Theorems · Theorem · functional analysis

NonUnitalStarAlgHom.norm_map

∀ {F : Type u_1} {A : Type u_2} {B : Type u_3} [inst : NonUnitalCStarAlgebra A] [inst_1 : NonUnitalCStarAlgebra B]
  [inst_2 : FunLike F A B] [NonUnitalAlgHomClass F ℂ A B] [StarHomClass F A B] (φ : F),
  Function.Injective ⇑φ → ∀ (a : A), ‖φ a‖ = ‖a‖

A non-unital star algebra monomorphism of complex C⋆-algebras is isometric.

Defined in

Mathlib.Analysis.CStarAlgebra.Hom

Cited by

2 results in Mathlib

Foundations

Depth 312 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NonUnitalCStarAlgebraNonUnitalCStarAlgebraFunLikeNonUnitalAlgHomClassStarHomClass