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Soumalya Joardar

Publications and source records attributed to Soumalya Joardar.

At least 19 recordsLinked to original sources
arXiv2026

Continuous family of compact quantum metric space structures from cocycle twisted crossed product $\textrm{C}^{\ast}$-algebras

We establish the existence of a three-parameter family of compact quantum metric space structures on cocycle twisted crossed products by discrete groups. We are mainly interested in the case where the acting group has exponential/subexponential growth. We prove that the family is jointly continuous with respect to the parameters when the acting group is exact. We obtain quantitative upper and lower bounds for the associated metric dimensions. In particular, the bounds are helpful to prove the failure of lower semicontinuity of the metric dimension with respect to the quantum Gromov-Hausdorff distance. We also prove invariance of metric dimension under zero quantum Gromov-Hausdorff distance.

math.OA↗
arXiv2026

Metric dimension of $C^{\ast}$-algebras of cocycle twisted transformation groupoids: Growth and dynamical complexity

We consider a natural CQMS structure on a twisted transformation groupoid $C^{\ast}$-algebra coming from stratified $_{\text {c}}$Lip-norm introduced by Austad. We obtain upper bounds of metric dimension of reduced $C^{\ast}$-algebra of a transformation groupoid $Γ\rtimes X$ and its cocycle twist for a suitably chosen CQMS structure, provided $(X,d)$ is a compact metric space of finite Kolmogorov dimension and $Γ$ is a discrete group of polynomial growth. When $Γ$ has exponential growth, we prove that the dimension is generically $+\infty$ proving that the dichotomy between polynomial growth and exponential growth of groups survive even after considering cocycle twists of transformation groupoids.

math.OA↗
arXiv2026

Metric dimension and product entropy of group $C^{\ast}$-algebras

We consider reduced group $C^{\ast}$-algebras of finitely generated discrete groups metrized by seminorms obtained from word length functions. We study the metric dimensions of such $C^{\ast}$-algebras as defined by David Kerr. We also study the product entropy of the automorphisms of group $C^{\ast}$-algebras induced by the automorphisms of the underlying groups. We get a lower bound and an upper bound of the product entropy of an automorphism in terms of the classical group theoretic algebraic and geometric entropy of the automorphisms, provided the group has polynomial growth property. For groups with exponential growth, we show that the metric dimension of the group $C^{\ast}$-algebras is generically $+\infty$.

math.OA↗
arXiv2025

On outer automorphisms of certain graph $C^{*}$-algebras

Given a countable abelian group $A$, we construct a row finite directed graph $Γ(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(Γ(A))$ is canonically isomorphic to $A$. Moreover, each element of $\textrm {Aut}(A)$ is a lift of an automorphism of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(Γ(A))$.

math.OA↗
arXiv2025

Metrics on $C^{\ast}$-algebras of Étale groupoids from length functions

We show that for an étale groupoid with compact unit space, the natural Dirac type operator from a continuous length function produces a natural pseudo-metric on the state space of the corresponding reduced $C^{\ast}$-algebra. For a transformation groupoid with a continuous, proper length function with rapid decay, the state space decomposes into genuine metric spaces with a uniform finite diameter fibred over the state space of the compact unit space. Moreover, when the unit space of the transformation groupoid has finitely many points, the metric on each fibre metrizes the weak*-topology.

math.OA↗
arXiv2024

Isometric actions of compact quantum groups on graph C*-algebras

For a strongly connected directed graph $Λ$, it is shown that the quantum automorphism group of $Λ$ acts isometrically in the sense of D. Goswami et al. on the spectral triple $(C^{\ast}(Λ), L^{2}(Λ^{\infty},M),D)$ constructed by Farsi et al.. It is also shown that the natural action of $U_{n}^{+}$ is not isometric on the Cuntz algebra $\mathcal{O}_{n}$.

math.OA↗
arXiv2024

Quantum isomorphism of $2$-graphs

We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a `non-trivial' example of a pair of $2$-graphs that are not quantum isomorphic to each other.

math.OA↗
arXiv2024

Braided quantum symmetries of graph $\mathrm{C}^*$-algebras

We prove the existence of a universal braided compact quantum group acting on a graph $\mathrm{C}^*$-algebra in the category of $\mathbb{T}$-$\mathrm{C}^*$-algebras with a twisted monoidal structure, in the spirit of the seminal work of S. Wang. To achieve this, we construct a braided analogue of the free unitary quantum group and study its bosonization. As a concrete example, we compute this universal braided compact quantum group for the Cuntz algebra.

