arXiv subjects

Keshab Chandra Bakshi

Publications and source records attributed to Keshab Chandra Bakshi.

At least 19 recordsLinked to original sources

arXiv2026

Structural and Dynamical Properties of Subfactor Commuting Squares

This paper explores the algebraic invariants and ergodic behavior of the finite-index inclusions forming commuting squares of $\text{II}_1$ factors. We begin by establishing how regularity, intermediate-subfactor lattices, and Weyl groups transfer across the commuting square under relative commutant and irreducibility conditions. Furthermore, we provide a complete characterization, via a novel factorization theorem, of the unitary normalizers of the upper inclusion when the lower inclusion is regular. Applying these findings to crossed-product inclusions induced by discrete groups, we introduce a relative eigenbasis property that preserves regular inclusions. Finally, we contrast this by proving that a relative weak mixing condition on the group action forces the crossed-product inclusion to be singular.

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arXiv2026

Regularity for Subfactors

We study three naturally associated notions of regularity for a unital inclusion of von Neumann algebras $\mathcal{B} \subseteq \mathcal{M}$: regularity, groupoid regularity, and unitary regularity. We first observe that regularity and groupoid regularity are unconditionally equivalent by showing that the standard normaliser and the groupoid normaliser of $\mathcal{B}$ in $\mathcal M$ generate the same linear span. The central question addressed in this work is whether regularity implies unitary regularity. For arbitrary von Neumann subalgebras, we demonstrate that this implication fails dramatically; we construct explicit examples to show that every factor (of type $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, and $\mathrm{III}$) contains a regular von Neumann subalgebra that is not unitarily regular. In stark contrast, we prove that the behaviour changes completely in the setting of subfactors. For subfactor inclusions across all these types, regularity always implies unitary regularity, and thus all three notions of regularity coincide. Finally, we establish an analogous result for irreducible unital inclusions of simple $C^*$-algebras admitting a conditional expectation of finite Watatani index. In this setting, we prove that regularity, unitary regularity, being a generalised Cartan subalgebra, and arising from a crossed product by a finite group are all mutually equivalent conditions.

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arXiv2026

A Complete Characterization of Regular Inclusions of Finite Dimensional $C^*$-algebras

We give a complete characterization of regular (in the sense of Kumjian and Renault) unital inclusions of finite-dimensional $C^*$-algebras. For subalgebras $\bigoplus_j( \mathbb{M}_{d_j}(\mathbb{C}) \otimes \mathbb{I}_{p_j})$ of $\mathbb{M}_n(\mathbb{C})$, we show that regularity depends only on equality of the multiplicities $p_j$, while unitary regularity---characterized recently by the first author and Silambarasan---additionally requires equality of the $d_j$; we recover the latter via a streamlined alternative proof. Extending this to inclusions of arbitrary finite-dimensional $C^*$-algebras, encoded by an inclusion matrix $Λ$, we show that regularity is equivalent to an explicit row/column condition on $Λ$---coinciding with the normalizer matrix introduced by the first author and Silambarasan ---so that the device used there to detect unitary regularity is shown to characterize regularity in general; unitary regularity is recovered by a further dimension-equality condition.

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arXiv2026

Regular diagonal subfactors

We show that a diagonal subfactor arising from a finite family of automorphisms of a $II_1$-factor $Q$ is regular precisely when the classes of the defining automorphisms occur with same cardinality and form a subgroup of $\mathrm{Out}(Q)$, a subgroup that happens to be isomorphic to the generalized Weyl group of the subfactor. Moreover, it turns out that the cleanest picture of regularity in diagonal subfactors is graph-theoretic, namely, a diagonal subfactor is regular precisely when its principal graph is a complete, regular, balanced bipartite multigraph, with the generalized Weyl group fixing its size and the common multiplicity of the defining automorphisms determining its regular edge multiplicity. Prior to the characterization of regularity, revisiting Bisch and Popa's observations on the standard invariant and depth of diagonal subfactors, we give an exact criterion for a diagonal subfactor to have any prescribed depth, in terms of a stabilizing sequence of subsets of $\mathrm{Out}(Q)$ consisting of non-reduced alternating words in the classes of the defining automorphisms, which proves useful in the characterization of regularity.

