Theorems · Theorem · functional analysis

CFC.spectrum_nonempty

Deprecated since 2026-03-08Mathlib marks this declaration as deprecated.

∀ (R : Type u_1) {A : Type u_2} {p : A → Prop} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
  [inst_7 : StarRing A] [inst_8 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] [Nontrivial A] (a : A),
  autoParam (p a) CFC.spectrum_nonempty._auto_1 → (spectrum R a).Nonempty

In an R-algebra with a continuous functional calculus, every element satisfying the predicate has nonempty R-spectrum.

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

0 results in Mathlib

Foundations

Depth 19 from the axioms · uses no axioms

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculusNontrivial