On the Stone-Weierstrass theorem

TL;DR

This paper generalizes the Stone-Weierstrass theorem to broader classes of C*-algebras by showing that subalgebras separating factorial states are equal to the entire algebra, extending prior results for separable cases.

Contribution

It extends the Stone-Weierstrass theorem to non-separable C*-algebras by using factorial states, broadening the theorem's applicability.

Findings

B contains the entire algebra if it separates factorial states

Generalization of Popa and Longo's result for non-separable algebras

Open problem remains for pure state separation in the general case

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 This work dates from 2006, and some references may be out of date.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