Contractive and completely contractive modules, matricial tangent vectors and distance decreasing metrics

Abstract

It is shown that a tangent vector v in T_ωΩ determines a finite dimensional Hilbert module over H^∞ (Ω) and that the module is contractive if and only if C_Ωω(v), the Caratheodory length of v, is less or equal to one. More generally, an element V of T_ωΩ ⊗ M_n also determines a finite dimensional Hilbert module over H^∞ (Ω) and if the norm on T_ωΩ⊗ M_n is understood to be the injective tensor product norm then again the module is contractive if and only if ||V|| is less or equal to one. The question of which contractive modules are completely contractive over H^∞ (Ω) is discussed in terms of the Caratheodory metric on Ω. The relationship of this question with certian aspects of the theory of tensor products for operator spaces is established.