Contractivity, Complete Contractivity and Curvature inequalities

Let $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathbf{A}}$ be a norm on ${\mathbb{C}}^m$ given by the formula $\mathrm{\parallel }(z_1,\dots ,z_m){\mathrm{\parallel }}_{\mathbf{A}}=\mathrm{\parallel }{z}_1{A}_1+\cdots +z_m{A}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ 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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathbf{A}}\mathrm{.}$ %For a holomorphic function $f$ on ${\mathrm{\Omega }}_{\mathbf{A}},$ 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$$p\times q$matrices$V_1, \ldots, V_m$andafunction$f \in \mathcal O(\Omega_\mathbf A),$ the algebra of function holomorphic on an open set $U$ containing the closed unit ball ${\bar{\mathrm{\Omega }}}_{\mathbf{A}}$ define $${\rho }_V(f):=\left(\begin{array}{cc}f(w)I_p& {\sum }_{i=1}^m{\mathrm{\partial }}_{i}f(w)V_i\\ 0& f(w)I_q\end{array}\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 ${\mathrm{\Omega }}_{\mathbf{A}}$ in ${\mathbb{C}}^2\mathrm{.}$ The class of homomorphisms of the form ${\rho }_V$ arise from localization of operators in the Cowen-Douglas class of $\mathrm{\Omega }\mathrm{.}$ 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.