The Stone-Weierstrass property in Banach algebras

Introduction* Let A be a semi-simple commutative Banach algebra with maximal ideal space . Regarding the elements of A as functions on , we call a subalgebra B of A self-adjoint if corresponding to every f e B the function / defined on by f(x) = f(x) is also in B; we call B separating if to every pair of distinct points x 0 , x e there is an feB such that f(x 0 ) = 0, f{x ) = 1. If every separating self-adjoint subalgebra of A is dense in A, we say that A has the Stone-Weierstrass property.