Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 497-512.

A non-self-adjoint Lebesgue decomposition

Matthew Kennedy and Dilian Yang

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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.

Article information

Anal. PDE, Volume 7, Number 2 (2014), 497-512.

Received: 4 July 2013
Revised: 27 October 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B04: Isometric theory of Banach spaces 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22] 47L50: Dual spaces of operator algebras 47L55: Representations of (nonselfadjoint) operator algebras

Lebesgue decomposition extended F. and M. Riesz theorem unique predual Drury–Arveson space


Kennedy, Matthew; Yang, Dilian. A non-self-adjoint Lebesgue decomposition. Anal. PDE 7 (2014), no. 2, 497--512. doi:10.2140/apde.2014.7.497.

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