# Journal of Operator Theory

Volume 82, Issue 1, Summer 2019  pp. 23-47.

Contractivity and complete contractivity for finite dimensional Banach spaces

Summary: It is known that if $m\geqslant 3$ and $\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say $\|\cdot\|_{\mathbb B},$ then there exists a linear map \mbox{$L:(\mathbb C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$} which is contractive but not completely contractive. The characterization of those balls in $\mathbb C^2$ for which contractive linear maps are always completely contractive, however, remains open. We answer this question for balls of the form $\Omega_\mathbf A$ in $\mathbb C^2$ and the balls in their norm dual, where $\Omega_\mathbf A = \{(z_1, z_2): \|z_1 A_1 + z_2 A_2 \|_{\rm Op} < 1 \}$ for some pair of $2\times 2$ matrices $A_1, A_2$.