nonUnitalStarAlgHom_precomp — Mathlib

English

If f is a homeomorphism X ≃ₜ Y sending 0 to 0, then the induced uniform equivalence provides a uniform equality between the actions of f and f⁻¹ on C(X, R)₀ and C(Y, R)₀.

Русский

Если f — гомоморфизм пространства X в Y, переводящий 0 в 0, то получаемое равномерное соответствие задаёт равномерное равенство действий f и f⁻¹ на C(X, R)₀ и C(Y, R)₀.

LaTeX

$$$\text{UniformEquiv.arrowCongrLeft₀}(f)\quad \text{and}\quad \text{UniformEquiv.arrowCongrLeft₀}(f^{-1})$$$

Lean4

/-- The functor `C(·, R)₀` from topological spaces with zero (and `ContinuousMapZero` maps) to
non-unital star algebras. -/
@[simps]
def nonUnitalStarAlgHom_precomp (f : C(X, Y)₀) : C(Y, R)₀ →⋆ₙₐ[R] C(X, R)₀
    where
  toFun g := g.comp f
  map_zero' := rfl
  map_add' _ _ := rfl
  map_mul' _ _ := rfl
  map_star' _ := rfl
  map_smul' _ _ := rfl

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