  The non-commutative theory of the Lebesgue-type decomposition of positive
functionals is originated with S. P. Gudder. Although H. Kosaki's
counterexample shows that the decomposition is not unique in general, the
complete characterization of uniqueness is still not known.
  Using the famous operator-decomposition of T. Ando, we give a necessary and
sufficient condition for uniqueness in the particular case when the underlying
algebra is $B(H)$, the $C^*$-algebra of all continuous linear operators on a
Hilbert space $H$. Namely, given a normal state $f$, the $f$-Lebesgue
decomposition of any other normal state is unique if and only if the
representing trace class operator of $f$ has finite rank.
  Some recent results tell that the decomposition is unique over a large class
of commutative algebras. Our characterization demonstrates that the lack of
commutativity is not the real cause of non-uniqueness.
