On the Stone-Weierstrass theorem

Abstract

This paper extends a version of the Stone-Weierstrass theorem to more general C*-algebras. Namely, assume that A is a unital, not necessarily separable, C*-algebra, and B is a C*-subalgebra containing the unit element. Then, I prove that: If B separates the factorial states of A, then B=A. This generalizes a result of Popa and Longo for the case when A is separable. A true Stone-Weierstrass theorem would state that, if B separates the pure states of A, then B=A. This problem is open even in the separable case. The present paper relies on the more technical, foundational results in the companion article 'on Maximal Measures'. This work dates from 2006, and some references may be out of date. Comments are welcome