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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: David P. Blecher and Louis E. Labuschagne
Journal: Trans. Amer. Math. Soc. 360 (2008), 6131-6147
MSC (2000): Primary 46L51, 46L52, 47L75; Secondary 46J15, 46K50, 47L45
Published electronically: June 26, 2008
MathSciNet review: 2425707
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Abstract: 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.


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Additional Information

David P. Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: dblecher@math.uh.edu

Louis E. Labuschagne
Affiliation: Department of Mathematical Sciences, P.O. Box 392, 0003 UNISA, South Africa
Email: labusle@unisa.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04506-6
PII: S 0002-9947(08)04506-6
Received by editor(s): September 20, 2006
Received by editor(s) in revised form: February 22, 2007
Published electronically: June 26, 2008
Additional Notes: The first author was partially supported by grant DMS 0400731 from the National Science Foundation
The second author was partially supported by a National Research Foundation Focus Area Grant
Article copyright: © Copyright 2008 American Mathematical Society