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