  A famous question of Halmos asks whether every operator on a separable
infinite dimensional Hilbert space is a norm limit of reducible operators. In
[30] , Voiculescu gave this problem an affirmative answer by his remarkable
non-commutative Weyl-von Neumann theorem. We investigate the existence or
non-existence of an analogue of Voiculescu's result in factors of type ${\rm
II}_1$.
  In the paper we prove that, in the operator norm topology, the set of
reducible operators is ${\it nowhere}$ dense in a non-$\Gamma$ factor $\mathcal
M$ of type ${\rm II}_1$, where separable and non-separable cases of $\mathcal
M$ are both considered. Main tools developed in the paper are a new
characterization of Murray and von Neumann's Property $\Gamma$ for a factor of
type ${\rm II}_1$ and a spectral gap property for a single operator in a
non-$\Gamma$ factor of type ${\rm II}_1$.
