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Valeriano Aiello

Publications and source records attributed to Valeriano Aiello.

At least 19 recordsLinked to original sources
arXiv2026

The Cuntz-Pimsner algebra of the simplest Motzkin subproduct system is 2-subhomogeneous

The Motzkin subproduct system (SPS) is constructed from the Jones Wenzl idempotents of the Motzkin algebras, which generalizes the Temperley Lieb SPS. The simplest Motzkin SPS, which is not a Temperley Lieb SPS, is constructed from a 3 dimensional Hilbert space. We explicitly describe the spectrum of the corresponding Cuntz Pimsner algebra, which remarkably admits only irreducible representations of dimensions 1 and 2, and can thus be viewed as a mildly quantum space. We moreover analyse representations of the Richard Thompson groups and of the Cuntz algebra that are associated to this spectrum.

math.OA↗
arXiv2026

Piecewise Linear Equivariant Maps for Compact Groups

Motivated by equivariant neural networks, we study piecewise linear equivariant maps between finite-dimensional real representations of compact groups. We show that all genuinely non-linear piecewise linear behaviour is confined to the subspaces on which the identity component of the group acts trivially, while equivariance forces linearity on the corresponding orthogonal complements. As a consequence, we obtain a compact-group analogue of the finite-group existence criterion of Gibson--Tubbenhauer--Williamson for non-zero equivariant piecewise linear maps between irreducible representations, with the identity component giving rise to a rigidity phenomenon absent from the finite-group case.

math.RT↗
arXiv2026

On the Ryll-Nardzewski Theorem for Quantum Stochastic Processes

We prove a Ryll-Nardzewski Theorem for quantum stochastic processes, that shows that under natural assumptions which generalize the classical probability setting, the distributional symmetries of exchangeability and spreadability are the same. We further show that product states on twisted tensor products of C^*-algebras provide a source of counterexamples to the Ryll-Nardzewski theorem, namely of quantum stochastic processes which are spreadable but not exchangeable. Furthermore, in this setting, we also analyze braidability of product states. We then prove an extended de Finetti Theorem for quantum stochastic processes whose distribution factorizes through twisted tensor products.

math.OA↗
arXiv2026

Remarks on some maximal subgroups of $F$ and on the $\vec{F}$-index of knots

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup \( K_{(2,2)} \) of the Thompson group \( F \), each containing Jones's \( 3 \)-colorable subgroup \( \mathcal{F} \), can be characterized as stabilizer subgroups. Additionally, we show that the \( \vec{F} \)-index, an elementary knot invariant introduced thanks to Jones's construction of knots from Thompson groups, may increase at most by $3$ after changing the orientation of a knot.

math.GT↗
arXiv2025

The Motzkin subproduct system

We introduce a subproduct system of finite-dimensional Hilbert spaces by using the Motzkin planar algebra and its Motzkin Jones-Wenzl idempotents, which generalizes the Temperley-Lieb subproduct system of Habbestad and Neshveyev. We provide a description of the corresponding Toeplitz and Cuntz-Pimsner C$^*$-algebras as universal C$^*$-algebras, defined in terms of generators and relations, and we highlight properties of their representation theory.

math.OA↗
arXiv2024

An extension of Krishnan's central limit theorem to the Brown-Thompson groups

We extend a central limit theorem, recently established for the Thompson group $F=F_2$ by Krishnan, to the Brown-Thompson groups $F_p$, where $p$ is any integer greater than or equal to $2$. The non-commutative probability space considered is the group algebra $\mathbb{C}[F_p]$, equipped with the canonical trace. The random variables in question are $a_n:= (x_n + x_n^{-1})/\sqrt{2}$, where $\{x_i\}_{i\geq 0}$ represents the standard family of infinite generators. Analogously to the case of $F=F_2$, it is established that the limit distribution of $s_n = (a_0 + \ldots + a_{n-1})/\sqrt{n}$ converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones's oriented subgroup $\vec{F}$, such a central limit theorem does not hold.

