arXiv subjects

Satyajit Guin

Publications and source records attributed to Satyajit Guin.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Gromov-Hausdorff Convergence for Extensions of $C^*$-Algebras

We investigate the lifting of quantum Gromov-Hausdorff convergence through Toeplitz type $C^*$-algebra extensions by stable ideals in the framework of noncommutative metric geometry. Working with the spectral metric space construction of Hawkins and Zacharias (Comm. Math. Phys. 350 (2017), 475-506), we consider a sequence of complete sub-operator systems of the quotient or the unital $C^*$-algebra underlying the stable ideal, converging in the quantum Gromov-Hausdorff distance. We study whether this induces a corresponding convergent sequence of complete sub-operator systems of the extension. To address this problem, we construct complete sub-operator systems of the extension associated with those of the quotient and the unital $C^*$-algebra underlying the stable ideal. We also introduce the notion of unital $2$-contractive approximation together with its Toeplitz type refinement to provide the compatibility required by the commutator structure of the Dirac operator on the extension. We prove that, under this approximation hypothesis on the convergent sequence in the quotient or the unital $C^*$-algebra underlying the stable ideal, quantum Gromov-Hausdorff convergence lifts to the extension.

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arXiv2026

Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves

We extend Atiyah's holomorphic jet bundle formalism to holomorphic vector bundles over noncommutative algebras endowed with a bigraded differential calculus truncated at bidegree $(1,1)$; such structures are referred to as noncommutative complex curves. For a holomorphic vector bundle $(E,\,\overline{\nabla}_E)$ over such an algebra $\mathcal{A}$, we construct a canonical holomorphic structure $\overline{\nabla}_J$ on the first jet module $J_E^1\,$, making the jet sequence \[ 0\longrightarrow \Omega^{1,0}(\mathcal{A})\otimes_{\mathcal A}E \longrightarrow J_E^1 \longrightarrow E \longrightarrow 0 \] exact in the holomorphic category. The assignment $(E,\,\overline\nabla_E)\,\rightsquigarrow\,(J_E^1,\,\overline\nabla_J)$ defines an endofunctor on the category of holomorphic vector bundles over $\mathcal{A}$. We define the notion of holomorphic connection in this setting and prove that a holomorphic vector bundle admits a holomorphic connection if and only if the above jet sequence splits in the holomorphic category, or equivalently, if and only if its Atiyah class vanishes. This yields a noncommutative analogue of Atiyah's classical correspondence for Riemann surfaces. Finally, we specialize to the quantum projective line $\mathbb{CP}_q^1\,$ and determine when $\overline{\nabla}_J$ defines a bimodule connection, assuming that $\overline{\nabla}_E$ does so.

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arXiv2026

Fixed points and Holomorphic Structures on Line Bundles over the Quantum Projective Line

It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line are not uniquely determined by degree. We formulate a fixed-point-theoretic framework for the analysis of flat $\bar\partial$-connections that define holomorphic structures on line bundles over the quantum projective line. Within this framework, we relate the existence of invertible solutions to the gauge equation associated with holomorphic structures precisely to the existence of fixed points, lying in the open unit ball, of certain nonlinear maps acting on an appropriate Banach space.

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arXiv2025

Intersection of conjugate Hadamard subfactors arising from Fourier matrices

Given two distinct complex Hadamard matrices belonging to the same equivalence class generated by the tensor products of Fourier matrices, we show that if the corresponding Hadamard subfactors are conjugate, then their intersection is a factor with finite Jones index. We compute the index of the intersection explicitly and determine its relative commutant. Furthermore, we precisely characterize when these intersections give rise to vertex model subfactors, thereby extending our earlier results in low dimensions. As an application, we derive an explicit formula for the Connes-Størmer relative entropy associated with these intersections. These results reveal how the internal algebraic structure of complex Hadamard matrices governs the relative position and entropic behaviour of the subfactors.

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arXiv2025

On two notions of torsion and metric compatibility of connections in noncommutative geometry

We compare the notions of metric-compatibility and torsion of a connection in the frameworks of Beggs-Majid and Mesland-Rennie. It follows that for $\ast$-preserving connections, compatibility with a real metric in the sense of Beggs-Majid corresponds to Hermitian connections in the sense of Mesland-Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland-Rennie for unitary cocycle deformations of a large class of Riemannian manifolds as well as the Heckenberger-Kolb calculi on all quantized irreducible flag manifolds.

