Reducible operators in non-$Γ$ type ${\rm II}_1$ factors
Junhao Shen, Rui Shi
TL;DR
The paper addresses Halmos' question on norm-limits of reducible operators within type $\mathrm{II}_1$ factors lacking property $\Gamma$. It develops a new single-operator criterion for $\Gamma$ and constructs a spectral-gap operator in non-$\Gamma$ factors, linking central sequences, ultrapowers, and commutator gaps. Using these tools, it proves that in non-$\Gamma$ type $\mathrm{II}_1$ factors, the set of reducible operators is nowhere dense and not closed in the operator norm, for both separable and non-separable cases. This work clarifies the boundary between reducible and irreducible operators in the II$_1$ setting and provides a robust framework for understanding how spectral-gap phenomena constrain norm-approximation by reducibles.
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-$Γ$ 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 $Γ$ for a factor of type ${\rm II}_1$ and a spectral gap property for a single operator in a non-$Γ$ factor of type ${\rm II}_1$.