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
Top images from around the web for Algebraic properties
Validity of Closed Ideals in Algebras of Series of Square Analytic Functions View original
Is this image relevant?
On the Commutativity of States in von Neumann Algebras | Results in Mathematics View original
Is this image relevant?
Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model View original
Is this image relevant?
Validity of Closed Ideals in Algebras of Series of Square Analytic Functions View original
Is this image relevant?
On the Commutativity of States in von Neumann Algebras | Results in Mathematics View original
Is this image relevant?
1 of 3
Top images from around the web for Algebraic properties
Validity of Closed Ideals in Algebras of Series of Square Analytic Functions View original
Is this image relevant?
On the Commutativity of States in von Neumann Algebras | Results in Mathematics View original
Is this image relevant?
Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model View original
Is this image relevant?
Validity of Closed Ideals in Algebras of Series of Square Analytic Functions View original
Is this image relevant?
On the Commutativity of States in von Neumann Algebras | Results in Mathematics View original
Is this image relevant?
1 of 3
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
Key Terms to Review (16)
Bicommutant: The bicommutant is the double commutant of a subset of a von Neumann algebra, which is essential for understanding the structure and properties of these algebras. Specifically, given a subset $M$ of a von Neumann algebra $A$, the bicommutant is denoted as $M''$ and consists of all operators in $A$ that commute with every operator in the commutant of $M$. This concept helps establish important results like the double commutant theorem, linking the algebraic properties of operators to topological ones.
Commutant: In the context of von Neumann algebras, the commutant of a set of operators is the set of all bounded operators that commute with each operator in the original set. This concept is fundamental in understanding the structure of algebras, as the relationship between a set and its commutant can reveal important properties about the underlying mathematical framework.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational work in various fields, including functional analysis, algebra, and mathematical logic. His contributions laid the groundwork for the development of Hilbert spaces, which are essential in quantum mechanics, noncommutative measure theory, and the mathematical formulation of physics, particularly in string theory.
Double commutant theorem: The double commutant theorem states that for a von Neumann algebra, the original algebra is equal to the double commutant of any of its subsets. This means that if you take a set of operators and find their commutant, and then find the commutant of that commutant, you will return to a larger structure that includes the original algebra. This theorem is foundational in understanding how algebras relate to their representations and the role of dual structures in operator theory.
Dual Space: The dual space of a vector space consists of all linear functionals defined on that space, essentially capturing the way vectors can be analyzed through their interactions with scalars. This concept is important because it connects various structures in functional analysis and plays a crucial role in understanding the behavior of operators and algebraic objects within various mathematical contexts.
Functional Analysis: Functional analysis is a branch of mathematical analysis that deals with the study of vector spaces and the linear operators acting upon them. It plays a crucial role in understanding how these operators can be utilized in various contexts, particularly in quantum mechanics and in the theory of differential equations. The concepts of weights, traces, commutants, and bicommutants are all foundational ideas within functional analysis that help characterize the structure and behavior of operators in von Neumann algebras.
Gelfand-Naimark Theorem: The Gelfand-Naimark Theorem states that every commutative C*-algebra is isometrically *-isomorphic to a continuous function algebra on a compact Hausdorff space. This theorem provides a crucial link between algebraic structures and topological spaces, helping to understand the dual nature of C*-algebras and their representations.
Hilbert Space Representation: Hilbert space representation refers to the mathematical framework in which linear operators act on Hilbert spaces, allowing the study of quantum mechanics and functional analysis through the lens of linear algebra. This representation is crucial for understanding the structure of operator algebras, as it connects algebraic concepts to geometric interpretations in infinite-dimensional spaces, playing a key role in both the commutant and bicommutant theories as well as in C*-dynamical systems.
John von Neumann: John von Neumann was a pioneering mathematician and physicist whose contributions laid the groundwork for various fields, including functional analysis, quantum mechanics, and game theory. His work on operator algebras led to the development of von Neumann algebras, which are critical in understanding quantum mechanics and statistical mechanics.
Kaplansky's Density Theorem: Kaplansky's Density Theorem states that for a von Neumann algebra, the weak closure of the set of all finite rank operators is dense in the algebra if and only if the algebra contains no non-zero finite-dimensional representations. This theorem connects to the structure of von Neumann algebras and highlights the importance of commutants and bicommutants in understanding the relationships between operators.
Norm closure: Norm closure refers to the smallest closed set containing a given subset in a normed vector space, where the closure is determined by the limits of sequences of points within that subset. In the context of functional analysis, understanding norm closure is crucial as it helps in identifying important properties of operators and algebras, especially in terms of their completeness and convergence behaviors. This concept is especially relevant when discussing commutants and bicommutants, as well as the dual spaces that characterize von Neumann algebras.
Quantum mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality, superposition, and entanglement, which challenge classical intuitions and have implications for various mathematical frameworks, including those found in operator algebras.
Strong Operator Topology: Strong operator topology (SOT) is a way to define convergence of sequences of bounded linear operators on a Hilbert space, where a sequence of operators converges if it converges pointwise on every vector in the space. This concept is crucial in understanding the structure and behavior of von Neumann algebras, as well as their applications in various areas such as quantum mechanics and noncommutative geometry.
Type I von Neumann algebra: A Type I von Neumann algebra is a special class of von Neumann algebras that can be represented on a Hilbert space with a decomposition into a direct sum of one-dimensional projections. This structure is closely related to the presence of states that can be represented by bounded operators, which reflects certain properties like amenability and is crucial in understanding representations in various mathematical and physical contexts.
Type II von Neumann Algebra: A Type II von Neumann algebra is a specific class of von Neumann algebras that can be characterized by their rich structure, including the existence of a faithful normal state and the presence of non-trivial projections that cannot be decomposed into a direct sum of smaller projections. These algebras are crucial in understanding various mathematical frameworks, as they exhibit properties that bridge the gap between classical and quantum mechanics, and are often involved in advanced concepts like amenability, local structures, and quantum field theories.
Weak closure: Weak closure refers to the smallest closed set in a topological space that contains a given set when considering a weaker topology, typically involving convergence in the sense of weak limits. In the context of operator algebras, weak closure is significant because it relates to how elements behave under limits of sequences and impacts the structure of von Neumann algebras. Understanding weak closure is essential for grasping concepts like the GNS construction and the relationships between commutants and bicommutants.