Publications

1. Published Paper

   Algebraic aspects and functoriality of the set of affiliated operators I. Ghosh and S. Nayak

   International Mathematics Research Notices, Volume 2024, Issue 21, Nov 2024, Pages 13525-13562

   IMRN arXiv ResearchGate

   Abstract

   In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let \(\mathscr{M}\) be a von Neumann algebra acting on a Hilbert space \(\mathcal{H}\), and let \(\mathscr{M}_{\text{aff}}\) denote the set of unbounded operators of the form \(T = AB^{\dagger}\) for \(A, B \in \mathscr{M}\) with \(\text{null} (B) \subseteq \text{null} (A)\), where \((\cdot)^{\dagger}\) denotes the Kaufman inverse. We show that \(\mathscr{M}_{\text{aff}}\) is closed under sum, product, Kaufman-inverse and adjoint, and has the structure of a (right) near-semiring; Moreover, the above quotient representation of an operator in \(\mathscr{M}_{\text{aff}}\) is essentially unique. Thus we may view \(\mathscr{M}_{\text{aff}}\) as the multiplicative monoid of unbounded operators on \(\mathcal{H}\) generated by \(\mathscr{M}\) and \(\mathscr{M} ^{\dagger}\). We further show that our definition of affiliation, as reflected in \(\mathscr{M}_{\text{aff}}\), subsumes the traditional one. Let \(\Phi\) be a unital normal \(*\)-homomorphism between represented von Neumann algebras \((\mathscr{M}; \mathcal{H})\) and \((\mathscr{N}; \mathcal{K})\). Using the quotient representation, we obtain a canonical extension of \(\Phi\) to a mapping \(\Phi_{\text{aff}} : \mathscr{M}_{\text{aff}} \to \mathscr{N}_{\text{aff}}\) which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, \(\Phi_{\text{aff}}\) respects Murray-von Neumann affiliation of operators and also respects strong-sum and strong-product. Thus \(\mathscr{M}_{\text{aff}}\) is intrinsically associated with \(\mathscr{M}\) and transforms functorially as we change representations of \(\mathscr{M}\). Furthermore, \(\Phi_{\text{aff}}\) preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein-von Neumann extensions of densely-defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via 'abstract nonsense'.

Preprints

1. Preprint

   An Operator Theoretic Characterization of Properly Infinite von Neumann Algebras I. Ghosh and K. K. Mukherjee

   pre-print • Feb 2026

   Abstract
2. Preprint

   A Complete Characterization of Regular Inclusions of Finite Dimensional \(C^*\)-algebras K. C. Bakshi, I. Ghosh and S. Kumar

   pre-print • Aug 2026

   arXiv ResearchGate

   Abstract

   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 \(\Lambda\), we show that regularity is equivalent to an explicit row/column condition on \(\Lambda\)—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.
3. Preprint

   A Complete Characterization of Cartan Inclusions of Finite Dimensional \(C^*\)-algebras I. Ghosh and S. Kumar

   pre-print • Aug 2026

   arXiv ResearchGate

   Abstract

   We give a complete characterization of Cartan inclusions of finite dimensional \(C^*\)-algebras in terms of their inclusion matrices. More precisely, for a unital inclusion \(\mathcal{B}\subseteq\mathcal{A}\) with inclusion matrix \(\Lambda=(\Lambda_{ij})\), where Cartan means that \(\mathcal{B}\) is a generalized Cartan subalgebra of \(\mathcal{A}\) in the sense of Exel, we prove that the inclusion is Cartan if and only if \[ \sum_i \Lambda_{ij}\leq 1 \] for every \(j\). We call matrices satisfying this condition multiplicity free. Thus, our characterization provides a purely combinatorial criterion for determining when a finite dimensional inclusion is Cartan. We further prove that every Cartan inclusion admits a unique conditional expectation from \(\mathcal{A}\) onto \(\mathcal{B}\). Conversely, we show that, for unital inclusions of finite dimensional \(C^*\)-algebras, the uniqueness of the conditional expectation is sufficient for the inclusion to be Cartan. Consequently, a unital inclusion \(\mathcal{B}\subseteq\mathcal{A}\) of finite dimensional \(C^*\)-algebras is Cartan if and only if there exists a unique conditional expectation from \(\mathcal{A}\) onto \(\mathcal{B}\).
4. Preprint

