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

We prove for an arbitrary complex ${}^{\ast }$-algebra $A$ that every topologically irreducible ${}^{\ast }$-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.