Factorization in analytic crossed products

Abstract

Let M be a von Neumann algebra, let α be a *-automorphism of M, and let M\ times_α\bm{Z} be the crossed product determined by M and α. In this paper, considering the Cholesky decomposition for a positive operator in M\ times_α\bm{Z}, we give a factorization theorem for positive operators in M\ times_α\bm{Z} with respect to analytic crossed product M\ times_α\bm{Z}₊ determined by M and α. And we give a necessary and sufficient condition that every positive operator in M\ times_α\bm{Z} can be factored by the form A^*A, where A belongs to M\ times_α\bm{Z}₊∩(M\ times_α\bm{Z}₊)^-1.