Vol. 7, No. 2, 2014

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A non-self-adjoint Lebesgue decomposition

Matthew Kennedy and Dilian Yang

Vol. 7 (2014), No. 2, 497–512

We study the structure of bounded linear functionals on a class of non-self-adjoint operator algebras that includes the multiplier algebra of every complete Nevanlinna–Pick space, and in particular the multiplier algebra of the Drury–Arveson space. Our main result is a Lebesgue decomposition expressing every linear functional as the sum of an absolutely continuous (i.e., weak-* continuous) linear functional and a singular linear functional that is far from being absolutely continuous. This is a non-self-adjoint analogue of Takesaki’s decomposition theorem for linear functionals on von Neumann algebras. We apply our decomposition theorem to prove that the predual of every algebra in this class is (strongly) unique.

Lebesgue decomposition, extended F. and M. Riesz theorem, unique predual, Drury–Arveson space
Mathematical Subject Classification 2010
Primary: 46B04, 47B32, 47L50, 47L55
Received: 4 July 2013
Revised: 27 October 2013
Accepted: 27 November 2013
Published: 30 May 2014
Matthew Kennedy
School of Mathematics and Statistics
Carleton University
1125 Colonel By Drive
Ottawa, ON K1S 5B6
Dilian Yang
Department of Mathematics and Statistics
University of Windsor
401 Sunset Avenue
Windsor, ON N9B 3P4