Vol. 15, No. 4, 1965

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ISSN: 0030-8730
On sub-algebras of a Cāˆ—-algebra

John R. Ringrose

Vol. 15 (1965), No. 4, 1377ā€“1382
Abstract

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 be a C-sub-algebra containing I. Suppose that separates the pure state space of 𝒜. Then = 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 be a C-sub-algebra of 𝒜, and defne

𝒩 = {f : f is a pure state of 𝒜 and f(B) = 0(B ∈ℬ)},
= {(g,h) : g,h ∈𝒫 and g(B) = h(B)(B ∈ℬ)},
= {A : A ∈𝒜,f(A) = 0(f ∈𝒩) and g(A) = h(A)((g,h) ∈ℰ)}.
Then = .

Mathematical Subject Classification
Primary: 46.65
Milestones
Received: 30 September 1964
Published: 1 December 1965
Authors
John R. Ringrose