We study homomorphisms $\rho_{V}$($\rho_{V}(f)=\left (
\begin{smallmatrix}
f(w)I_n& \sum_{i=1}^{m} \partial_if(w)V_{i} \\
0 & f(w)I_n
\end{smallmatrix}\right ), f \in \mathcal O(\Omega_\mathbf A)$
) defined on
$\mathcal
O(\Omega_\mathbf A)$
, where $\Omega_\mathbf A$
is a bounded
domain of the form:
\(\begin{eqnarray*}
\Omega_\mathbf A & := &\{(z_1 ,z_2, \ldots, z_m) :\|z_1 A_1
+\cdots + z_mA_m \|_{\rm op} < 1\}
\end{eqnarray*}\)
for some choice of a linearly independent set of $n\times n$
matrices $\{A_1, \ldots, A_m\}.$
From the work of V. Paulsen and E. Ricard, it follows that if
$n\geq 3$
and $\mathbb B$
is any ball in $\mathbb C^m$
, then there exists
a contractive linear map which is not complete
contractivity. It is known that contractive homomorphisms of the
disc and the bi-disc algebra are completely contractive, thanks
to the dilation theorem of B. Sz.-Nagy and Ando. However, an
example of a contractive homomorphism of the (Euclidean) ball
algebra which is not completely contractive was given by G. Misra. The
characterization of those balls in $\mathbb C^2$
for which
contractive linear maps which are always comletely contractive
remained open. We answer this question.
The class of homomorphism 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.