Theorems · Theorem · functional analysis

Commute.cfc

∀ {𝕜 : Type u_1} {A : Type u_2} {p : A → Prop} [inst : RCLike 𝕜] [inst_1 : Ring A] [inst_2 : StarRing A]
  [inst_3 : Algebra 𝕜 A] [inst_4 : TopologicalSpace A] [inst_5 : ContinuousFunctionalCalculus 𝕜 A p]
  [IsSemitopologicalRing A] [T2Space A] {a b : A}, Commute a b → Commute (star a) b → ∀ (f : 𝕜 → 𝕜), Commute (cfc f a) b

An element commutes with cfc f a if it commutes with both a and star a. If the base ring is ℝ or ℝ≥0, see Commute.cfc_real or Commute.cfc_nnreal which don't require the Commute (star a) b hypothesis.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute

Cited by

2 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeRingStarRingAlgebraTopologicalSpaceContinuousFunctionalCalculusIsSemitopologicalRingT2Space