The following noncommutative
extension of the Stone-Weierstrass approximation theorem has been obtained by
Glimm.
Theorem. Let 𝒜 be a C∗-algebra with identity I, and let ℬ be a C∗-sub-algebra
containing I. Suppose that ℬ separates the pure state space of 𝒜. Then
ℬ = 𝒜.
In the present paper, we apply Glimm’s theorem to obtain the following
noncommutative generalisation of another result of Stone.
Let 𝒜 be a C∗-algebra with identity I and pure state space 𝒫. Let ℬ be a
C∗-sub-algebra of 𝒜, and defne
𝒩
= {f : f is a pure state of 𝒜 and f(B) = 0(B ∈ℬ)},
ℰ
= {(g,h) : g,h ∈𝒫 and g(B) = h(B)(B ∈ℬ)},
ℋℬ
= {A : A ∈𝒜,f(A) = 0(f ∈𝒩) and g(A) = h(A)((g,h) ∈ℰ)}.
