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

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

2006-09-25

arxiv.org/abs/math/0609662v2

Mathematics

Operator Algebras

16 pages

Scientific paper

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.

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