Applications of the Fuglede-Kadison determinant: Szegö's theorem and outers for noncommutative $H^p$

We first use properties of the Fuglede-Kadison determinant on $L^p(M)$, for a finite von Neumann algebra $M$, to give several useful variants of the noncommutative Szegö theorem for $L^p(M)$, including the one usually attributed to Kolmogorov and Krein. As an application, we solve the longstanding open problem concerning the noncommutative generalization, to Arveson's noncommutative $H^p$ spaces, of the famous `outer factorization' of functions $f$ with $\log \vert f\vert$ integrable. Using the Fuglede-Kadison determinant, we also generalize many other classical results concerning outer functions.