🧮Von Neumann Algebras Unit 7 – Subfactors and Jones Index Theory

Key Concepts and Definitions

• Von Neumann algebras non-commutative generalizations of classical measure theory and complex analysis
• Factors specific types of von Neumann algebras with a trivial center (only scalar multiples of the identity commute with all elements)
• Subfactors subalgebras of factors that are factors themselves
• Jones index $[M:N]$ measures the "relative size" of a subfactor $N$ inside a larger factor $M$
• Bimodules generalize the notion of vector spaces with left and right actions of algebras
• Frobenius reciprocity relates induction and restriction of representations between algebras and subalgebras
  • Adjunction between the functors of induction and restriction
  • Allows for transfer of problems between different levels
• Planar algebras diagrammatic approach to studying subfactors using tangles and strings

Historical Context and Development

• Von Neumann algebras originated in the 1930s from the work of Murray and von Neumann on rings of operators
• Factors classified into types I, II, and III based on their projections and traces
  • Type I factors isomorphic to bounded operators on a Hilbert space
  • Type II factors have a unique trace (finite or infinite)
  • Type III factors have no non-trivial traces
• Subfactor theory emerged in the 1980s with the work of Jones on index theory
• Jones discovered a surprising connection between subfactors and knot theory
  • Temperley-Lieb algebras and Jones polynomial invariants for knots
• Ocneanu's paragroup theory provided a categorical framework for studying subfactors
• Popa's work on classification of subfactors and rigidity properties expanded the field

Subfactor Theory Fundamentals

• Subfactors capture the notion of "sub-symmetries" within a larger symmetry group
• Basic construction given a subfactor $N \subset M$, one can construct a tower of factors $M \subset M_1 \subset M_2 \subset \cdots$
  • Jones projections $e_i$ implement the conditional expectations from $M_i$ to $M_{i-1}$
• Finite index subfactors have a well-behaved theory
  • Index values constrained to the range $[4, \infty) \cup \{4 \cos^2(\pi/n) : n \geq 3\}$
• Standard invariant of a subfactor encodes its structure and properties
  • Consists of higher relative commutants and fusion algebra
• Bimodules and fusion categories provide a categorical perspective on subfactors
  • Bimodules over subfactors form a rigid C*-tensor category
• Minimal index subfactors are building blocks for more complex subfactors

Jones Index Theory Basics

• Jones index $[M:N]$ quantifies the relative size of a subfactor $N$ inside a factor $M$
• Index can be computed using the trace of the Jones projection $Tr(e_1)$
  • Formula: $[M:N] = (Tr(e_1))^{-1}$
• Pimsner-Popa inequality relates the index to the conditional expectation $E_N: M \to N$
  • $E_N(x) \geq [M:N]^{-1} x$ for all positive elements $x \in M$
• Watatani's approach extends index theory to arbitrary C*-algebras using quasi-bases
• Index values have a deep connection to statistical mechanics and quantum field theory
  • Appearance of the $A-D-E$ classification of modular invariant partition functions
• Finite index subfactors give rise to braid group representations and link invariants

Algebraic Structures and Properties

• Standard invariant of a subfactor forms a powerful algebraic object
  • Higher relative commutants capture the "hidden symmetries" of the subfactor
  • Fusion algebra encodes the multiplication rules for bimodules
• Ocneanu's paragroup theory associates a unique paragroup to each finite index subfactor
  • Paragroups generalize the notion of groups and provide a complete invariant
• Popa's symmetric enveloping algebra allows for a "linearization" of the subfactor
  • Reduces the study of subfactors to the study of certain algebraic objects
• Planar algebras axiomatize the diagrammatic calculus for subfactors
  • Tangles and strings represent morphisms and compositions
  • Efficient computational tool for studying subfactors and their invariants
• Subfactor planar algebras have a rich structure and classification
  • Temperley-Lieb, Fuss-Catalan, and other families arise naturally

Applications in Operator Algebras

• Subfactor theory has deep connections to various areas of operator algebras
• Jones' work on actions of finite groups on factors led to the development of orbifold constructions
  • Relates the fixed point algebra to the crossed product via a subfactor
• Ocneanu's asymptotic inclusion relates subfactors to amenable group actions
  • Provides a powerful tool for constructing and studying certain subfactors
• Popa's work on classification of amenable subfactors up to isomorphism
  • Rigidity properties and techniques for distinguishing subfactors
• Applications to quantum field theory and conformal field theory
  • Subfactors give models for rational CFTs and their defects
• Connections to free probability and random matrices
  • Subfactor planar algebras related to free product of planar algebras

Advanced Topics and Current Research

• Subfactor theory is an active area of research with many open problems and directions
• Classification of subfactors with small index values (e.g., index less than 5)
  • Haagerup subfactor first example with index greater than 4 but not of the form $4 \cos^2(\pi/n)$
• Planar algebras and their classification
  • Universal planar algebra construction and free composition
  • Connections to random matrix theory and free probability
• Conformal inclusions and modular invariants in conformal field theory
  • Subfactors provide a operator algebraic approach to studying CFTs
• Noncommutative geometry and spectral triples from subfactors
  • Relates subfactor theory to Connes' noncommutative geometry program
• Quantum symmetries and quantum groups arising from subfactors
  • Subfactors give examples of compact quantum group actions and coactions

Problem-Solving Techniques

• Diagrammatic techniques using planar algebras and tangles
  • Representing morphisms and computations graphically
  • Efficient for proving identities and relations in subfactor theory
• Categorical methods using bimodules and fusion categories
  • Translating problems into the language of tensor categories
  • Applying powerful results and constructions from category theory
• Algebraic techniques using the symmetric enveloping algebra and Popa's methods
  • Reducing questions about subfactors to algebraic problems
  • Utilizing results from the theory of finite-dimensional algebras
• Analytic techniques using Jones' index theory and Pimsner-Popa inequality
  • Estimating dimensions and norms using the index value
  • Deriving inequalities and constraints on subfactor data
• Combinatorial methods using fusion graphs and principal graphs
  • Encoding subfactor data into graph-theoretic objects
  • Applying graph theory and combinatorics to study subfactors