  For any complex Banach space $X$, let $J$ denote the duality mapping of $X$.
For any unit vector $x$ in $X$ and any ($C_0$) contraction semigroup
$(T_t)_{t>0}$ on $X$, Baillon and Guerre-Delabriere proved that if $X$ is a
smooth reflexive Banach space and if there is $x^* \in J(x)$ such that
$|\langle T(t) \, x,J(x)\rangle| \to 1 $ as $t \to \infty$, then there is a
unit vector $y\in X$ which is an eigenvector of the generator $A$ of
$(T_t)_{t>0}$ associated with a purely imaginary eigenvalue. They asked whether
this result is still true if $X$ is replaced by $c_o$. In this article, we show
the answer is negative.
