Operators with disconnected spectrum in von Neumann algebras
Minghui Ma, Rui Shi, Tianze Wang
TL;DR
This work characterizes when every operator in a von Neumann algebra $\mathcal{M}$ can be approximated, in the sense of a unitarily invariant norm on a dense ideal $\mathcal{I}$, by a perturbation that yields a disconnected spectrum. A central sup–inf criterion involving $c_{\Phi}$ (and $f_{\Phi}$) is shown to be equivalent to the perturbation property: for all $T\in\mathcal{M}$ and $\varepsilon>0$ there exists $X\in\mathcal{I}$ with $\Phi(X)<\varepsilon$ so that $\sigma(T+X)$ is disconnected, and a complementary converse when $c_{\Phi}=\infty$. The results extend to unital $C^*$-algebras of real rank zero with essential ideals, giving an analogous perturbation theorem and showing that the set of elements with disconnected spectrum is open and dense. Consequently, in factors with dim$>1$, the set of weakly reducible (hence disconnected-spectral) operators is norm-dense, while questions about density in commutative $C(X)$-type algebras are linked to the topology of $X$ via spectral-density considerations.
Abstract
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.