math.OA↗
arXiv2024

Operator on Operator Regression in Quantum Probability

This article introduces operator on operator regression in quantum probability. Here in the regression model, the response and the independent variables are certain operator valued observables, and they are linearly associated with unknown scalar coefficient (denoted by $β$), and the error is a random operator. In the course of this study, we propose a quantum version of a class of estimators (denoted by $M$ estimator) of $β$, and the large sample behaviour of those quantum version of the estimators are derived, given the fact that the true model is also linear and the samples are observed eigenvalue pairs of the operator valued observables.

stat.ME↗
arXiv2024

Twisted edge Laplacians on finite graphs from a Kähler structure

In this paper we study a Kahler structure on finite points. In particular, we study the edge Laplacian of a graph twisted by the Kahler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic Dolbeault-Dirac spectral triple and show that the points have a finite diameter with respect to Connes' distance.

math.QA↗
arXiv2024

Equivariant $\mathrm{C}^*$-correspondences and compact quantum group actions on Pimsner algebras

Let $G$ be a compact quantum group. We show that given a $G$-equivariant $\mathrm{C}^*$-correspondence $E$, the Pimsner algebra $\mathcal{O}_E$ can be naturally made into a $G$-$\mathrm{C}^*$-algebra. We also provide sufficient conditions under which it is guaranteed that a $G$-action on the Pimsner algebra $\mathcal{O}_E$ arises in this way, in a suitable precise sense. When $G$ is of Kac type, a $\mathrm{KMS}$ state on the Pimsner algebra, arising from a quasi-free dynamics, is $G$-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is $G$-equivariant, under a natural condition. We apply these results to the situation when the $\mathrm{C}^*$-correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.

math.OA↗
arXiv2023

Almost complex structure on finite points from bidirected graphs

We show that there is an almost complex structure on a differential calculus on finite points coming from a bidirected finite graph without multiple edges or loops. We concentrate on a polygon as a concrete case. In particular, a `holomorphic structure on the exterior bundle' built from the polygon is studied. Also a positive Hochschild 2-cocycle on the vertex set of the polygon, albeit a trivial one, is shown to arise naturally from the almost complex structure.

math.QA↗
arXiv2022

Levi-Civita connections from toral actions

We construct tame differential calculi coming from toral actions on a class of $\mathrm{C}^*$-algebras. Relying on the existence of a unique Levi-Civita connection on such a calculus, we prove a version of the Bianchi identity. A Gauss-Bonnet theorem for the canonical tame calculus of rank two is studied.

math.QA↗
arXiv2021

A new look at Levi-Civita connection in noncommutative geometry

We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in references 9 and 10. Star-compatibility of Levi-Civita connections for bilinear pseudo-Riemannian metrics are also discussed.

math.QA↗
arXiv2021

Levi-Civita connections for conformally deformed metrics on tame differential calculi

Given a tame differential calculus over a noncommutative algebra $\mathcal{A}$ and an $\mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$ consider the conformal deformation $ g = k. g_0, $ $k$ being an invertible element of $\mathcal{A}.$We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the pseudo-Riemannian metric $g_0.$ As an application, we compute the Ricci and scalar curvature for a general conformal perturbation of the canonical metric on the noncommutative $2$-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

math.QA↗
arXiv2019

Scalar Curvature of a Levi-Civita Connection on Cuntz algebra with three generators

A differential calculus on Cuntz algebra with three generators coming from the action of rotation group in three dimensions is introduced. The differential calculus is shown to satisfy Assumptions I-IV of [1] so that Levi-Civita Connection exists uniquely for any pseudo-Riemannian metric in the sense of [1]. Scalar curvature is computed for the Levi-Civita connection corresponding to the canonical bilinear metric.

math.OA↗
arXiv2018

An example of explicit dependence of quantum symmetry on KMS states

We compute all the quantum symmetries of a graph with n- disjoint loops at the critical inverse temperature. We show that the set of non-isomorphic CQG's appearing as quantum symmetry at the critical inverse temperature has a one to one correspondence with the cardinality of the set of partitions of the number of vertices.

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