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.