

We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. Finiteness of the order zero dimension implies that the dimension of the target algebra can be bounded by the dimension of the domain. The dimension can be, among others, decomposition rank, nuclear dimension, or Rokhlin dimension. As a consequence, we obtain new inequalities for these quantities. As a third application we obtain the following result: if a C*-algebra $A$ absorbs a strongly self-absorbing C*-algebra $D$, and $α$ is an action of a compact group $G$ on $A$ with finite Rokhlin dimension with commuting towers, then $α$ absorbs any strongly self-absorbing action of $G$ on $% D$. This has a number of interesting consequences, already in the case of the trivial action on $D$. For example, we deduce that $D$-stability passes from $A$ to the crossed product. Additionally, in many cases of interest, our result restricts the possible values of the Rokhlin dimension to $0, 1$ and $\infty$, showing a striking parallel to the behavior of the nuclear dimension for simple C*-algebras. We also show that an action of a finite group with finite Rokhlin dimension with commuting towers automatically has the Rokhlin property if the algebra is UHF-absorbing.

math.OA↗
arXiv2018

Nonlocal Games and Quantum Permutation Groups

We present a strong connection between quantum information and quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs $X$ and $Y$ are quantum isomorphic if and only if there exists $x \in V(X)$ and $y \in V(Y)$ that are in the same orbit of the quantum automorphism group of the disjoint union of $X$ and $Y$. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. We exploit this link by using ideas and results from quantum information in order to prove new results about quantum automorphism groups, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum group literature. Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other.

quant-ph↗
arXiv2018

Actions of nonamenable groups on $\mathcal{Z}$-stable $C^*$-algebras

We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if $G$ contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of $G$ on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann algebras. We moreover show that $G$ is amenable if and only if the Bernoulli shift on a finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods consist in a careful study of weak containments of the Koopman representations of different Bernoulli-type actions.

math.OA↗
arXiv2018

Rokhlin dimension for compact quantum group actions

We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for $G$-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when $G$ is second countable and coexact, to an arbitrary compact quantum group $G$. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

math.OA↗
arXiv2017

Actions of rigid groups on UHF-algebras

Let $Λ$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $Λ$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $Λ$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement: the relations of conjugacy, cocycle conjugacy, and weak cocycle conjugacy of strongly outer actions of $Λ$ on $D$ are complete analytic sets, and in particular not Borel. The same conclusions hold more generally when $Λ$ is only assumed to contain an infinite subgroup with relative property (T), and for actions on (not necessarily simple) separable, nuclear, UHF-absorbing, self-absorbing C*-algebras with at least one trace. Finally, we use the techniques of this paper to construct outer actions on $R$ with prescribed cohomology. Precisely, for every infinite property (T) group $Λ$, and for every countable abelian group $Γ$, we construct an outer action of $Λ$ on $R$ whose 1-cohomology is isomorphic to $Γ$.

math.OA↗
arXiv2017

The complexity of conjugacy, orbit equivalence, and von Neumann equivalence of actions of nonamenable groups

Building on work of Popa, Ioana, and Epstein--Törnquist, we show that, for every nonamenable countable discrete group $Γ$, the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (or weak mixing) measure preserving actions of $Γ$ on the standard atomless probability space are not Borel, thus answering questions of Kechris. This is an optimal and definitive result, which establishes a neat dichotomy with the amenable case, since any two free ergodic actions of an amenable group on the standard atomless probability space are orbit equivalent by classical results of Dye and Ornstein--Weiss. The statement about conjugacy solves the nonamenable case of Halmos' conjugacy problem in Ergodic Theory, originally posed by Halmos in 1956 for ergodic transformations. In order to obtain these results, we study ergodic (or weak mixing) class-bijective extensions of a given ergodic countable probability measure preserving equivalence relation $R$. When $R$ is nonamenable, we show that the relations of isomorphism and von Neumann equivalence of extensions of $R$ are not Borel. When $R$ is amenable, all the extensions of $R$ are again amenable, and hence isomorphic by classical results of Dye and Connes--Feldman--Weiss. This approach allows us to extend the results about group actions mentioned above to the case of nonamenable locally compact unimodular groups, via the study of their cross-section equivalence relations.

math.DS↗
arXiv2017

Boundary representations of operator spaces, and compact rectangular matrix convex sets

We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.

math.OA↗
arXiv2017

Representations of étale groupoids on $L^p$-spaces

For $p\in (1,\infty)$, we study representations of étale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of étale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an étale groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight regular $L^{p}$-representations of any countable inverse semigroup of open slices of $G$ that is a basis for the topology of $G$. We define analogs $F^{p}(G)$ and $F_{\mathrm{red}}^{p}(G)$ of the full and reduced groupoid C*-algebras using representations on $L^{p}$-spaces. As a consequence of our main result, we deduce that every contractive representation of $F^{p}(G)$ or $F_{\mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on $L^{p}$-spaces, and the analogs of AF-algebras on $L^{p}$-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid $L^{p}$-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras.

math.OA↗