arXiv subjects

Roberto Conti

Publications and source records attributed to Roberto Conti.

At least 37 records · Page 2Linked to original sources

arXiv2019

Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebras

Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behaviour at infinity of their matrix coefficients, thus showing that these representations do not contain any finite type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of F. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the first non-trivial index value for which the corresponding subgroup is isomorphic to the Brown-Thompson's group $F_3$, we show that when the index is large enough this subgroup is always trivial.

math.GR↗

arXiv2019

Asymptotic morphisms and superselection theory in the scaling limit II: analysis of some models

We introduced in a previous paper a general notion of asymptotic morphism of a given local net of observables, which allows to describe the sectors of a corresponding scaling limit net. Here, as an application, we illustrate the general framework by analyzing the Schwinger model, which features confined charges. In particular, we explicitly construct asymptotic morphisms for these sectors in restriction to the subnet generated by the derivatives of the field and momentum at time zero. As a consequence, the confined charges of the Schwinger model are in principle accessible to observation. We also study the obstructions, that can be traced back to the infrared singular nature of the massless free field in d=2, to perform the same construction for the complete Schwinger model net. Finally, we exhibit asymptotic morphisms for the net generated by the massive free charged scalar field in four dimensions, where no infrared problems appear in the scaling limit.

math-ph↗

arXiv2018

The Jones polynomial and functions of positive type on the oriented Jones-Thompson groups $\vec{F}$ and $\vec{T}$

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones-Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups $\vec{F}$ and $\vec{T}$. Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on $T$.

math.GR↗

arXiv2018

Graph polynomials and link invariants as positive type functions on Thompson's group F

In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of $F$. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of $N$-colourings and the Tutte polynomial, can be viewed as positive definite functions on $F$.

math.GR↗

arXiv2018

The inner structure of boundary quotients of right LCM semigroups

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,θ)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.

math.OA↗

arXiv2018

Permutative representations of the $2$-adic ring $C^*$-algebra

The notion of permutative representation is generalized to the $2$-adic ring $C^*$-algebra $\mathcal{Q}_{2}$. Permutative representations of $\mathcal{Q}_2$ are then investigated with a particular focus on the inclusion of the Cuntz algebra $\mathcal{O}_2\subset\mathcal{Q}_2$. Notably, every permutative representation of $\mathcal{O}_2$ is shown to extend automatically to a permutative representation of $\mathcal{Q}_2$ provided that an extension whatever exists. Moreover, all permutative extensions of a given representation of $\mathcal{O}_2$ are proved to be unitarily equivalent to one another. Irreducible permutative representations of $\mathcal{Q}_2$ are classified in terms of irreducible permutative representations of the Cuntz algebra. Apart from the canonical representation of $\mathcal{Q}_2$, every irreducible representation of $\mathcal{Q}_2$ is the unique extension of an irreducible permutative representation of $\mathcal{O}_2$. Furthermore, a permutative representation of $\mathcal{Q}_2$ will decompose into a direct sum of irreducible permutative subrepresentations if and only if it restricts to $\mathcal{O}_2$ as a regular representation in the sense of Bratteli-Jorgensen. As a result, a vast class of pure states of $\mathcal{O}_2$ is shown to enjoy the unique pure extension property with respect to the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$.

math.OA↗

arXiv2017

Diagonal automorphisms of the $2$-adic ring $C^*$-algebra

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the group is amounts to understanding when an automorphism of $\mathcal{O}_2$ that fixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.

math.OA↗

arXiv2017

The Homflypt polynomial and the oriented Thompson group

We show how to construct unitary representations of the oriented Thompson group $\vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec{F}$.

math.GR↗

arXiv2017

A look at the inner structure of the $2$-adic ring $C^*$-algebra and its automorphism groups

We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphisms of $\mathcal{Q}_2$ that restrict to the identity on $\mathcal{O}_2$ are actually the identity on the whole $\mathcal{Q}_2$. Moreover, there is no conditional expectation from $\mathcal{Q}_2$ onto $\mathcal{O}_2$. As for the inner structure of $\mathcal{Q}_2$, the diagonal subalgebra $\mathcal{D}_2$ and $C^*(U)$ are both proved to be maximal abelian in $\mathcal{Q}_2$. The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of $\mathcal{Q}_2$. In particular, the semigroup of the endomorphisms fixing $U$ turns out to be a maximal abelian subgroup of ${\rm Aut}(\mathcal{Q}_2)$ topologically isomorphic with $C(\mathbb{T},\mathbb{T})$. Finally, it is shown by an explicit construction that ${\rm Out}(\mathcal{Q}_2)$ is uncountable and non-abelian.

