Contractivity and complete contractivity for finite dimensional Banach Spaces

Abstract: 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 $\mathbb C^2.$