Finite dimensional irreducible representations and the uniqueness of the Lebesgue decomposition of positive functionals

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.