  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 Szeg\"{o} 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 $\log |f|$ integrable. Using the Fuglede-Kadison
determinant, we also generalize many other classical results concerning outer
functions.
