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A compendium of research in operator algebras and operator theory

Jaydeb Sarkar

TL;DR

In this chapter, we survey a decade of Indian contributions to operator algebras and operator theory, spanning $20$ sections from curvature invariants in the Cowen–Douglas class $B_k(\Omega)$ to $E_0$-semigroups and $K$-theory. It showcases methods including dilation theory, subfactor techniques with $2$-sided Pimsner–Popa bases, quotient-domain models, trace formulas via spectral shift functions, and noncommutative dynamics, as well as applications to quantum Gaussian states and operator systems. The work provides explicit dilation theorems, unitary invariants such as curvature and imprimitivity, and new realizations on quotient domains, with several open problems and conjectures. It positions Indian researchers at the forefront of global developments and reveals deep links between operator algebras, Banach-space geometry, and quantum information, e.g., through $C^*$-envelopes of operator systems and Rokhlin-dimension phenomena. Overall, the chapter offers an integrative view of theory, methods, and directions for future research.

Abstract

This chapter surveys the advances of the past decade arising from the contributions of Indian mathematicians in the broad areas of operator algebras and operator theory. It brings together the work of twenty mathematicians and their collaborators, each writing from the perspective of their respective research fields and beyond. Several problems highlighted here are expected to shape the future development of the subject at a global level.

A compendium of research in operator algebras and operator theory

TL;DR

In this chapter, we survey a decade of Indian contributions to operator algebras and operator theory, spanning sections from curvature invariants in the Cowen–Douglas class to -semigroups and -theory. It showcases methods including dilation theory, subfactor techniques with -sided Pimsner–Popa bases, quotient-domain models, trace formulas via spectral shift functions, and noncommutative dynamics, as well as applications to quantum Gaussian states and operator systems. The work provides explicit dilation theorems, unitary invariants such as curvature and imprimitivity, and new realizations on quotient domains, with several open problems and conjectures. It positions Indian researchers at the forefront of global developments and reveals deep links between operator algebras, Banach-space geometry, and quantum information, e.g., through -envelopes of operator systems and Rokhlin-dimension phenomena. Overall, the chapter offers an integrative view of theory, methods, and directions for future research.

Abstract

This chapter surveys the advances of the past decade arising from the contributions of Indian mathematicians in the broad areas of operator algebras and operator theory. It brings together the work of twenty mathematicians and their collaborators, each writing from the perspective of their respective research fields and beyond. Several problems highlighted here are expected to shape the future development of the subject at a global level.
Paper Structure (20 sections, 9 theorems, 41 equations)

This paper contains 20 sections, 9 theorems, 41 equations.

Table of Contents

  1. Some problems in operator theory and related topics
  2. Fourier theory, angle, Pimsner-Popa bases and regularity for inclusions of $C^*$-algebras
  3. Contributions in Banach space theory
  4. Quantum dynamical maps and quantum Gaussian states
  5. On Carathéodory approximation
  6. Operator theory on quotient domains
  7. Trace formulas in operator theory
  8. Dilation of operators and algebraic varieties
  9. The Weighted Discrete Semigroup Algebra $\ell^1(S, \omega)$
  10. Cartan isometries and Cartan contractions
  11. Norms on operator space tensor products and $C^*$-envelopes of operator systems
  12. Bohr's Inequality and the Gleason--Kahane--Żelazko Theorem
  13. von Neumann Algebras
  14. Applications of Birkhoff-James orthogonality in studying the geometry of operators between Banach spaces
  15. Commutators and Lie algebras of compact operators
  16. ...and 5 more sections

Key Result

Theorem 8.1

Let $(T_1, T_2)$ be a pair of commuting contractions such that $T_1$ is pure and the defect spaces of $T_1$ and $T_2$ are finite-dimensional. Then:

Theorems & Definitions (9)

  • Theorem 8.1: DS
  • Theorem 8.2: DSS
  • Theorem 8.3: DS2
  • Theorem 10.1
  • Corollary 10.2
  • Theorem 12.1
  • Theorem 12.2
  • Theorem 19.1: Sundar_Existence
  • Theorem 19.2: Namitha_Sundar, Injectivity