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Contractivity and complete contractivity for finite dimensional Banach Spaces | | Gadadhar Misra
; Avijit Pal
; Cherian Varughese
; | | Date: |
7 Apr 2016 | | Abstract: | Choose an arbitrary but fixed set of $n imes n$ matrices $A_1, ldots, A_m$
and let $Omega_mathbf Asubset mathbb C^m$ be the unit ball with respect to
the norm $|cdot|_{mathbf A},$ where $|(z_1,ldots ,z_m)|_{mathbf
A}=|z_1A_1+ cdots+z_mA_m|_{
m op}.$ 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 $L:(mathbb
C^m,|cdot|^*_{mathbb B}) o mathcal M_k$ 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 $mathbb C^2.$ | | Source: | arXiv, 1604.1872 | | Services: | Forum | Review | PDF | Favorites | |
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