  Let $\|\cdot\|_{\mathbf A}$ be a norm on $\mathbb C^m$ given by the formula
$\|(z_1,\ldots,z_m)\|_{\mathbf A}=\|z_1A_1+\cdots+z_mA_m\|_{\rm op}$ for some
choice of an $m$-tuple of $n\times n$ linearly independent matrices $\mathbf
A=(A_1, \ldots, A_m).$ Let $\Omega_\mathbf A\subset \mathbb C^m$ be the unit
ball with respect to the norm $\|\cdot\|_{\mathbf A}.$ %For a holomorphic
function $f$ on $\Omega_\mathbf A,$ let %$\rho_{V}(f):=\left (
%\begin{smallmatrix} %f(w)I_p& \sum_{i=1}^{m} \partial_if(w)V_{i} \\ %0 &
f(w)I_q %\end{smallmatrix}\right ),$ where $V_1, \ldots, V_m$ are $p\times q$
%matrices. Given $p\times q$ matrices $V_1, \ldots, V_m$ and a function $f \in
\mathcal O(\Omega_\mathbf A),$ the algebra of function holomorphic on an open
set $U$ containing the closed unit ball $\bar{\Omega}_\mathbf A$ define
$$\rho_{V}(f):=\left ( \begin{smallmatrix} f(w)I_p& \sum_{i=1}^{m}
\partial_if(w)V_{i} \\ 0 & f(w)I_q \end{smallmatrix}\right ),$$ $w\in
\Omega_\mathbf A.$ Clearly, $\rho_{V}$ defines an algebra homomorphism. We
study contractivity (resp. complete contractivity) of such homomorphisms.
  The characterization of those balls in $\mathbb C^2$ for which contractive
linear maps are always completely contractive remained open. We answer this
question for balls of the form $\Omega_\mathbf A$ in $\mathbb C^2.$
  The class of homomorphisms of the form $\rho_V$ arise from localization of
operators in the Cowen-Douglas class of $\Omega.$ The (complete) contractivity
of a homomorphism in this class naturally produces inequalities for the
curvature of the corresponding Cowen-Douglas bundle. This connection and some
of its very interesting consequences are discussed.
