commutants

Commutants are a fundamental concept in algebra and functional analysis describing elements that commute with a given set. In the context of operator algebras on a Hilbert space, let H be a Hilbert space and B(H) the algebra of all bounded linear operators on H. For a subset S ⊆ B(H), the commutant of S is defined as S' = { T ∈ B(H) : TS = ST for all S ∈ S }. The commutant S' is a unital *-subalgebra of B(H) and, in fact, a von Neumann algebra (closed in the weak operator topology).

A related construction is the bicommutant. For a subset S, its bicommutant is S'' = (S')'. The double commutant theorem states that if A ⊆ B(H) is a *-subalgebra containing the identity, then A'' is the von Neumann algebra generated by A, and A is a von Neumann algebra if and only if A = A'' (equivalently, A is closed in the weak operator topology).

Typical examples illustrate the concept. The commutant of the whole B(H) is the set of scalar multiples of the identity. The commutant of a single operator A consists of all operators that commute with A. The commutant construction is central to the study and classification of von Neumann algebras, representations, and quantum observables.

Outside operator algebras, the idea generalizes to centralizers in rings and groups: the commutant (or centralizer) of a subset S in a ring R or group G is the set of elements that commute with every element of S. In many contexts, the commutant provides a convenient way to capture symmetry and invariance properties.