2.2 Commutants and bicommutants

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 bicommutant theorem is a cornerstone result, showing that for self-adjoint subsets, the von Neumann algebra they generate equals their double commutant. This powerful idea links algebraic structure to weak closure, 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 $A \subseteq A''$ (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, $A'' = \overline{A}^{WOT}$ (weak operator topology closure of $A$)
• Implies von Neumann algebras uniquely determined by their algebraic structure

Proof outline

• Show $A'' \subseteq \overline{A}^{WOT}$ using density argument and properties of weak topology
• Prove $\overline{A}^{WOT} \subseteq A''$ using Kaplansky density theorem
• Combine results to establish equality $A'' = \overline{A}^{WOT}$

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 $\overline{A}^{WOT} = A''$ 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 ($Z(M) = M \cap M' = \mathbb{C}I$)
• 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 $M \subseteq B(H)$ and $N \subseteq B(K)$, $(M \otimes N)' = M' \otimes N'$
• 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 $L^\infty(X, \mu)$ for some measure space $(X, \mu)$
• Correspond to classical observables in quantum mechanics
• Examples include multiplication operators on $L^2$ spaces

Non-abelian examples

• Matrix algebras $M_n(\mathbb{C})$ (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 $N \subseteq M$, relative commutant $N' \cap M$ 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 $N \subseteq M$ of II₁ factors central in subfactor theory
• Jones index $[M:N]$ related to properties of relative commutant $N' \cap M$
• 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 $\pi$ of group $G$ on Hilbert space $H$ induces von Neumann algebra $\pi(G)''$
• Commutant $\pi(G)'$ contains operators commuting with all group elements
• Structure of $\pi(G)'$ 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 $\pi(G)' = \mathbb{C}I$ for irreducible $\pi$
• 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 $K \supseteq H$
• Any operator $X$ commuting with $T$ can be "lifted" to operator $Y$ commuting with $V$
• Formally, $\forall X \in T', \exists Y \in V'$ with $P_H Y|_H = X$ 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