Factorization in analytic crossed products

Abstract

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