CommutantEdit
Commutants are a central tool in the study of operator algebras, providing a way to isolate the set of operators that act independently of a given family of actions. In the standard setting, one works with a Hilbert space and the bounded linear operators on it. The commutant of a set A of operators is the collection of all operators that commute with every element of A. This simple idea leads to deep structure theorems and connections to symmetry, dynamics, and representation theory.
When a *-algebra of operators is considered, the commutant inherits a rich structure. If A is a subset of the bounded operators on a Hilbert space Hilbert space, the commutant A' consists of all operators T that satisfy TA = AT for every A in A. The notion is dual in a precise sense: the larger the family you start with, the smaller its commutant tends to be, and vice versa. For a single operator, the commutant captures the operators that preserve the same spectral or invariant structure; for a whole algebra, the commutant often encodes the symmetries that the algebra fails to see directly.
Definition and basic notions
Definition. Let H be a complex Hilbert space and let A be a set of bounded operators on H. The commutant A' is
- A' = { T ∈ B(H) : TA = AT for all A ∈ A }. In this language, B(H) denotes the algebra of all bounded operators on H.
Key properties.
- A' is always a *-subalgebra of B(H): it is closed under adjoints and under addition and multiplication.
- The center of a given algebra A is the commutant of A within A itself, i.e., Z(A) = A ∩ A', consisting of those elements that commute with every element of A.
- Taking commutants reverses inclusion: if A ⊆ B(H) then B(H)′ ⊆ A′.
The bicommutant and von Neumann algebras.
- If A is a *-subalgebra of B(H), then the bicommutant A'' (the commutant of A') is an important object. By the bicommutant theorem, A'' is the weak operator topology closure of the von Neumann algebra generated by A. This is a cornerstone result linking algebraic generation to topological closure, and it sits at the heart of the theory of von Neumann algebras.
- In particular, for a large class of algebras arising in practice, A'' is the canonical ambient von Neumann algebra that contains A.
Commutant and structure.
- If A is a commutative (abelian) *-algebra acting on H, then A ⊆ A' and A' contains A. In favorable circumstances, A' is as large as possible while still respecting the commutation with A, revealing how much freedom remains outside A’s direct action.
- A maximal abelian self-adjoint algebra (MASA) is one that is abelian and equal to its own commutant in a precise sense: A' is as small as possible given A’s commutative structure. MASAs are central to decompositions and spectral theory in operator algebras.
Examples
Whole algebra of bounded operators.
- The commutant of the full algebra B(H) is just the scalar multiples of the identity: B(H)' = {λ I : λ ∈ ℂ}. This reflects the fact that the only operators commuting with every operator are the trivial scalars.
A single diagonal operator.
- Let A be a diagonal operator on a finite-dimensional H with respect to some basis and with all eigenvalues distinct. Then A' is the full algebra of diagonal operators in that basis, i.e., the set of all operators that are diagonal in the same eigenbasis. If eigenvalues have multiplicities, A' becomes the block-diagonal algebra that preserves each eigenspace.
Commutant of a normal operator via spectral theory.
- If N is normal and has a spectral decomposition N = ∑ λ_j P_j, then an operator T commutes with N if and only if T preserves each spectral subspace P_j(H). In this sense, the commutant encodes the invariant subspace structure dictated by N.
Representations and symmetries.
- Given a unitary representation {U_g} of a group G, the algebra generated by these unitaries has a commutant consisting of operators that intertwine the representation. This is a fundamental viewpoint in studying symmetries in quantum systems and in harmonic analysis.
Structure, invariants, and applications
Decompositions by commutants.
- The commutant framework provides a route to decompose a Hilbert space into invariant parts under a family of actions. When paired with the bicommutant theorem, this leads to a powerful description of the associated von Neumann algebra and its representations.
Symmetries and observables.
- In mathematical physics, the commutant captures observables that are compatible with a given set of measurements. Operators in the commutant commute with all elements of a chosen set, meaning they can be measured simultaneously without disturbing the outcomes determined by that set.
Classification and factors.
- The study of commutants is closely tied to the classification of von Neumann algebras into types, and to the analysis of factors (algebras with trivial centers). These ideas have influenced developments in representation theory, quantum statistical mechanics, and noncommutative geometry.
Connections to spectral and functional calculus.
- In many settings, the commutant interacts with the spectral theorem: operators that are functions of a given normal operator belong to its commutant. The precise description depends on the spectrum and multiplicities, but the overarching theme is that commutation reflects functional dependence.
Remarks and contexts
Philosophical and methodological notes.
- The commutant concept emphasizes a dual viewpoint: rather than focusing solely on a set of operators, one studies what commutes with them to reveal hidden structure and symmetries. This duality is a recurring theme in operator theory and its applications to physics and geometry.
Variants and related notions.
- The double commutant operation, A → A'', produces canonical closures that are central objects in the theory of C*-algebras and von Neumann algebras.
- The idea extends beyond operators on a single Hilbert space to more general representations and to modules over algebras, where commutants track intertwiners and commuting actions.