A note on lattice ordered $C^{\ast }$-algebras and Perron--Frobenius theory

A classical result of Sherman says that if the space of self-adjoint elements in a $C^{\ast }$-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 $C^{\ast }$-algebra.