Contractivity, Complete Contractivity and Curvature Inequalities

Abstract

Let ???A\| \cdot \|_A???A? be a norm on Cm\mathbb{C}^mCm given by the formula ?(z1,…,zm)?A=?z1A1+?+zmAm?\|(z_1, \ldots, z_m)\|_A = \|z_1 A_1 + \cdots + z_m A_m\|?(z1?,…,zm?)?A?=?z1?A1?+?+zm?Am?? for some choice of an mmm-tuple of n×nn \times nn×n linearly independent matrices A=(A1,…,Am)A = (A_1, \ldots, A_m)A=(A1?,…,Am?). Let ?A?Cm\Omega_A \subset \mathbb{C}^m?A??Cm be the unit ball with respect to the norm ???A\| \cdot \|_A???A?. Given p×qp \times qp×q matrices and a function f?O(?A)f \in \mathcal{O}(\Omega_A)f?O(?A?), the algebra of functions holomorphic on an open set UUU containing the closed unit ball ?A\Omega_A?A?, define (?V(f))i,j=f(vij)(\rho_V(f))_{i,j} = f(v_{ij})(?V?(f))i,j?=f(vij?) Clearly, ?V\rho_V?V? defines an algebra homomorphism. We study contractivity (resp. complete contractivity) of such homomorphisms. The homomorphism ?V\rho_V?V? induces a linear map LV:(Cm,???A)?Cp×q,LV(w)=w1V1+?+wmVmL_V : (\mathbb{C}^m, \| \cdot \|_A) \to \mathbb{C}^{p \times q}, \quad L_V(w) = w_1 V_1 + \cdots + w_m V_mLV?:(Cm,???A?)?Cp×q,LV?(w)=w1?V1?+?+wm?Vm? The contractivity (resp. complete contractivity) of the homomorphism ?V\rho_V?V? determines the contractivity (resp. complete contractivity) of the linear map LVL_VLV? and vice versa. It is known that contractive homomorphisms of the disc and the bi-disc algebra are completely contractive, thanks to the dilation theorems of B. Sz.-Nagy and Ando respectively. However, examples of contractive homomorphisms ?V\rho_V?V? of the (Euclidean) ball algebra which are not completely contractive were given by G. Misra. From the work of V. Paulsen and E. Ricard, it follows that if m>3m > 3m>3 and BBB is any ball in Cm\mathbb{C}^mCm with respect to some norm, say ???B\| \cdot \|_B???B?, then there exists a contractive linear map L:(Cm,???B)?B(V)L : (\mathbb{C}^m, \| \cdot \|_B) \to B(\mathcal{V})L:(Cm,???B?)?B(V) which is not completely contractive. The characterization of those balls in Cm\mathbb{C}^mCm for which contractive linear maps are always completely contractive remained open. We answer this question for balls of the form ?A\Omega_A?A?. The class of homomorphisms of the form ?V\rho_V?V? arise from localization of operators in the Cowen–Douglas class of rank nnn. 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