  We analyse $C_0$-semigroups of contractive operators on real-valued
$L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We
show that, under some regularity assumptions on the semigroup, the geometry of
the unit ball of those spaces forces the semigroup's generator to have only
trivial (point) spectrum on the imaginary axis. This has interesting
consequences for the asymptotic behaviour as $t \to \infty$. For example, we
can show that a contractive and eventually norm continuous $C_0$-semigroup on a
real-valued $L^p$-space automatically converges strongly if $p \not\in
\{1,2,\infty\}$.
