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Martino Lupini

Publications and source records attributed to Martino Lupini.

At least 55 records · Page 3Linked to original sources
arXiv2017

The classification problem for operator algebraic varieties and their multiplier algebras

We study from the perspective of Borel complexity theory the classification problem for multiplier algebras associated with operator algebraic varieties. These algebras are precisely the multiplier algebras of irreducible complete Nevanlinna-Pick spaces. We prove that these algebras are not classifiable up to algebraic isomorphism using countable structures as invariants. In order to prove such a result, we develop the theory of turbulence for Polish groupoids, which generalizes Hjorth's turbulence theory for Polish group actions. We also prove that the classification problem for multiplier algebras associated with varieties in a finite dimensional ball up to isometric isomorphism has maximum complexity among the essentially countable classification problems. In particular, this shows that Blaschke sequences are not smoothly classifiable up to conformal equivalence via automorphisms of the disc.

math.OA↗
arXiv2017

An intrinsic order-theoretic characterization of the weak expectation property

We prove the following characterization of the weak expectation property for operator systems in terms of Wittstock's matricial Riesz separation property: an operator system $S$ satisfies the weak expectation property if and only if $M_{q}(S)$ satisfies the matricial Riesz separation property for every $q\in \mathbb{N}$. This can be seen as the noncommutative analog of the characterization of simplex spaces among function systems in terms of the classical Riesz separation property.

math.OA↗
arXiv2017

A von Neumann algebra characterization of property (T) for groupoids

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusion $L\left( H\right) \subset L\left( G\right) $.

math.OA↗
arXiv2017

A universal nuclear operator system

By means of Fraïssé theory for metric structures developed by Ben Yaacov, we show that there exists a separable $1$-exact operator system $\mathbb{GS}$---which we call the Gurarij operator system---of almost universal disposition. This means that whenever $E\subset F$ are finite-dimensional $1$-exact operator systems, $ϕ:E\rightarrow \mathbb{GS}$ is a unital complete isometry, and $\varepsilon >0$, there is a linear extension $\widehat{ϕ}:F\rightarrow \mathbb{GS}$ of $ϕ$ such that $||\widehat{ϕ}||_{cb}{}||\widehat{ϕ}^{-1}||_{cb}\leq 1+\varepsilon $. Such an operator system is unique up to complete order isomorphism. Furthermore it is nuclear, homogeneous, and any separable $1$-exact operator system admits a complete order embedding into $\mathbb{GS}$. The space $\mathbb{GS}$ can be regarded as the operator system analog of the Gurarij operator space $\mathbb{NG}$ introduced by Oikhberg, which is in turn a canonical operator space structure on the Gurarij Banach space. We also show that the canonical $\ast $-homomorphism from the universal C*-algebra of $\mathbb{GS}$ to the C*-envelope of $\mathbb{GS}$ is a $\ast $-isomorphism. This implies that $\mathbb{GS}$ does not admit any complete order embedding into a unital exact C*-algebra. In particular $\mathbb{GS}$ is not completely order isomorphic to a unital C*-algebra. With similar methods we show that the Gurarij operator space $\mathbb{NG}$ does not admit any completely isometric embedding into an exact C*-algebra, and in particular $\mathbb{NG}$ is not completely isometric to a C*-algebra. This answers a question of Timur Oikhberg.

math.OA↗
arXiv2016

Actions of trees on semigroups, and an infinitary Gowers--Hales--Jewett Ramsey theorem

We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales--Jewett theorems (for both located and nonlocated words), and the Farah--Hindman--McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken--Taylor theorem of Bergelson--Hindman--Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of delta sets in amenable groups.

math.CO↗
arXiv2016

Fraïssé limits of C*-algebras

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.

math.LO↗
arXiv2016

Weak equivalence of stationary actions and the entropy realization problem

We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and moreover that it descends to a continuous function on the space of weak equivalence classes.

math.DS↗
arXiv2016

Fraïssé limits in functional analysis

We provide a unified approach to Fra\"ıssé limits in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogs. We obtain in this general framework many known and new results about the Gurarij space and the Poulsen simplex, and at the same time establish their noncommutative analogs. Particularly, we construct noncommutative analogs of universal operators in the sense of Rota.

math.FA↗
arXiv2016

Gowers' Ramsey theorem for generalized tetris operations

We prove a generalization of Gowers' theorem for $\mathrm{FIN}_{k}$ where, instead of the single tetris operation $T:\mathrm{FIN}_{k}\rightarrow \mathrm{FIN}_{k-1}$, one considers all maps from $\mathrm{FIN}_{k}$ to $\mathrm{FIN}_{j}$ for $0\leq j\leq k$ arising from nondecreasing surjections $f:\left\{ 0,1,\ldots ,k+1\right\} \rightarrow \left\{ 0,1,\ldots ,j+1\right\} $. This answers a question of Bartošová and Kwiatkowska. We also prove a common generalization of such a result and the Galvin--Glazer--Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.

math.CO↗
arXiv2015

A López-Escobar theorem for metric structures, and the topological Vaught conjecture