math.OA↗
arXiv2024

The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part

We study the planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part $\mathcal{E}_{\rm EVEN}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of $F$ isomorphic to $F$, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part $\mathcal{E}_{\rm EVEN}$ of the planar $3$-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence $\mathcal{E}_{\rm EVEN}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with $\mathcal{E}_{\rm EVEN}$: two are shown to be irreducible and one to be reducible.

math.GR↗
arXiv2024

On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$

In his work on representations of Thompson's group $F$, Vaughan Jones defined and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper we continue with the study of the $3$-colorable subgroup and prove that the quasi-regular representation of $F$ associated with the $3$-colorable subgroup is irreducible. We show moreover that the preimage of $\mathcal{F}$ under a certain injective endomorphism of $F$ is contained in three (explicit) maximal subgroups of $F$ of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of $F$, namely the parabolic subgroups that fix a point in $(0,1)$, (up to isomorphism) the Jones' oriented subgroup $\vec{F}$, and the explicit examples found by Golan.

math.GR↗
arXiv2022

A computational study of the number of connected components of positive Thompson links

Almost a decade ago Vaughan Jones introduced a method to produce knots from elements of the Thompson groups $F$, which was later extended to the Brown-Thompson group $F_3$. In this article we define a way to produce permutations out of elements of the $F$ and $F_3$ that we call Thompson permutations. The number of orbits of each Thompson permutation coincides with the number of connected components of the link. We explore the positive elements of $F_3$ of fixed \emph{width} and \emph{height} and make some conjectures based on numerical experiments. In order to define the Thompson permutations we need to assign an orientation to each link produced from elements of $F$ and $F_3$. We prove that all oriented links can be produced in this way.

math.GT↗
arXiv2022

On the oriented Thompson subgroup $\vec{F}_3$ and its relatives in higher Brown-Thompson groups

A few years ago the so-called oriented subgroup $\vec F$ of the Thompson group $F$ was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of $F$ it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, $\vec F$ provided the first example of a maximal subgroup of infinite index in $F$ different from the parabolic subgroups that fix a point in $(0,1)$. In this paper we investigate possible analogues of $\vec F$ in higher Thompson groups $F_k, k\geq 2$, with $F=F_2$, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup $\vec{F}_3$ of $F_3$, as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in $F_3$ and that the corresponding quasi-regular representation is irreducible.

math.GR↗
arXiv2022

On the Jones polynomial modulo primes

We derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$, within a sufficiently large degree range: $4/p^7$. As an application, we classify knot Jones polynomials modulo two of span up to eight.

math.GT↗
arXiv2021

On the entropy and index of the winding endomorphisms of p-adic ring C$^*$-algebras

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $χ_k\in{\rm End}(\mathcal{Q}_p)$ by setting $χ_k(U):=U^k$ and $χ_k(S_p):=S_p$. We then compute the entropy of $χ_k$, which turns out to be $\log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $χ_k$ showing that the entropy is the natural logarithm of the index.

math.OA↗
arXiv2021

Spectral triples on irreversible $C^*$-dynamical systems

Given a spectral triple on a $C^*$-algebra $\mathcal A$ together with a unital injective endomorphism $α$, the problem of defining a suitable crossed product $C^*$-algebra endowed with a spectral triple is addressed. The proposed construction is mainly based on the works of Cuntz and of Hawkins, Skalski, White and Zacharias, and on our previous papers. The embedding of $α(\mathcal A)$ in $\mathcal A$ can be considered as the dual form of a covering projection between noncommutative spaces. A main assumption is the expansiveness of the endomorphism, which takes the form of the local isometricity of the covering projection and is expressed via the compatibility of the Lip-norms on $\mathcal A$ and $α(\mathcal A)$.

math.OA↗
arXiv2021

On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra

We investigate the structure of the fixed-point algebra of $\mathcal{O}_n$ under the action of the cyclic permutation of the generating isometries. We prove that it is $*$-isomorphic with $\mathcal{O}_n$, thus generalizing a result of Choi and Latrémolière on $\mathcal{O}_2$. As an application of the technique employed, we also describe the fixed-point algebra of $\mathcal{O}_{2n}$ under the exchange automorphism.

math.OA↗