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arXiv2026

Regular irreducible inclusions of simple $C^*$-algebras and crossed product structure

We study regular irreducible inclusions $B\subset A$ of simple unital $C^*$-algebras admitting a conditional expectation. We introduce a generalized notion of quasi-basis extending Watatani's framework and show that such inclusions admit a unitary orthonormal generalized quasi-basis. As a consequence, we prove that every regular irreducible inclusion in this setting is canonically isomorphic to a reduced twisted crossed product of $B$ by its Weyl group. This extends earlier crossed product characterizations beyond the finite-index setting.

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arXiv2026

Unitary normalizers in finite-dimensional inclusions

We study regular inclusions of finite-dimensional von Neumann algebras from a matrix-theoretic perspective. To this end, we introduce a new combinatorial invariant of an inclusion, called the normalizer matrix, which encodes the structure of the normalizer purely at the level of the inclusion matrix. Using this invariant, we obtain a complete characterization of regular inclusions of finite-dimensional von Neumann algebras. As consequences, we show that every regular inclusion decomposes into finite direct sums and tensor products of basic building blocks, and that regular inclusions are necessarily of depth two. We further investigate the existence of unitary orthonormal bases in the sense of Pimsner-Popa and prove that, under a natural spectral condition, regularity is equivalent to the existence of a unitary orthonormal basis contained in the normalizer. These results provide a unified description of regularity, unitary bases, and depth through the normalizer matrix formalism.

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arXiv2026

Weak quantum hypergroups from finite index C*-inclusions

We study a finite index inclusion of simple unital C*-algebras and construct a canonical completely positive coproduct on the second relative commutant, thereby endowing it with a natural coalgebra structure. Motivated by this construction, we introduce the notion of a weak quantum hypergroup, a generalization of the quantum hypergroups of Chapovsky and Vainerman. We show that every finite index inclusion gives rise to such a weak quantum hypergroup, and that the corresponding weak quantum hypergroup possesses a Haar integral. In the irreducible case, this structure satisfies the axioms of a quantum hypergroup in the sense of Chapovsky and Vainerman, while in the depth 2 setting our framework yields the associated weak Hopf algebra constructed by Nikshych and Vainerman. These results provide a unified and intrinsically C*-algebraic framework for generalized quantum symmetries associated with finite index inclusions.

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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

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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arXiv2025

Unitary orthonormal bases of finite dimensional inclusions

We study unitary orthonormal bases in the sense of Pimsner and Popa for inclusions $(\mathcal{B}\subseteq \mathcal{A}, E),$ where $\mathcal{A}, \mathcal{B}$ are finite dimensional von Neumann algebras and $E$ is a conditional expectation map from $\mathcal{A}$ onto $\mathcal{B}$. It is shown that existence of such bases requires that the associated inclusion matrix satisfies a spectral condition forcing dimension vectors to be Perron-Frobenius eigenvectors and the conditional expectation map preserves the Markov trace. Subject to these conditions, explicit unitary orthonormal bases are constructed if either one of the algebras is abelian or simple. They generalize complex Hadamard matrices, Weyl unitary bases, and a recent work of Crann et al which correspond to the special cases of $\mathcal{A}$ being abelian, simple, and general multi-matrix algebras respectively with $\mathcal{B}$ being the algebra of complex numbers. For the first time $\mathcal{B}$ is more general. As an application of these results it is shown that if $(\mathcal{B}\subseteq \mathcal{A}, E),$ admits a unitary orthonormal basis then the Connes-Størmer relative entropy $H(\mathcal{A}_1|\mathcal{A})$ equals the logarithm of the square of the norm of the inclusion matrix, where $\mathcal{A}_1$ denotes the Jones basic construction of the inclusion. As a further application, we prove the existence of unitary orthonormal bases for a large class of depth 2 subfactors with abelian relative commutant.

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arXiv2024

Regular inclusions of simple unital $C^*$-algebras

We prove that an inclusion $\mathcal{B} \subset \mathcal{A}$ of simple unital $C^*$-algebras with a finite-index conditional expectation is regular if and only if there exists a finite group $G$ that admits a cocycle action $(α,σ)$ on the intermediate $C^*$-subalgebra $\mathcal{C}$ generated by $\mathcal{B}$ and its centralizer $\mathcal{C}_\mathcal{A}(\mathcal{B})$ such that $\mathcal{B}$ is outerly $α$-invariant and $(\mathcal{B} \subset \mathcal{A}) \cong ( \mathcal{B} \subset \mathcal{C}\rtimes^r_{α, σ} G)$. Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most $2$.

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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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