Semi-Perfect C∗ -Algebras and the Stone-Weierstrass Problem
C. J. K. Batty · Journal of the London Mathematical Society · 1986
A separable unital C∗-algebra A is semi-perfect if each member of Ac is annihilated by each boundary affine dependence on the state space of A, or equivalently if there is a natural embedding of Ac into A∗∗, where Ac (introduced by Akemann and Shultz) is the largest C∗-subalgebra of the atomic part of A∗∗ whose elements are continuous on the pure states of A. Separable nuclear C∗-algebras are semi-perfect. If A is a semi-perfect C∗-subalgebra of B, separating the pure states of B, then A = B.