Factorization in Subdiagonal Algebras

Let M be aσ-finite von Neumann algebra and let A be a maximal subdiagonal algebra of M with respect to a faithful normal conditional expectationΦ. We show that ifSis an invertible operator in M, then there exists an isometryWin M such that bothS−1WandW*Sbelong to A. We also give several characterizations of a maximal subdiagonal algebra A such that every invertible operatorSin M can be factored asUA, whereUis a unitary operator inMand bothAandA−1are in A.