Commutants and bicommutants are key concepts in von Neumann algebra theory. They define sets of operators that commute with given subsets, helping us understand the structure of operator algebras. These tools connect algebraic properties with topological ones.
The theorem is a cornerstone result, showing that for self-adjoint subsets, the von Neumann algebra they generate equals their double . This powerful idea links algebraic structure to , providing a unique characterization of von Neumann algebras.
Definition of commutant
Fundamental concept in von Neumann algebra theory defines set of operators commuting with given subset
Plays crucial role in understanding structure and properties of operator algebras
Algebraic properties
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Commutant A′ of subset A in algebra B consists of elements commuting with every element in A
Forms algebra itself closed under addition, scalar multiplication, and operator multiplication
Contains identity operator and closed under taking adjoints
Satisfies A⊆A′′ (double commutant always contains original set)
Topological properties
Commutant A′ always weakly closed in B(H) (bounded operators on Hilbert space)
Weak closure of A contained in double commutant A′′
Commutant A′ forms von Neumann algebra when A is self-adjoint
Bicommutant theorem
Cornerstone result in von Neumann algebra theory connects algebraic and topological properties
Provides characterization of von Neumann algebras in terms of their commutants
Statement of theorem
For self-adjoint subset A of B(H), von Neumann algebra generated by A equals its double commutant A′′
Equivalently, A′′=AWOT (weak operator topology closure of A)
Implies von Neumann algebras uniquely determined by their algebraic structure
Proof outline
Show A′′⊆AWOT using density argument and properties of weak topology
Prove AWOT⊆A′′ using Kaplansky density theorem
Combine results to establish equality A′′=AWOT
Double commutant
Powerful tool for studying von Neumann algebras and their properties
Connects algebraic structure with topological closure
Relationship to von Neumann algebras
Von Neumann algebra M satisfies M=M′′ (equal to its double commutant)
Provides algebraic characterization of von Neumann algebras
Allows study of von Neumann algebras through commutation relations
Weak closure properties
Double commutant A′′ always weakly closed in B(H)
Equals weak closure of algebra generated by A and identity operator
Preserves important structural properties (self-adjointness, unitarity) of original set A
Commutant in B(H)
Specific case of commutant in context of bounded operators on Hilbert space
Crucial for understanding operator algebras and their properties
Bounded operators on Hilbert space
B(H) denotes algebra of all bounded linear operators on Hilbert space H
Forms von Neumann algebra itself with operator norm topology
Contains all von Neumann algebras as subalgebras
Commutant vs weak closure
Commutant A′ in B(H) always weakly closed
Weak closure of A generally smaller than A′′ (strict inclusion possible)
Equality AWOT=A′′ holds for self-adjoint sets (bicommutant theorem)
Applications of commutants
Commutants provide powerful tools for analyzing structure of von Neumann algebras
Enable decomposition and classification of operator algebras
Factor decomposition
Factors defined as von Neumann algebras with trivial center (Z(M)=M∩M′=CI)
Commutants used to decompose general von Neumann algebras into direct integrals of factors
Allows reduction of many problems to study of simpler factor algebras
Tensor products
Commutants play crucial role in defining and studying tensor products of von Neumann algebras
For von Neumann algebras M⊆B(H) and N⊆B(K), (M⊗N)′=M′⊗N′
Enables construction of new von Neumann algebras from simpler ones
Commutative vs non-commutative
Distinction between commutative and non-commutative von Neumann algebras fundamental in theory
Reflects different mathematical and physical structures
Abelian von Neumann algebras
Commutative von Neumann algebras (all elements commute with each other)
Isomorphic to L∞(X,μ) for some measure space (X,μ)
Correspond to classical observables in
Examples include multiplication operators on L2 spaces
Non-abelian examples
Matrix algebras Mn(C) (finite-dimensional case)
B(H) for infinite-dimensional Hilbert space H
Group von Neumann algebras associated with non-abelian groups
Represent quantum observables and symmetries in physics
Relative commutants
Generalization of commutant concept to subalgebras
Provides finer structure analysis of von Neumann algebras
Definition and properties
For von Neumann algebras N⊆M, relative commutant N′∩M consists of elements in M commuting with all of N
Forms von Neumann subalgebra of M
Measures "how much larger" M compared to N
Connection to subfactors
Study of inclusions N⊆M of II₁ factors central in subfactor theory
Jones index [M:N] related to properties of relative commutant N′∩M
Leads to classification of subfactors and discovery of new mathematical structures (planar algebras)
Commutant in representation theory
Commutants provide important tools for analyzing group representations
Connect representation theory with von Neumann algebra theory
Group representations
Representation π of group G on Hilbert space H induces von Neumann algebra π(G)′′
Commutant π(G)′ contains operators commuting with all group elements
Structure of π(G)′ reflects properties of representation (irreducibility, decomposition)
Schur's lemma
Fundamental result states commutant of irreducible representation consists only of scalar multiples of identity
Equivalent to π(G)′=CI for irreducible π
Generalizes to von Neumann algebra setting (factors have trivial center)
Commutant lifting theorem
Important result in operator theory relating commutants of operators and their compressions
Has applications in control theory and function theory
Statement of theorem
Given contraction T on Hilbert space H and its minimal isometric dilation V on K⊇H
Any operator X commuting with T can be "lifted" to operator Y commuting with V
Formally, ∀X∈T′,∃Y∈V′ with PHY∣H=X and ∥Y∥=∥X∥
Applications in operator theory
Provides tool for studying contractions through their isometric dilations
Used in interpolation problems for analytic functions
Connects operator theory with function theory on unit disk
Commutant in quantum mechanics
Commutants play fundamental role in mathematical formulation of quantum mechanics
Connect algebraic structure with physical observables and symmetries
Observables and symmetries
Observables represented by self-adjoint operators in von Neumann algebra
Symmetries correspond to unitary operators in commutant of observables
Commuting observables can be simultaneously measured (uncertainty principle)
Heisenberg picture vs Schrödinger picture
Heisenberg picture: observables evolve in time, states fixed
Schrödinger picture: states evolve, observables fixed
Commutant of time evolution operator determines constants of motion in both pictures