  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.
