Reducible operators in non-$\hat{I}“$ type ${\mathrm{II}}_1$ 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 ${\mathrm{II}}_1$. In the paper, we prove that, in the operator norm topology, the set of reducible operators is $nowhere$ dense in a non-$\hat{I}“$ factor $\mathcal{M}$ of type ${\mathrm{II}}_1$, 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 $\hat{I}“$ for a factor of type ${\mathrm{II}}_1$ and a spectral gap property for a single operator in a non-$\hat{I}“$ factor of type ${\mathrm{II}}_1$.

34 pages