A note on lattice ordered $C^*$-algebras and Perron--Frobenius theory

TL;DR

This paper provides a new proof of Sherman’s classical result linking lattice structures in $C^*$-algebras to commutativity, and explores the limitations of Perron--Frobenius spectral results in non-commutative cases.

Contribution

It introduces a novel proof connecting lattice properties with spectral theory and demonstrates the failure of certain Perron--Frobenius results in non-commutative $C^*$-algebras.

Findings

Lattice structure in self-adjoint parts implies commutativity.

Spectral results analogous to Perron--Frobenius do not extend to non-commutative $C^*$-algebras.

New proof techniques link lattice theory with spectral analysis.

Abstract

A classical result of Sherman says that if the space of self-adjoint elements in a $C^*$-algebra $\mathcal{A}$ is a lattice with respect to its canonical order, then $\mathcal{A}$ is commutative. We give a new proof of this theorem which shows that it is intrinsically connected with the spectral theory of positive operator semigroups. Our methods also show that some important Perron--Frobenius like spectral results fail to hold in any non-commutative $C^*$-algebra.