Contractivity and complete contractivity for finite dimensional Banach Spaces

TL;DR

This paper investigates the conditions under which contractive linear maps on certain finite-dimensional Banach spaces, specifically those defined by matrix-induced norms in two dimensions, are also completely contractive, addressing an open problem.

Contribution

It characterizes when contractive linear maps on matrix-induced norm balls in 2 are necessarily completely contractive, solving an open question in the field.

Findings

For 2, contractive maps are always completely contractive under certain conditions.

Identifies specific properties of 2 balls where contractivity implies complete contractivity.

Provides a complete characterization for 2 case, contrasting with higher dimensions.

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 $\mathbb C^2.$