Operators with disconnected spectrum in von Neumann algebras

TL;DR

This paper characterizes conditions under which operators in von Neumann algebras and certain $C^*$-algebras can be decomposed into parts with disconnected spectra and small-norm operators, advancing spectral theory understanding.

Contribution

It provides a necessary and sufficient condition for such spectral decompositions in von Neumann algebras and real rank zero $C^*$-algebras, extending spectral decomposition techniques.

Findings

Operators can be decomposed into disconnected spectrum parts and small-norm parts under specified conditions.

The results apply to von Neumann algebras and unital $C^*$-algebras of real rank zero.

The paper establishes a spectral decomposition framework for these algebraic structures.

Abstract

Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and $\Phi$ a unitarily invariant $\|\cdot\|$-dominating norm on $\mathcal{I}$. In this paper, we provide a necessary and sufficient condition on $\Phi$ 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 $\Phi$-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.