Character space of a topological algebra #
The character space of a topological algebra is the subset of elements of the weak dual that are also algebra homomorphisms. This space is used in the Gelfand transform, which gives an isomorphism between a commutative C⋆-algebra and continuous functions on the character space of the algebra. This, in turn, is used to construct the continuous functional calculus on C⋆-algebras.
Implementation notes #
WeakDual.characterSpace 𝕜 A as a subset of the weak dual, which automatically puts the
correct topology on the space. We then define
WeakDual.CharacterSpace.toAlgHom which provides the
algebra homomorphism corresponding to any element. We also provide
which provides the element as a continuous linear map. (Even though
WeakDual 𝕜 A is a type copy of
A →L[𝕜] 𝕜, this is often more convenient.)
character space, Gelfand transform, functional calculus
The character space of a topological algebra is the subset of elements of the weak dual that are also algebra homomorphisms.
Elements of the character space are continuous linear maps.
An element of the character space, as a continuous linear map.
Elements of the character space are non-unital algebra homomorphisms.
An element of the character space, as a non-unital algebra homomorphism.
In a unital algebra, elements of the character space are algebra homomorphisms.
An element of the character space of a unital algebra, as an algebra homomorphism.
under suitable mild assumptions on
𝕜, the character space is a closed set in
WeakDual 𝕜 A.
The Gelfand transform is an algebra homomorphism (over
𝕜) from a topological
A into the
𝕜-algebra of continuous
𝕜-valued functions on the
characterSpace 𝕜 A.
The character space itself consists of all algebra homomorphisms from