Editing Stone–Weierstrass theorem (section)

== Stone–Weierstrass theorem, C*-algebra version ==
The space of complex-valued continuous functions on a compact Hausdorff space <math>X</math> i.e. <math>C(X, \Complex)</math> is the canonical example of a unital [[C*-algebra#Commutative C.2A-algebras|commutative C*-algebra]] <math>\mathfrak{A}</math>. The space ''X'' may be viewed as the space of pure states on <math>\mathfrak{A}</math>, with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved, is as follows:

{{math theorem | name = Conjecture | math_statement = If a unital [[C*-algebra]] <math>\mathfrak{A}</math> has a C*-subalgebra <math>\mathfrak{B}</math> which separates the pure states of <math>\mathfrak{A}</math>, then <math>\mathfrak{A} = \mathfrak{B}</math>.}}

In 1960, [[James Glimm|Jim Glimm]] proved a weaker version of the above conjecture.

{{math theorem | name = Stone–Weierstrass theorem (C*-algebras)<ref>{{cite journal |first=James |last=Glimm |author-link=James Glimm |title=A Stone–Weierstrass Theorem for C*-algebras |journal=[[Annals of Mathematics]] |series=Second Series |volume=72 |issue=2 |year=1960 |pages=216–244 [Theorem 1] |jstor=1970133 |doi=10.2307/1970133}}</ref> | math_statement = If a unital C*-algebra <math>\mathfrak{A}</math> has a C*-subalgebra <math>\mathfrak{B}</math> which separates the pure state space (i.e. the weak-* closure of the pure states) of <math>\mathfrak{A}</math>, then <math> \mathfrak{A}= \mathfrak{B}</math>.}}