We show that a version of López-Escobar's theorem holds in the setting of logic for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let $\mathrm{Mod}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\mathrm{Iso}(\mathbb{U})$-invariant Borel function $f\colon \mathrm{Mod}(\mathcal{L}, \mathbb{U})\rightarrow \lbrack 0,1]$, there exists a sentence $ϕ$ of $\mathcal{L}_{ω_{1}ω}$ such that for all $M\in \mathrm{Mod}(\mathcal{L},\mathbb{U})$ we have $f(M)=ϕ^{M}$. At the same time we introduce a variant $\mathcal{L}_{ω_1ω}^\ast$ of $\mathcal{L}_{ω_1ω}$ in which the usual quantifiers are replaced with category quantifiers, and establish the analogous theorem for $\mathcal{L}_{ω_1ω}^\ast$. This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group action is Borel isomorphic to the isomorphism relation on the set of models of a given $\mathcal{L}_{ω_{1}ω}$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.

math.LO↗
arXiv2015

Approximate polynomial structure in additively large sets

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on subsets of the natural numbers that imply the existence of approximate powers of arithmetic progressions are developed and explored.

math.CO↗
arXiv2015

High density piecewise syndeticity of product sets in amenable groups

M. Beiglböck, V. Bergelson, and A. Fish proved that if $G$ is a countable amenable group and $A$ and $B$ are subsets of $G$ with positive Banach density, then the product set $AB$ is piecewise syndetic. This means that there is a finite subset $E$ of $G$ such that $EAB$ is thick, that is, $EAB$ contains translates of any finite subset of $G$. When $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of $% EAB$. When $G=\mathbb{Z}^d$, this result was first proven by the current set of authors using completely different techniques.

math.CO↗
arXiv2015

Strongly productive ultrafilters on semigroups

We prove that if S is a commutative semigroup with well founded universal semilattice or a solvable inverse semigroup with well founded semilattice of idempotents, then every strongly productive ultrafilter on S is idempotent. Moreover we show that any very strongly productive ultrafilter on the free semigroup with countably many generators is sparse, answering a question of Hindman and Legette Jones.

math.LO↗
arXiv2015

Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture

Sofic and hyperlinear groups are the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several longstanding conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II_1 factors. In this case the famous conjecture due to Connes (commonly known as the Connes embedding conjecture) that any II_1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II_1 factors, and stands out today as one of the major open problems in the field. The aim of these notes is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and the Connes embedding conjecture. The presentation is nonetheless self contained and accessible to any student or researcher with a graduate level mathematical background. An appendix by V. Pestov provides a pedagogically new introduction to the concepts of ultrafilters, ultralimits, and ultraproducts for those mathematicians who are not familiar with them, and aiming to make these concepts appear very natural.

math.GR↗
arXiv2015

Model-theoretic aspects of the Gurarij operator system

We establish some of the basic model theoretic facts about the Gurarij operator system $\mathbb{GS}$ recently constructed by the second-named author. In particular, we show: (1) $\mathbb{GS}$ is the unique separable 1-exact existentially closed operator system; (2) $\mathbb{GS}$ is the unique separable nuclear model of its theory; (3) every embedding of $\mathbb{GS}$ into its ultrapower is elementary; (4) $\mathbb{GS}$ is the prime model of its theory; and (5) $\mathbb{GS}$ does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any $q\in \mathbb{N}$, the theories of $M_q$-spaces and $M_q$-systems do have a model companion, namely the Fraïssé limit of the class of finite-dimensional $M_q$-spaces and $M_q$-systems respectively; moreover we show that the model companion is separably categorical. We conclude the paper by showing that no C$^*$ algebra can be existentially closed as an operator system.

math.LO↗
arXiv2015

The classification problem for finitely generated operator systems and spaces

The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for arbitrary separable operator systems and spaces are intractable. On the other hand we show that the finitely generated operator systems and spaces are completely classifiable (or smooth); in fact a finitely generated operator system is classified by its complete theory when regarded as a structure in continuous logic. In the particular case of operator systems generated by a single unitary, a complete invariant is given by the spectrum of the unitary up to a rigid motion of the circle, provided that the spectrum contains at least 5 points. As a consequence of these results we show that the relation on compact subsets of $\mathbb{C}^{n}$, given by homeomorphism via a degree 1 polynomial, is smooth.

math.OA↗
arXiv2015

Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

The Gurarij operator space $\mathbb{NG}$ introduced by Oikhberg is the unique separable $1$-exact operator space that is approximately injective in the category of $1$-exact operator spaces and completely isometric linear maps. We prove that a separable operator space $X$ is nuclear if and only if there exist a linear complete isometry $φ:X\rightarrow \mathbb{NG}$ and a completely contractive projection from $\mathbb{NG}$ onto the range of $φ$. This can be seen as the operator space analog of Kirchberg's nuclear embedding theorem. We also establish the natural operator system analog of Kirchberg's nuclear embedding theorem involving the Gurarij operator system $\mathbb{GS}$.

math.OA↗