math.OA↗

arXiv2016

Negative definite functions for C*-dynamical systems

Given an action $α$ of a discrete group $G$ on a unital C*-algebra $A$, we introduce a natural concept of $α$-negative definiteness for functions from $G$ to $A$, and examine some of the first consequences of such a notion. In particular, we prove analogs of theorems due to Delorme-Guichardet and Schoenberg in the classical case where $A$ is trivial. We also give a characterization of the Haagerup property for the action $α$ when $G$ is countable.

math.OA↗

arXiv2016

The Fourier-Stieltjes algebra of a C*-dynamical system

In analogy with the Fourier-Stieltjes algebra of a group, we associate to a unital discrete twisted C*-dynamical system a Banach algebra whose elements are coefficients of equivariants representations of the system. Building upon our previous work, we show that this Fourier-Stieltjes algebra embeds continuously in the Banach algebra of completely bounded multipliers of the (reduced or full) C*-crossed product of the system. We also introduce a notion of positive definiteness and prove a Gelfand-Raikov type theorem allowing us to describe the Fourier-Stieltjes algebra of a system in a more intrinsic way. After a study of some of its natural commutative subalgebras, we end with a characterization of the Fourier-Stieltjes algebra involving C*-correspondences over the (reduced or full) C*-crossed product.

math.OA↗

arXiv2014

Categorical Non-commutative Geometry

The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes' non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.

math.OA↗

arXiv2014

On maximal ideals in certain reduced twisted C*-crossed products

We consider a twisted action of a discrete group G on a unital C*-algebra A and give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals of A and the maximal ideals in the associated reduced C*-crossed product.

math.OA↗

arXiv2013

Fourier series and twisted C*-crossed products

This paper is an invitation to Fourier analysis in the context of reduced twisted C*-crossed products associated with discrete unital twisted C*-dynamical systems. We discuss norm-convergence of Fourier series, multipliers and summation processes. Our study relies in an essential way on the (covariant and equivariant) representation theory of C*-dynamical systems on Hilbert C*-modules. It also yields some information on the ideal structure of reduced twisted C*-crossed products.

math.OA↗

arXiv2013

On Conjugacy of MASAs and the Outer Automorphism Group of the Cuntz Algebra

We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of MASAa in O_n. In particular, we exhibit an uncountable family of MASAs, conjugate to the standard MASA D_n via Bogolubov automorphisms, that are not inner conjugate to D_n.

math.OA↗

arXiv2013

The Weyl group of the Cuntz algebra

The Weyl group of the Cuntz algebra O_n, with n finite, is investigated. This is (isomorphic to) the group of polynomial automorphisms of O_n, namely those induced by unitaries that can be written as finite sums of words in the canonical generating isometries and their adjoints. A necessary and sufficient algorithmic combinatorial condition is found for deciding when a polynomial endomorphism restricts to an automorphism of the canonical diagonal MASA. Some steps towards a general criterion for invertibility of such endomorphisms on the whole of O_n are also taken. A condition for verifying invertibility of a certain subclass of polynomial endomorphisms is given. First examples of polynomial automorphisms of O_n not inner related to permutative ones are exhibited, for every n. In particular, the image of the Weyl group in the outer automorphism group of O_n is strictly larger than the image of the reduced Weyl group analyzed in previous papers. Results about the action of the Weyl group on the spectrum of the diagonal are also included.

math.OA↗

arXiv2012

Representations of Conformal Nets, Universal C*-Algebras and K-Theory

We study the representation theory of a conformal net A on the circle from a K-theoretical point of view using its universal C*-algebra C*(A). We prove that if A satisfies the split property then, for every representation πof A with finite statistical dimension, π(C*(A)) is weakly closed and hence a finite direct sum of type I_\infty factors. We define the more manageable locally normal universal C*-algebra C*_ln(A) as the quotient of C*(A) by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if A is completely rational with n sectors, then C*_ln(A) is a direct sum of n type I_\infty factors. Its ideal K_A of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of C*(A) with finite statistical dimension act on K_A, giving rise to an action of the fusion semiring of DHR sectors on K_0(K_A)$. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.

math.OA↗

arXiv2012

Endomorphisms of graph algebras

We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D_E of C*(E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. In particular, criteria of outerness for automorphisms in the restricted Weyl group are found. We also show that the restriction to the diagonal MASA of an automorphism which globally preserves both the diagonal and the core AF-subalgebra eventually commutes with the corresponding one-sided shift. Secondly, we exhibit several properties of proper endomorphisms, investigate invertibility of localized endomorphisms both on C*(E) and in restriction to D_E, and develop a combinatorial approach to analysis of permutative endomorphisms.

math.OA↗