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arXiv2025

Higher reflections and entropy of canonical shifts for inclusions of $C^*$-algebras with finite Watatani index

Given a unital inclusion of simple $C^*$-algebras equipped with a conditional expectation of index-finite type, we study Fourier transforms and rotation operators and introduce the reflection operators on the relative commutants. We prove that the reflections are unital, involutive, $*$-preserving anti-homomorphisms that preserve certain Markov-type traces. As an application, we prove Fourier theoretic inequalities on the higher relative commutants and refine the existing constant in Young's inequality as presented in the current literature. By employing the reflection operators, we define a canonical shift on the von Neumann algebra generated by the relative commutants. We establish a connection between the Connes-Størmer entropy of the canonical shift and the minimal Watatani index.

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arXiv2025

A noncommutative construction of families of biunitary matrices and application to subfactors

We introduce a construction that, given a pair (u,v) of complex Hadamard matrices of the same order, generates infinitely many biunitary matrices of varying (and distinct) orders. As a key application, this framework yields nested sequences of vertex model subfactors that are not a tower of downward basic construction. Notably, the construction is noncommutative: interchanging the matrices (i.e., considering (v,u) instead of (u,v)) can lead to non-isomorphic subfactors. Focusing on the Hadamard equivalence class of the Fourier matrix, we provide a full characterization of the resulting vertex model subfactors, along with explicit computations of their relative commutants. Along the way, we conduct a detailed study of certain naturally arising inner and outer automorphisms that play a key role in the structure of these subfactors.

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arXiv2025

On Pimsner-Popa orthonormal basis and Popa's relative dimension of projections

We show that any depth 2 subfactor with a simple first relative commutant has a unitary orthonormal basis. As a pleasant consequence, we produce new elements in the set of Popa's relative dimension of projections for such subfactors. We also construct infinitely many new elements in the set of relative dimension of projections for subfactors arising from complex Hadamard matrices and bi-unitary matrices.

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arXiv2025

Conjugate pairs of Hadamard subfactors and vertex models

We show that any two Hadamard subfactors arising from a pair of distinct complex Hadamard matrices of order 3 are either equal or conjugate by a unitary in the relative commutant of their intersection. Moreover, when the Hadamard subfactors are not equal, we prove the factoriality of their intersection, and it turns out to be a vertex model subfactor. We compute the first relative commutant and characterize this subfactor by identifying it with a particular type of Krishnan-Sunder subfactor. A few key invariants, including the Pimsner-Popa probabilistic number, the angle, and the Connes-Størmer relative entropy for the pair of Hadamard subfactors are computed to understand their relative position.

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arXiv2025

A few remarks on intermediate subalgebras of an inclusion of C*-algebras

We show that the angle between intermediate $C^*$-subalgebras of an inclusion of simple $C^*$-algebras with finite Watatani index is stable. The notion of angle is instrumental in providing a bound for the cardinality of the lattice of intermediate subalgebras for an irreducible inclusion of simple $C^*$-algebras. We improve the existing upper bound for the cardinality of this set.

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arXiv2024

Von Neumann entropy of the angle operator between a pair of intermediate subalgebras

Given a pair of intermediate $C^*$-subalgebras of a unital inclusion of simple $C^*$-algebras with a conditional expectation of finite Watatani index, we discuss the corresponding angle operator and its Fourier transform. We provide a calculable formula for the von Neumann entropy of the (Fourier) dual angle operator for a large class of quadruple of simple $C^*$-algebras.

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arXiv2023

Relative position between a pair of spin model subfactors

Jones proposed the study of two subfactors of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. The Pimsner-Popa probabilistic constant, Sano-Watatani angle, interior and exterior angle, and Connes-Størmer relative entropy (along with a slight variant of it) are a few key invariants for pair of subfactors that analyze their relative position. In practice, however, the explicit computation of these invariants is often difficult. In this article, we provide an in-depth analysis of a special class of two subfactors, namely a pair of spin model subfactors of the hyperfinite type $II_1$ factor $R$. We first characterize when two distinct $n\times n$ complex Hadamard matrices give rise to distinct spin model subfactors. Then, a detailed investigation has been carried out for pairs of (Hadamard equivalent) complex Hadamard matrices of order $2\times 2$ as well as Hadamard inequivalent complex Hadamard matrices of order $4\times 4$. To the best of our knowledge, this article is the first instance in the literature where the exact value of the Pimsner-Popa probabilistic constant and the noncommutative relative entropy for pairs of (non-trivial) subfactors have been obtained. Furthermore, we prove the factoriality of the intersection of the corresponding pair of subfactors using the `commuting square technique'. En route, we construct an infinite family of potentially new subfactors of $R$. All these subfactors are irreducible with Jones index $4n,n\geq 2$. As a consequence, the rigidity of the interior angle between the spin model subfactors is established. Last but not least, we explicitly compute the Sano-Watatani angle between the spin model subfactors.

