[Merged by Bors] - feat: define NonUnitalSubalgebra and develop basic API
#5512
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This continues the non-unital-ization of mathlib. - [x] depends on: #5151
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NonUnitalSubalgebra and develop basic APINonUnitalSubalgebra and develop basic API
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This continues the non-unital-ization of mathlib. - [x] depends on: #5151
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This continues the non-unital-ization of mathlib This PR also redefines `StarSubalgebra.centralizer` so that it no longer requires the set `s` provided to be closed under `star`, and instead the carrier is just the `Set.centralizer (s ∪ star s)`. Consequently, this changes some things in von Neumann algebras, where we now need to see that `Set.centralizer (↑S ∪ star ↑S) = Set.centralizer ↑S`, where `S` is a `StarSubalgebra`. Therefore we add the `simp` lemma `StarMemClass.star_coe_eq`. - [x] depends on: #5151 - [x] depends on: #5512
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… `Algebra.adjoin` (#5602) If `S` is non-unital subalgebra of a unital `R`-algebra `A`, there is a natural surjective map `Unitization R S →ₐ[R] Algebra.adjoin R (S : Set A)`. When `1 ∉ S` and `R` is a field, this becomes and `AlgEquiv`. We specialize this to the `ℕ`-unitization of a non-unital subsemiring and its `Subsemiring.closure`, as well as the `ℤ`-unitization of a non-unital subring and its `Subring.closure`. We also extend the above map to a `StarAlgHom` in the case of `NonUnitalStarSubalgebra`s. This continues the non-unital-ization of mathlib. - [x] depends on: #5151 - [x] depends on: #5512 - [x] depends on: #5537
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This continues the non-unital-ization of mathlib. - [x] depends on: #5151
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This continues the non-unital-ization of mathlib This PR also redefines `StarSubalgebra.centralizer` so that it no longer requires the set `s` provided to be closed under `star`, and instead the carrier is just the `Set.centralizer (s ∪ star s)`. Consequently, this changes some things in von Neumann algebras, where we now need to see that `Set.centralizer (↑S ∪ star ↑S) = Set.centralizer ↑S`, where `S` is a `StarSubalgebra`. Therefore we add the `simp` lemma `StarMemClass.star_coe_eq`. - [x] depends on: #5151 - [x] depends on: #5512
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… `Algebra.adjoin` (#5602) If `S` is non-unital subalgebra of a unital `R`-algebra `A`, there is a natural surjective map `Unitization R S →ₐ[R] Algebra.adjoin R (S : Set A)`. When `1 ∉ S` and `R` is a field, this becomes and `AlgEquiv`. We specialize this to the `ℕ`-unitization of a non-unital subsemiring and its `Subsemiring.closure`, as well as the `ℤ`-unitization of a non-unital subring and its `Subring.closure`. We also extend the above map to a `StarAlgHom` in the case of `NonUnitalStarSubalgebra`s. This continues the non-unital-ization of mathlib. - [x] depends on: #5151 - [x] depends on: #5512 - [x] depends on: #5537
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This continues the non-unital-ization of mathlib.
NonUnitalSubrings #5151