Theorems · Theorem · functional analysis

Unitization.norm_splitMul_snd_sq

∀ (𝕜 : Type u_1) {E : Type u_2} [inst : DenselyNormedField 𝕜] [inst_1 : NonUnitalNormedRing E] [inst_2 : StarRing E]
  [CStarRing E] [inst_4 : NormedSpace 𝕜 E] [inst_5 : IsScalarTower 𝕜 E E] [inst_6 : SMulCommClass 𝕜 E E]
  [inst_7 : StarRing 𝕜] [StarModule 𝕜 E] (x : Unitization 𝕜 E),
  ‖((Unitization.splitMul 𝕜 E) x).2‖ ^ 2 ≤ ‖((Unitization.splitMul 𝕜 E) (star x * x)).2‖

This is the key lemma used to establish the instance Unitization.instCStarRing (i.e., proving that the norm on Unitization 𝕜 E satisfies the C⋆-property). We split this one out so that declaring the CStarRing instance doesn't time out.

Defined in

Mathlib.Analysis.CStarAlgebra.Unitization

Cited by

0 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DenselyNormedFieldNonUnitalNormedRingStarRingCStarRingNormedSpaceIsScalarTowerSMulCommClassStarRingStarModule