Disjointness and quasi-equivalence of representations #
Building on UnitaryEquiv, this file introduces the two coarser comparisons of
*-representations from superselection theory:
- an intertwiner is a bounded operator
T : H₁ →L H₂withT π₁(a) = π₂(a) T; - two representations are disjoint (
AreDisjoint) when the only intertwiner is0— equivalently, they share no unitarily equivalent subrepresentation; - two representations are quasi-equivalent (
QuasiEquiv) when there is a*-isomorphism of the generated von Neumann algebrasπ₁(A)'' ≃⋆ₐ π₂(A)''carryingπ₁(a)toπ₂(a).
We prove the basic algebra of intertwiners, that disjointness is symmetric and irreflexive (unitarily equivalent representations are never disjoint), that quasi-equivalence is an equivalence relation, and that unitary equivalence implies quasi-equivalence.
Intertwiners #
T : H₁ →L H₂ intertwines π₁ and π₂ when T π₁(a) = π₂(a) T.
Equations
- Physicslib4.GNS.Intertwines π₁ π₂ T = ∀ (a : A) (x : H₁), T ((π₁ a) x) = (π₂ a) (T x)
Instances For
The composition of intertwiners is an intertwiner.
The adjoint of an intertwiner π₁ → π₂ is an intertwiner π₂ → π₁.
Disjointness #
Two representations are disjoint when the only operator intertwining them
is 0.
Equations
- Physicslib4.GNS.AreDisjoint π₁ π₂ = ∀ (T : H₁ →L[ℂ] H₂), Physicslib4.GNS.Intertwines π₁ π₂ T → T = 0
Instances For
Disjointness is symmetric (take adjoints of intertwiners).
A representation on a nonzero Hilbert space is never disjoint from itself (the identity is a nonzero intertwiner).
Unitarily equivalent representations are not disjoint (on nonzero spaces): the implementing unitary is a nonzero intertwiner.
Schur's lemma and the irreducible dichotomy #
Schur's lemma. A nonzero intertwiner between two irreducible representations
rescales to a unitary: π₁ and π₂ are unitarily equivalent. The operator T⋆T
commutes with π₁, hence is a positive scalar r · 1; the normalisation
(√r)⁻¹ · T is then a linear isometry, and T T⋆ being a nonzero scalar makes it
surjective, i.e. a unitary equivalence.
The irreducible dichotomy. Two irreducible representations are either disjoint or unitarily equivalent.
Schur multiplicity. The space of intertwiners between two irreducible
representations is at most one-dimensional: any two intertwiners with S ≠ 0 are
proportional, T = λ • S.
The endomorphism algebra of an irreducible representation #
A self-intertwiner is exactly an operator in the commutant of the representation.
The endomorphism algebra of an irreducible representation is ℂ · 1. Every
self-intertwiner of an irreducible representation is a scalar multiple of the
identity.
The commutant (self-intertwiner) von Neumann algebra #
The commutant π(A)' as a von Neumann algebra: the algebra of
self-intertwiners of π — its "gauge"/intertwiner algebra. The centralizer of
π(A) is self-adjoint, and a commutant is always a von Neumann algebra
(S''' = S').
Equations
Instances For
Membership in the commutant von Neumann algebra is exactly being a
self-intertwiner of π.
A representation is irreducible iff its commutant von Neumann algebra is
trivial, π(A)' = ℂ · 1. This is the von Neumann form of Schur's lemma: the
gauge/intertwiner algebra collapses to the scalars exactly for irreducibles.
Double-commutant duality #
Double-commutant duality (I). The commutant of the generated von Neumann
algebra π(A)'' is the commutant von Neumann algebra π(A)'. This is the
triple-commutant collapse S''' = S' applied to the self-adjoint image π(A).
Double-commutant duality (II). The commutant of the commutant von Neumann
algebra π(A)' is the generated von Neumann algebra π(A)'' — this is exactly the
definition of the bicommutant. So π(A)'' and π(A)' are each other's commutants.
A factor and its commutant. A von Neumann algebra and its commutant share the
same center (their intersection is symmetric), so the generated algebra π(A)'' is a
factor if and only if its commutant π(A)' is a factor.
Triviality duality. The commutant collapses to the scalars π(A)' = ℂ · 1 if
and only if the generated algebra is everything π(A)'' = B(H). This is the commutant
form of the equivalence "irreducible ⟺ generates B(H)".
The pure-state dichotomy #
The pure-state dichotomy. The GNS representations of two pure states are either disjoint or unitarily equivalent. Pure states thus fall into superselection sectors: the sector of a pure state is the unitary-equivalence class of its (irreducible) GNS representation.
Quasi-equivalence #
Conjugation by a unitary preserves the adjoint (star).
Conjugation by a unitary as a *-algebra isomorphism of the operator
algebras.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A *-isomorphism carrying the underlying set of one star-subalgebra onto that
of another restricts to a *-isomorphism between them (cross-space form).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The generators π(a) lie in the generated von Neumann algebra π(A)''.
Two representations are quasi-equivalent when there is a *-isomorphism of
their generated von Neumann algebras carrying π₁(a) to π₂(a).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Quasi-equivalence is reflexive.
Quasi-equivalence is symmetric.
Quasi-equivalence is transitive.
Unitary equivalence implies quasi-equivalence. The conjugation
*-isomorphism restricts to a *-isomorphism of the generated von Neumann
algebras carrying π₁(a) to π₂(a).