Operators with disconnected spectrum in von Neumann algebras

Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and $Φ$ a unitarily invariant $|\cdot|$-dominating norm on $\mathcal{I}$. In this paper, we provide a necessary and sufficient condition on $Φ$ such that every operator in $\mathcal{M}$ can be expressed as the sum of an operator in $\mathcal{M}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose $Φ$-norm is arbitrarily small. Similarly, if $\mathcal{A}$ is a unital $C^*$-algebra of real rank zero with dimension greater than one and $\mathcal{I}$ is an essential ideal in $\mathcal{A}$, then every element in $\mathcal{A}$ can be written as the sum of an operator in $\mathcal{A}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose norm is arbitrarily small.

💡 Research Summary

The paper investigates the density of operators whose spectra are disconnected within von Neumann algebras and, analogously, within unital C*‑algebras of real rank zero. The authors work with a weak‑operator dense two‑sided ideal ( \mathcal I ) inside a von Neumann algebra ( \mathcal M ) and a norm ( \Phi ) defined on ( \mathcal I ) that is both unitarily invariant and (|\cdot|)-dominating (i.e., ( \Phi(X) \ge |X| ) for all ( X\in\mathcal I)). The central question is: under what condition on ( \Phi ) can every element ( T\in\mathcal M ) be approximated, up to an arbitrarily small ( \Phi )-norm perturbation, by an operator whose spectrum is not connected?

To answer this, the authors introduce for each non‑zero projection ( P\in\mathcal M ) the quantity \