Ideals of factors

Every primitive C*-algebra is prime. An old theorem of Dixmier shows that the converse is true if the algebra is separable. Recently Weaver has shown that without separability this is false. AW* -factors are easily seen to be prime. Are all their ideals primitive? A number of partial results have appeared recently. It turns out that the answer is always positive and that this follows immediately from a theorem of FB Wright proved nearly 50 years ago. His proof was hard and complicated. He works in a more general setting but when his result is specialised to the class of operator algebras investigated here, we are able to give a new, easy proof that the ideals of an AW* -factor form a well-ordered set; from this, it follows swiftly that every proper ideal of an AW*-factor is primitive.