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Contractivity, Complete Contractivity and Curvature inequalities

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 $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A}.$ %For a holomorphic function $f$ on $Ω_\mathbf A,$ let %$ρ_{V}(f):=\left ( %\begin{smallmatrix} %f(w)I_p& \sum_{i=1}^{m} \partial_if(w)V_{i} \\ %0 & f(w)I_q %\end{smallmatrix}\right ),$ where $V_1, \ldots, V_m$ are $p\times q$ %matrices. Given $p\times q$ matrices $V_1, \ldots, V_m$ and a function $f \in \mathcal O(Ω_\mathbf A),$ the algebra of function holomorphic on an open set $U$ containing the closed unit ball $\barΩ_\mathbf A$ define $$ρ_{V}(f):=\left ( \begin{smallmatrix} f(w)I_p& \sum_{i=1}^{m} \partial_if(w)V_{i} \\ 0 & f(w)I_q \end{smallmatrix}\right ),$$ $w\in Ω_\mathbf A.$ Clearly, $ρ_{V}$ 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 $Ω_\mathbf A$ in $\mathbb C^2.$ The class of homomorphisms of the form $ρ_V$ arise from localization of operators in the Cowen-Douglas class of $Ω.$ 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.