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

Reducible operators in non-$\Gamma$ type ${\rm II}_1$ factors