Theorems · Theorem · functional analysis

Commute.mul_nonneg

∀ {A : Type u_1} [inst : NonUnitalRing A] [inst_1 : PartialOrder A] [inst_2 : StarRing A] [StarOrderedRing A]
  [inst_4 : TopologicalSpace A] [inst_5 : Module ℝ A] [inst_6 : IsScalarTower ℝ A A] [inst_7 : SMulCommClass ℝ A A]
  [NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [NonnegSpectrumClass ℝ A] {a b : A},
  0 ≤ a → 0 ≤ b → Commute a b → 0 ≤ a * b

In a C⋆-algebra, commuting nonnegative elements have nonnegative products.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Cited by

3 results in Mathlib

Foundations

Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonUnitalRingPartialOrderStarRingStarOrderedRingTopologicalSpaceModuleIsScalarTowerSMulCommClassNonUnitalContinuousFunctionalCalculusNonnegSpectrumClass