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

Abstract

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 ${\rm II}_1$.