Research Interests

Operator algebras, particularly von Neumann algebras, offer a powerful framework for analyzing 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]

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 Dr. 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].

Current and Future Work

Motivated by these developments, I extended the framework of affiliated operators beyond von Neumann algebras to a larger class of operator algebras—namely, unital monotone \( \sigma \)-complete \( C^* \)-algebras. In this setting, I showed that one can define a set \( \mathfrak{A}_{\mathrm{aff}} \) of unbounded operators affiliated to \( \mathfrak{A} \), satisfying similar algebraic properties and functorial behavior.

My future work aims to generalize this further to all unital \( C^* \)-algebras. Since these algebras lack certain order completeness properties, this requires identifying structural tools such as the Pedersen–Baire envelope to develop a meaningful and functorial theory of affiliation.

In particular, I intend to:

1. Develop a theory of unbounded affiliated operators for general \( C^* \)-algebras, with an emphasis on functoriality and algebraic rigor.
2. Bridge with Woronowicz affiliation, which plays a foundational role in quantum groups, to potentially unify disparate notions of operator affiliation under a broader categorical framework.
3. Explore topological aspects of affiliated operators, including the identification of canonical topologies compatible with Kaufman inverse and analytic operations.

Matrix Decomposition Results in \(M_n(\mathscr{R})\)

I am also interested in matrix decomposition phenomena over rings of operators. It is well known that any normal matrix in \(M_n(\mathbb{C})\) can be unitarily "diagonalized", that is, there is a unitary matrix \(U\) such that \(U A U^{-1}\) is diagonal. For a \(C^*\)-algebra \(\mathfrak{A}\), the ring of \(n \times n\) matrices over \(\mathfrak{A}\), \(M_n(\mathfrak{A})\), naturally has the structure of a \(C^*\)-algebra. For a von Neumann algebra \(\mathscr{R}\), \(M_n(\mathscr{R})\) is a von Neumann algebra. In [R84b], R. V. Kadison proved that each normal element in \(M_n(\mathscr{R})\) can be diagonalized via conjugation with a unitary matrix in \(M_n(\mathscr{R})\). Furthermore, using basic homotopy theoretic techniques, he showed that this is not true in general in the context of \(M_2(C(\mathbb{S}^4))\) where \(\mathbb{S}^4\) denotes the 4-sphere, and \(C(\mathbb{S}^4)\) is the \(C^*\)-algebra of complex-valued continuous functions on \(\mathbb{S}^4\).

In response to one of the questions posed by R. V. Kadison, in [R84a], K. Grove and G. Pedersen provided a complete topological characterization of those compact Hausdorff spaces \(X\) for which every normal matrix in \(M_n(C(X))\) is diagonalizable (for all \(n \in \mathbb{N}\)).

I have two main interests:

1. Other Decomposition Results

   I'm interested in exploring other decomposition results over \(M_n(C(X))\), such as QR decomposition or polar decomposition. Interestingly, I've discovered that the Polar Decomposition holds over \(C(X)\) whenever \(X\) is a (sub)-Stonean space.

2. Decomposition over Type \(II_1\) Factor

   I'm investigating decomposition results over a Type \(II_1\) factor and their implications.

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.
• [R84a] Grove, Karsten and Pedersen, Gert Kjærgård (1984). Diagonalizing matrices over \(C(X)\) . Journal of Functional Analysis, 59, 65–89.
• [R84b] Kadison, Richard V. (1984). Diagonalizing Matrices . American Journal of Mathematics, 106, 1451–1468.
• [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.