   Regularity for Subfactors K. C. Bakshi and I. Ghosh

   pre-print • Sep 2026

   arXiv ResearchGate

   Abstract

   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.
5. Preprint

   Kadison-Kastler Distance and Interior Angle for Masas: Connections with Entropy and Probabilistic Index I. Ghosh and S. Kumar

   pre-print • Sep 2026

   arXiv ResearchGate

   Abstract

   We study the Kadison--Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In $\mathbb{M}_2(\mathbb{C})$, we obtain an explicit formula for the Kadison--Kastler distance between two masas and show that every value in $[0,1]$ is attained by $\mathrm{d}_{\textrm{KK}}(\Delta,u\Delta u^*)$. We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. We also establish the identity \[ \mathrm{d}_{\textrm{KK}}(\mathcal{A},\mathcal{B})=\sin\alpha(\mathcal{A},\mathcal{B}) \] for any two intermediate subalgebras $\mathbb{C}\subsetneq\mathcal{A},\mathcal{B}\subsetneq\mathbb{M}_2(\mathbb{C})$. Most notably, for any unitaries $u,v\in\mathbb{M}_2(\mathbb{C})$, we establish the equivalence of maximal Kadison--Kastler distance, maximal interior angle, maximal Connes--St\o rmer modified entropy, and minimal Popa probabilistic index: \[ \mathrm{d}_{\textrm{KK}}(u\Delta u^*,v\Delta v^*)=1 \iff h(u\Delta u^*,v\Delta v^*)=\log 2 \iff \alpha(u\Delta u^*,v\Delta v^*)=\frac{\pi}{2} \iff \lambda(u\Delta u^*,v\Delta v^*)=\frac{1}{2}. \] For general $\mathbb{M}_n(\mathbb{C})$, we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form $u\Delta^{(n)}u^*$ and $v\Delta^{(n)}v^*$. As a consequence, the interior angle is $\frac{\pi}{2}$ precisely when the relative unitary $u^*v$ is a Hadamard unitary. We further show that every value in $\left[0,\frac{\pi}{2}\right]$ is attained as the interior angle between a pair of masas in $\mathbb{M}_n(\mathbb{C})$. Finally, we demonstrate that, for $n\geq 3$, the interior angle need not distinguish between distinct intermediate subalgebras: it can vanish even when the subalgebras are distinct. In particular, we construct distinct simple intermediate $C^*$-subalgebras with zero interior angle, thereby answering in the negative the question raised by Gupta and Sharma.
6. Preprint

   Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors I. Ghosh and S. Kumar

   pre-print • Sep 2026

   arXiv ResearchGate

   Abstract

   We investigate the relative position of von Neumann subalgebras by studying their Mashood--Taylor ($\mathrm{d}_{\mathrm{MT}}$) and Kadison--Kastler ($\mathrm{d}_{\mathrm{KK}}$) distances, alongside the interior angle. For each $n\in\mathbb{N}$, we show that the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\times n$ spin model subfactors. In particular, this yields a continuous family $(\mathscr{R}_{\mathsf{H}_\alpha})_{\alpha\in[0,\pi)}$ of $2\times2$ spin model subfactors. Moreover, equipped with the Mashood--Taylor metric, this family is homeomorphic to the circle group $\mathbb{T}$. More precisely, we establish the exact formula \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_\alpha},\mathscr{R}_{\mathsf{H}_\beta}) = \vert\sin(\alpha-\beta)\vert. \] Furthermore, we prove that a pair of such spin model subfactors forms a commuting square over their intersection if and only if they are maximally distant ($\mathrm{d}_{\mathrm{MT}}=1$). Motivated by this, we establish general structural results showing that commuting squares of $\mathrm{II}_1$-factors (under natural index conditions) force maximal distance, yielding $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$. We also show that two diffuse subalgebras that are orthogonal in the sense of Popa must be maximally distant. In contrast, we observe that no two subfactors in the family $(\mathscr{R}_{\mathsf{H}_\alpha})_{\alpha\in[0,\pi)}$ are orthogonal in the sense of Popa. Nevertheless, whenever two such spin model subfactors are maximally distant, their interior angle over their intersection is $\pi/2$, showing that ``orthogonality'' in terms of the interior angle is distinct from orthogonality in the sense of Popa. Finally, in the free group factor $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, we establish the explicit formula \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) = \sqrt{1-\vert\tau(u)\vert^4} \] for $u\in L(\langle b\rangle)$. We also construct maximally distant masas in $L(\mathbb{F}_2)$ that do not arise from subgroups of $\mathbb{F}_2$.

Ongoing Work

• Work in Progress
  A Royal Road to the Basic Theory of Unbounded Operators on Hilbert Space I. Ghosh, G. Maiti and S. Nayak
• Work in Progress
  On the Value of Connes-Størmer Entropy K. C. Bakshi, I. Ghosh and S. Nayak
• Work in Progress
  On Interior Angle Between Intermediate Subalgebras K. C. Bakshi, I. Ghosh and V. P. Gupta