Theorems · Theorem · functional analysis
IsSelfAdjoint.isConnected_spectrum_compl
∀ {A : Type u_1} [inst : CStarAlgebra A] {a : A}, IsSelfAdjoint a → IsConnected (spectrum ℂ a)ᶜThe complement of the spectrum of a selfadjoint element in a C⋆-algebra is connected.
- Defined in
- Mathlib.Analysis.CStarAlgebra.Spectrum
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CStarAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites37
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realproof · cited by 25,697
- TopologicalSpaceproof · cited by 24,529
- Set.ofPredproof · cited by 6,101
- Complexstatement and proof · cited by 5,565
- Set.preimageproof · cited by 4,946
- Set.univproof · cited by 3,945
- Compl.complstatement and proof · cited by 2,925
- Set.Nonemptyproof · cited by 2,627
- Complex.ofRealproof · cited by 1,654
- le_of_ltproof · cited by 1,175
- Complex.improof · cited by 591
Cited by1
Results whose statement or proof uses this declaration.
- StarSubalgebra.coe_isUnitproof · cited by 1