Theorems · Definition · ring theory

Unitization.starMap

{R : Type u_1} →
  {A : Type u_2} →
    {B : Type u_3} →
      [inst : CommSemiring R] →
        [inst_1 : StarRing R] →
          [inst_2 : NonUnitalSemiring A] →
            [inst_3 : StarRing A] →
              [inst_4 : Module R A] →
                [inst_5 : SMulCommClass R A A] →
                  [inst_6 : IsScalarTower R A A] →
                    [inst_7 : NonUnitalSemiring B] →
                      [inst_8 : StarRing B] →
                        [inst_9 : Module R B] →
                          [inst_10 : SMulCommClass R B B] →
                            [inst_11 : IsScalarTower R B B] →
                              [StarModule R B] → (A →⋆ₙₐ[R] B) → Unitization R A →⋆ₐ[R] Unitization R B

The functorial map on morphisms between the category of non-unital C⋆-algebras with non-unital star homomorphisms and unital C⋆-algebras with unital star homomorphisms. This sends φ : A →⋆ₙₐ[R] B to a map Unitization R A →⋆ₐ[R] Unitization R B given by the formula (r, a) ↦ (r, φ a) (or perhaps more precisely, algebraMap R _ r + ↑a ↦ algebraMap R _ r + ↑(φ a)).

Defined in

Mathlib.Algebra.Algebra.Unitization

Cited by

9 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringStarRingNonUnitalSemiringStarRingModuleSMulCommClassIsScalarTowerNonUnitalSemiringStarRingModuleSMulCommClassIsScalarTowerStarModule