  Choose an arbitrary but fixed set of $n\times n$ matrices $A_1, \ldots, A_m$
and let $\Omega_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to
the norm $\|\cdot\|_{\mathbf A},$ where $\|(z_1,\ldots ,z_m)\|_{\mathbf
A}=\|z_1A_1+ \cdots+z_mA_m\|_{\rm op}.$ It is known that if $m\geq 3$ and
$\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say
$\|\cdot\|_{\mathbb B},$ then there exists a contractive linear map $L:(\mathbb
C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$ which is not completely
contractive. The characterization of those balls in $\mathbb C^2$ for which
contractive linear maps are always completely contractive thus remains open. We
answer this question for balls of the form $\Omega_\mathbf A$ in $\mathbb C^2.$
