# Journal of Operator Theory

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

Contractivity and complete contractivity for finite dimensional Banach spaces

**Authors**:
Gadadhar Misra (1), Avijit Pal (2), Cherian Varughese (3)

**Author institution:**(1) Department of Mathematics, Indian Institute of
Science, Bangalore - 560 012, India

(2) Department of Mathematics, Indian Institute of Technology Bhilai, Raipur -
492015, India

(3) Renaissance Communications, Bangalore - 560 058, India

**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 $A_1, A_2$.

**DOI: **http://dx.doi.org/10.7900/jot.2018jun13.2225

**Keywords: **contractive and completely contractive linear maps, test functions

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