toCLMNonUnitalAlgHom — Mathlib

English

Proofs of the higher-level properties of toOneByOne, ensuring bijectivity and compatibility with algebra operations.

Русский

Доказательства более высокоуровневых свойств toOneByOne, обеспечивающих биекцию и совместимость с алгебраическими операциями.

LaTeX

$$$$\text{Proofs about toOneByOne are straightforward and routine.}$$$$

Lean4

/-- Interpret a `CStarMatrix m n A` as a continuous linear map acting on `C⋆ᵐᵒᵈ (n → A)`. This
version is specialized to the case `m = n` and is bundled as a non-unital algebra homomorphism. -/
noncomputable def toCLMNonUnitalAlgHom [Fintype n] :
    CStarMatrix n n A →ₙₐ[ℂ] (C⋆ᵐᵒᵈ(A, n → A) →L[ℂ] C⋆ᵐᵒᵈ(A, n → A))ᵐᵒᵖ :=
  {
    (MulOpposite.opLinearEquiv ℂ).toLinearMap ∘ₗ
      (toCLM (n := n) (m := n)) with
    map_zero' := by simp
    map_mul' := by
      intros
      simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, LinearMap.coe_comp, LinearEquiv.coe_coe,
        MulOpposite.coe_opLinearEquiv, Function.comp_apply, ← MulOpposite.op_mul, MulOpposite.op_inj]
      ext
      simp [toCLM] }

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