Research

Operator algebras—particularly von Neumann algebras—offer a powerful framework for studying infinite-dimensional linear operators, with deep connections to quantum mechanics, functional analysis, and noncommutative geometry. Originally introduced by von Neumann to formalize quantum theory, these algebras now play central roles in ergodic theory, quantum probability, and operator K-theory.

I recently submitted my PhD thesis titled On Algebraic Aspects and Functoriality of the Set of Unbounded Operators Affiliated with a von Neumann Algebra at the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore. My work investigates algebraic structures underlying unbounded operators affiliated with von Neumann algebras and proposes a functorial framework for understanding these operators in broader categorical and analytical settings.

1. 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
2. An Operator Theoretic Characterization of Properply Infinite von Neumann Algebras
   I. Ghosh and K. K. Mukherjee

   pre-print, Feb 2026
3. A Complete Characterization of Regular Inclusions of Finite Dimensional \(C^*\)-algebras
   K. C. Bakshi, I. Ghosh and S. Kumar

   pre-print, Aug 2026

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

   pre-print, Aug 2026

   arXiv

Doctoral Research

The classical notion of affiliated operators with a von Neumann algebra—in the sense of Murray and von Neumann—has long been studied in the literature (see [R36, 37, 40, 43], [R97], [R14]). However, these constructions lack a fully satisfactory algebraic formulation, especially for general von Neumann algebras and independent of Hilbert space representations.

In joint work with Prof. Soumyashant Nayak [R24], I introduced a new class of unbounded operators denoted \( \mathscr{M}_{\mathrm{aff}} \), defined for a von Neumann algebra \( \mathscr{M} \subseteq \mathcal{B}(\mathcal{H}) \) as the set of operators expressible as quotients \( T = AB^\dagger \) where \( A, B \in \mathscr{M} \) and the null space of \( B \) is contained in that of \( A \). The Kaufman inverse \( B^\dagger \) is used in place of the standard Moore–Penrose inverse to retain operator-theoretic meaning.

The key findings include:

• \( \mathscr{M}_{\mathrm{aff}} \) forms a right near-semiring under addition, product, adjoint, and Kaufman inverse.
• Each quotient representation is essentially unique.
• The construction \( \mathscr{M}_{\mathrm{aff}} \) is functorial: a unital normal *-homomorphism \( \Phi: \mathscr{M} \to \mathscr{N} \) induces \( \Phi_{\mathrm{aff}}: \mathscr{M}_{\mathrm{aff}} \to \mathscr{N}_{\mathrm{aff}} \) preserving structure and operator-theoretic properties.
• The Murray–von Neumann affiliated operators form a subcollection of this larger set: \( \mathscr{M}_{\mathrm{aff}}^{\mathrm{MvN}} \subseteq \mathscr{M}_{\mathrm{aff}} \).
• We also characterized closed operators via factorizations involving projections and positive operators within \( \mathcal{B}(\mathcal{H}) \).
• Furthermore, Krein and Friedrichs extensions of positive operators behave functorially under this framework: \( \Phi_{\mathrm{aff}}(S^{\mathrm{Fr}}) = \Phi_{\mathrm{aff}}(S)^{\mathrm{Fr}} \), and similarly for Krein extensions.

These results were published in International Mathematics Research Notices [R24].

References

• [R36] Murray, F. J. and von Neumann, J. (1936). On Rings of Operators . Annals of Mathematics, Second Series, 37(1), 116–229.
• [R37] Murray, F. J. and von Neumann, J. (1937). On Rings of Operators. II . Transactions of the American Mathematical Society, 41(2), 208–248.
• [R40] von Neumann, J. (1940). On Rings of Operators. III . Annals of Mathematics, 41(1), 94–161.
• [R43] Murray, F. J. and von Neumann, J. (1943). On Rings of Operators. IV . Annals of Mathematics, 44(4), 716–808.
• [R97] Kadison, Richard V. and Ringrose, John R. (1997). Fundamentals of the Theory of Operator Algebras. Vol. I: Elementary Theory . Graduate Studies in Mathematics, 15. 2nd printing with corrections. Providence, RI: American Mathematical Society.
• [R14] Kadison, Richard V. and Liu, Zhe (2014). The Heisenberg relation — mathematical formulations . SIGMA, 10, paper 009, 40 pages.
• [R24] Ghosh, Indrajit and Nayak, Soumyashant (2024). Algebraic aspects and functoriality of the set of affiliated operators . International Mathematics Research Notices, 2024(21), 13525–13562.