Theorems · Theorem · functional analysis

CStarAlgebra.isStrictlyPositive_iff_exists_isUnit_and_isSelfAdjoint_and_eq_mul_self

∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : TopologicalSpace A]
  [StarOrderedRing A] [inst_5 : Algebra ℝ A] [ContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [NonnegSpectrumClass ℝ A]
  [IsSemitopologicalRing A] [T2Space A] {a : A}, IsStrictlyPositive a ↔ ∃ b, IsUnit b ∧ IsSelfAdjoint b ∧ a = b * b

Defined in: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic
Cited by: 0 results in Mathlib
Foundations: Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: PartialOrder Ring StarRing TopologicalSpace StarOrderedRing Algebra ContinuousFunctionalCalculus NonnegSpectrumClass IsSemitopologicalRing T2Space

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Realstatement and proof · cited by 25,697
TopologicalSpacestatement and proof · cited by 24,529
Algebrastatement and proof · cited by 11,388
Ringstatement and proof · cited by 7,463
PartialOrderstatement and proof · cited by 6,410
StarRingstatement and proof · cited by 1,686
IsUnitstatement and proof · cited by 1,602
T2Spacestatement and proof · cited by 1,351
Star.starproof · cited by 1,082
StarOrderedRingstatement and proof · cited by 587
IsSelfAdjointstatement and proof · cited by 545
spectrumproof · cited by 510

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