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chore: rename cfc_map_quasispectrum (#12640)
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j-loreaux committed May 3, 2024
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Expand Up @@ -173,7 +173,7 @@ open Classical in
/-- This is the *continuous functional calculus* of an element `a : A` in a non-unital algebra
applied to bare functions. When either `a` does not satisfy the predicate `p` (i.e., `a` is not
`IsStarNormal`, `IsSelfAdjoint`, or `0 ≤ a` when `R` is `ℂ`, `ℝ`, or `ℝ≥0`, respectively), or when
`f : R → R` is not continuous on the quasispectrum of `a` or `f 0 ≠ 0`, then `cfc f a` returns the
`f : R → R` is not continuous on the quasispectrum of `a` or `f 0 ≠ 0`, then `cfcₙ f a` returns the
junk value `0`.
This is the primary declaration intended for widespread use of the continuous functional calculus
Expand Down Expand Up @@ -247,7 +247,7 @@ variable (R) in
lemma cfcₙ_id' : cfcₙ (fun x : R ↦ x) a = a := cfcₙ_id R a

/-- The **spectral mapping theorem** for the non-unital continuous functional calculus. -/
lemma cfc_map_quasispectrum : σₙ R (cfcₙ f a) = f '' σₙ R a := by
lemma cfcₙ_map_quasispectrum : σₙ R (cfcₙ f a) = f '' σₙ R a := by
simp [cfcₙ_apply f a, cfcₙHom_map_quasispectrum (p := p)]

lemma cfcₙ_predicate : p (cfcₙ f a) :=
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