Gelfand Duality #
gelfandTransform is an algebra homomorphism from a topological
C(character_space 𝕜 A, 𝕜). In the case where
A is a commutative complex Banach algebra, then
the Gelfand transform is actually spectrum-preserving (
A is a commutative C⋆-algebra over
ℂ, then the Gelfand transform is a surjective isometry,
and even an equivalence between C⋆-algebras.
Main definitions #
Ideal.toCharacterSpace: constructs an element of the character space from a maximal ideal in a commutative complex Banach algebra
WeakDual.CharacterSpace.compContinuousMap: The functorial map taking
ψ : A →⋆ₐ[ℂ] Bto a continuous function
characterSpace ℂ B → characterSpace ℂ Agiven by pre-composition with
Main statements #
spectrum.gelfandTransform_eq: the Gelfand transform is spectrum-preserving when the algebra is a commutative complex Banach algebra.
gelfandTransform_isometry: the Gelfand transform is an isometry when the algebra is a commutative (unital) C⋆-algebra over
gelfandTransform_bijective: the Gelfand transform is bijective when the algebra is a commutative (unital) C⋆-algebra over
StarAlgEquivis defined, realize
- Prove that if
Ais the unital C⋆-algebra over
ℂgenerated by a fixed normal element
xin a larger C⋆-algebra
characterSpace ℂ Ais homeomorphic to
spectrum ℂ x.
- From the previous result, construct the continuous functional calculus.
- Show that if
Xis a compact Hausdorff space, then
Xis (canonically) homeomorphic to
characterSpace ℂ C(X, ℂ).
- Conclude using the previous fact that the functors
characterSpace ℂ ·along with the canonical homeomorphisms described above constitute a natural contravariant equivalence of the categories of compact Hausdorff spaces (with continuous maps) and commutative unital C⋆-algebras (with unital ⋆-algebra homomorphisms); this is known as Gelfand duality.
Gelfand transform, character space, C⋆-algebra
Every maximal ideal in a commutative complex Banach algebra gives rise to a character on that
algebra. In particular, the character, which may be identified as an algebra homomorphism due to
WeakDual.CharacterSpace.equivAlgHom, is given by the composition of the quotient map and
the Gelfand-Mazur isomorphism
a : A is not a unit, then some character takes the value zero at
a. This is equivalent
gelfandTransform ℂ A a takes the value zero at some character.
The Gelfand transform as a
StarAlgEquiv between a commutative unital C⋆-algebra over
and the continuous functions on its
The functorial map taking
ψ : A →⋆ₐ[ℂ] B to a continuous function
characterSpace ℂ B → characterSpace ℂ A obtained by pre-composition with
WeakDual.CharacterSpace.compContinuousMap sends the identity to the identity.
WeakDual.CharacterSpace.compContinuousMap is functorial.