FACTORIZATION OF POSITIVE INVERTIBLE OPERATORS IN AF ALGEBRAS

We examine the problem of factoring a positive invertible operator in an AF C-algebra as T*T for some invertible operator T with both T and T-1 in a triangular AF subalgebra. A factorization theorem for a cert ain class of positive invertible operators in AF algebras is proven. H owever, we explicitly construct a positive invertible operator in the CAR algebra which cannot be factored with respect to the 2(infinity) r efinement algebra. Our main result generalizes this example, showing t hat in any AF algebra, there exist positive invertible operators which fail to factor with respect to a given triangular AF subalgebra. We a lso show that in the context of AF algebras, the notions of having a f actorization and having a weak factorization are the same.