Contractivity and complete contractivity for finite dimensional Banach spaces

摘要

Choose an arbitrary but fixed set of $nimes n$ matrices $A_1, ldots, A_m$and let $Omega_mathbf Asubset mathbb C^m$ be the unit ball with respect tothe norm $|cdot|_{mathbf A},$ where $|(z_1,ldots ,z_m)|_{mathbfA}=|z_1A_1+ cdots+z_mA_m|_{m op}.$ It is known that if $mgeq 3$ 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 $L:(mathbbC^m,|cdot|^*_{mathbb B})o mathcal M_k$ which is not completelycontractive. The characterization of those balls in $mathbb C^2$ for whichcontractive linear maps are always completely contractive thus remains open. Weanswer this question for balls of the form $Omega_mathbf A$ in $mathbb C^2.$