🧮Von Neumann Algebras Unit 4 – Representations and states
Representations and states form the backbone of von Neumann algebra theory. They bridge abstract algebraic structures with concrete realizations on Hilbert spaces, enabling rigorous mathematical analysis of quantum systems.
The GNS construction links states to representations, while pure and mixed states model quantum superposition and statistical ensembles. Normal states and faithful representations provide crucial tools for studying von Neumann algebras and their applications in quantum mechanics and beyond.
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 L2(G) is a fundamental example of a representation
The GNS representation associated to a state ϕ on a C*-algebra A acts on the Hilbert space obtained by completing A with respect to the inner product ⟨a,b⟩=ϕ(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,α) are pairs (π,U) where π is a representation of A and U is a unitary representation of G satisfying the covariance condition π(αg(a))=Ugπ(a)Ug∗
States and Their Properties
States are positive linear functionals of norm one on a C*-algebra
Positivity means ϕ(a∗a)≥0 for all a in the algebra
The norm condition ∥ϕ∥=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 πϕ to each state ϕ
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 aΩ↦a∗Ω for the cyclic vector Ω
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 II1 factor embeds into an ultrapower of the hyperfinite II1 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 II1 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