We prove for an arbitrary complex ∗-algebra A that every topologically
irreducible ∗-representation of A on a Hilbert space is finite dimensional
precisely when the Lebesgue decomposition of representable positive functionals
over A is unique. In particular, the uniqueness of the Lebesgue decomposition
of positive functionals over the L1-algebras of locally compact groups
provides a new characterization of Moore groups.Comment: To appear in: Journal of Operator Theor