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arXiv2023

Fourier theoretic inequalities for inclusion of simple C*-algebras

This paper originates from a naive attempt to establish various non-commutative Fourier theoretic inequalities for an inclusion of simple C*-algebras equipped with a conditional expectation of index-finite type. In this setting, we discuss the Hausdorff-Young inequality and Young's inequality. As a consequence, we prove the Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle. Our results generalize some of the results of Jiang, Liu and Wu [Noncommutative uncertainty principle, J. Funct. Anal., 270(1): 264--311, 2016].

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arXiv2022

Equivariant spectral triple for the quantum group $U_q(2)$ for complex deformation parameters

Let $q=|q|e^{iπθ},\,θ\in(-1,1],$ be a nonzero complex number such that $|q|\neq 1$ and consider the compact quantum group $U_q(2)$. For $θ\notin\mathbb{Q}\setminus\{0,1\}$, we obtain the $K$-theory of the $C^*$-algebra $C(U_q(2))$. We construct a spectral triple on $U_q(2)$ which is equivariant under its own comultiplication action. The spectral triple obtained here is even, $4^+$-summable, non-degenerate, and the Dirac operator acts on two copies of the $L^2$-space of $U_q(2)$. The $K$-homology class of the associated Fredholm module is shown to be nontrivial.

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arXiv2021

Representations and Classification of the compact quantum groups $U_q(2)$ for complex deformation parameters

In this article, we obtain a complete list of inequivalent irreducible representations of the compact quantum group $U_q(2)$ for non-zero complex deformation parameters $q$, which are not roots of unity. The matrix coefficients of these representations are described in terms of the little $q$-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of $L^2(U_q(2))$ is obtained. Thus, we have an explicit description of the Peter-Weyl decomposition of $U_q(2)$. As an application, we discuss the Fourier transform and establish the Plancherel formula. We also describe the decomposition of the tensor product of two irreducible representations into irreducible components. Finally, we classify the compact quantum groups $U_q(2)$.

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arXiv2017

Subadditivity and additivity of the Yang-Mills action functional in Noncommutative Geometry

We formulate notions of subadditivity and additivity of the Yang-Mills action functional in noncommutative geometry. We identify a suitable hypothesis on spectral triples which proves that the Yang-Mills functional is always subadditive, as per expectation. The additivity property is much stronger in the sense that it implies the subadditivity property. Under this hypothesis we obtain a necessary and sufficient condition for the additivity of the Yang-Mills functional. An instance of additivity is shown for the case of noncommutative $n$-tori. We also investigate the behaviour of critical points of the Yang-Mills functional under additivity. At the end we discuss few examples involving compact spin manifolds, matrix algebras, noncommutative $n$-torus and the quantum Heisenberg manifolds which validate our hypothesis.

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arXiv2017

Kähler structure on certain $C^*$-dynamical systems and the noncommutative even dimensional tori

Let $G$ be an even dimensional, connected, abelian Lie group and $(\mathcal{A}^\infty,G,α,τ)$ be a $C^*$-dynamical system equipped with a faithful $G$-invariant trace $τ$. We show that whenever it determines a $\varTheta$-summable even spectral triple, $\mathcal{A}^\infty$ inherits a Kähler structure. Moreover, there are at least $\prod_{j=1,\,j\,odd}^{\,dim(G)}(dim(G)-j)$ different Kähler structures. In particular, whenever $\mathbb{T}^{2k}$ acts ergodically on the algebra, it inherits a Kähler strcture. This gives a class of examples of noncommutative Kähler manifolds. As a corollary, we obtain that all the noncommutative even dimensional tori, like their classical counterpart the complex tori, are noncommutative Kähler manifolds. We explicitly compute the space of complex differential forms for the noncommutative even dimensional tori and show that the category of holomorphic vector bundle over it is an abelian category. We also explain how the earlier set-up of Polishchuk-Schwarz for the holomorphic structure on noncommutative two-torus follows as a special case of our general framework.

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arXiv2017

On the supersymmetric $N=1$ and $N=(1,1)$ spectral data : A multiplicativity property

We show that there are six different choice of tensor product of supersymmetric N=(1,1) spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We also show that the procedure of extending a supersymmetric N=1 spectral data to N=(1,1) spectral data respects only one tensor product among these. We refer this as the multiplicativity property of the extension procedure. Therefore, if we demand that the extension procedure is multiplicative then there is a unique choice of tensor product of N=(1,1) spectral data.

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