Theorems · Definition · functional analysis

NonUnitalStarSubalgebra.nonUnitalCommRingTopologicalClosure

{R : Type u_1} →
  {A : Type u_2} →
    [inst : CommRing R] →
      [inst_1 : TopologicalSpace A] →
        [inst_2 : NonUnitalRing A] →
          [inst_3 : Module R A] →
            [inst_4 : Star A] →
              [inst_5 : ContinuousStar A] →
                [inst_6 : ContinuousConstSMul R A] →
                  [inst_7 : IsSemitopologicalRing A] →
                    [T2Space A] →
                      (s : NonUnitalStarSubalgebra R A) →
                        (∀ (x y : ↥s), x * y = y * x) → NonUnitalCommRing ↥s.topologicalClosure

If a non-unital star subalgebra of a non-unital topological star algebra is commutative, then so is its topological closure. See note [reducible non-instances].

Defined in

Mathlib.Topology.Algebra.NonUnitalStarAlgebra

Cited by

1 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingTopologicalSpaceNonUnitalRingModuleStarContinuousStarContinuousConstSMulIsSemitopologicalRingT2Space