🧮Von Neumann Algebras Unit 4 – Representations and states

Key Concepts and Definitions

• Von Neumann algebras are self-adjoint algebras of bounded operators on a Hilbert space that are closed in the weak operator topology and contain the identity operator
• Representations are -homomorphisms from a C-algebra to the bounded operators on a Hilbert space preserving the algebraic and *-operation structures
• States are positive linear functionals of norm one on a C*-algebra
  • Pure states are extreme points of the convex set of states
  • Mixed states are convex combinations of pure states
• The GNS (Gelfand-Naimark-Segal) construction associates a representation to each state
• Cyclic vectors are vectors in a Hilbert space such that the orbit under the action of the algebra is dense in the Hilbert space
• Normal states are ultraweakly continuous positive linear functionals of norm one
• Faithful representations have trivial kernel and injectively map the algebra into bounded operators

Historical Context and Development

• Von Neumann algebras originated in the work of John von Neumann on operator algebras in the 1930s and 1940s
  • von Neumann was motivated by questions in quantum mechanics and the mathematical foundations of quantum theory
• The study of operator algebras was further developed by Murray and von Neumann in a series of papers titled "On Rings of Operators"
• Factors, which are von Neumann algebras with trivial center, were classified into types I, II, and III by Murray and von Neumann
• The modular theory of von Neumann algebras, introduced by Tomita and Takesaki in the 1970s, revolutionized the field by connecting the algebraic structure to the geometry of the Hilbert space
• Connes' classification of injective factors in the 1970s was a major breakthrough in the structure theory of type II and III factors
• The theory of von Neumann algebras has since found applications in various areas of mathematics (operator theory, ergodic theory) and mathematical physics (quantum statistical mechanics, quantum field theory)

Types of Representations

• The left regular representation of a group G on $L^2(G)$ is a fundamental example of a representation
• The GNS representation associated to a state $\phi$ on a C*-algebra $A$ acts on the Hilbert space obtained by completing $A$ with respect to the inner product $\langle a,b \rangle = \phi(b^*a)$
• Irreducible representations are those with no non-trivial invariant closed subspaces
  • Irreducible representations are crucial in the study of C*-algebras and their ideal structure
• Cyclic representations are those possessing a cyclic vector
  • The GNS representation is always cyclic
• Faithful representations injectively map the algebra into bounded operators
• Standard representations are faithful representations associated with faithful normal states
• Covariant representations of dynamical systems $(A,G,\alpha)$ are pairs $(\pi,U)$ where $\pi$ is a representation of $A$ and $U$ is a unitary representation of $G$ satisfying the covariance condition $\pi(\alpha_g(a)) = U_g \pi(a) U_g^*$

States and Their Properties

• States are positive linear functionals of norm one on a C*-algebra
  • Positivity means $\phi(a^*a) \geq 0$ for all $a$ in the algebra
  • The norm condition $\|\phi\| = 1$ ensures states are normalized
• Pure states are extreme points of the convex set of states and cannot be written as non-trivial convex combinations of other states
• Mixed states are convex combinations of pure states and represent statistical ensembles in quantum mechanics
• The GNS construction associates a cyclic representation $\pi_\phi$ to each state $\phi$
  • The cyclic vector in the GNS representation is given by the equivalence class of the identity
• Normal states on a von Neumann algebra are ultraweakly continuous and correspond bijectively to density operators (positive trace-class operators with unit trace)
• Faithful states have GNS representations that are faithful
• The set of normal states is a weakly-* compact convex subset of the dual of the von Neumann algebra

Connections to C*-Algebras

• Every von Neumann algebra is a C*-algebra, but not every C*-algebra is a von Neumann algebra
  • von Neumann algebras have additional topological closure properties (weak operator topology) and always contain the identity operator
• The double commutant theorem characterizes von Neumann algebras as C*-algebras that are equal to their double commutant (the commutant of the commutant)
• The bidual of a C*-algebra, which is its double dual equipped with Arens multiplication, is a von Neumann algebra
• States on a C*-algebra extend uniquely to normal states on its bidual
• The GNS representation of a state on a C*-algebra extends to a normal representation of the bidual
• von Neumann's bicommutant theorem shows that a *-closed unital subalgebra of bounded operators is a von Neumann algebra iff it is equal to its double commutant

Applications in Quantum Mechanics

• von Neumann algebras provide a rigorous mathematical framework for quantum mechanics
  • Observables are modeled as self-adjoint elements of a von Neumann algebra
  • States are modeled as positive linear functionals of norm one (normal states)
• The Heisenberg uncertainty principle can be formulated in terms of noncommutativity of observables in a von Neumann algebra
• In quantum statistical mechanics, equilibrium states are characterized as KMS (Kubo-Martin-Schwinger) states with respect to the modular automorphism group
• Quantum dynamical systems are modeled by one-parameter groups of automorphisms of a von Neumann algebra
  • KMS states are equilibrium states in this context
• Algebraic quantum field theory uses nets of von Neumann algebras to model local observables in relativistic quantum field theories
  • The Tomita-Takesaki modular theory is crucial in understanding the interplay between algebra and geometry in this setting

Theorems and Proofs

• The double commutant theorem: a *-closed unital subalgebra of bounded operators is a von Neumann algebra iff it equals its double commutant
  • Proof uses the Kaplansky density theorem and the strong operator topology
• von Neumann's bicommutant theorem: a *-subalgebra of bounded operators is weakly closed iff it equals its double commutant
• The Kaplansky density theorem: the unit ball of a C*-algebra is strongly dense in the unit ball of its double commutant
  • Used in the proof of the double commutant theorem
• The Tomita-Takesaki theorem: for a faithful normal state on a von Neumann algebra, there exists a modular automorphism group and a modular conjugation operator satisfying the modular relation
  • Proof involves studying the closure of the map $a \Omega \mapsto a^* \Omega$ for the cyclic vector $\Omega$
• Takesaki's theorem: a state is a KMS state iff it is invariant under the modular automorphism group
  • Key result connecting equilibrium states in quantum statistical mechanics to modular theory

Advanced Topics and Current Research

• The Connes embedding problem asks whether every separable type II$_1$ factor embeds into an ultrapower of the hyperfinite II$_1$ factor
  • Equivalent to several major open problems in operator algebras and has implications in quantum information theory
• Free probability theory, initiated by Voiculescu, studies noncommutative probability spaces and has deep connections with the theory of von Neumann algebras
  • Free entropy and free Fisher information have been used to study the fine structure of von Neumann algebras
• Popa's deformation/rigidity theory has revolutionized the structure and classification theory of type II$_1$ factors
  • Powerful techniques such as intertwining-by-bimodules and s-malleable deformations have led to the first examples of non-isomorphic factors with the same invariants
• Quantum information theory has inspired the study of new properties of von Neumann algebras (factorizability, quantum Markov chains) and their relevance to entanglement and quantum communication
• Jones' theory of subfactors, which studies inclusions of von Neumann algebras with finite index, has led to deep connections with knot theory, statistical mechanics, and conformal field theory
  • Planar algebras, introduced by Jones, provide a powerful pictorial framework for studying subfactors and their invariants