Theorems · Definition · functional analysis

WeakDual.gelfandTransform

(𝕜 : Type u_1) →
  (A : Type u_2) →
    [inst : CommRing 𝕜] →
      [NoZeroDivisors 𝕜] →
        [inst_2 : TopologicalSpace 𝕜] →
          [inst_3 : IsTopologicalRing 𝕜] →
            [inst_4 : TopologicalSpace A] →
              [inst_5 : Semiring A] → [inst_6 : Algebra 𝕜 A] → A →ₐ[𝕜] C(↑(WeakDual.characterSpace 𝕜 A), 𝕜)

The Gelfand transform is an algebra homomorphism (over 𝕜) from a topological 𝕜-algebra 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 A to 𝕜.

Defined in

Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace

Cited by

8 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingNoZeroDivisorsTopologicalSpaceIsTopologicalRingTopologicalSpaceSemiringAlgebra