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

For an arbitrary complex *-algebra A, we prove 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 L-1-algebras of locally compact groups provides a new characterization of Moore groups.