Completely bounded modules and associated extremal problems

Abstract

In this paper we continue our study of certain finite dimensional Hilbert modules over the function algebra A (Ω), Ω ⊆ C^m. We show that these modules are always completely bounded with the bound obtained as the matrix valued analogue of a certain scalar valued extremal problem. In particular, we obtain a necessary and sufficient condition for our module to be completely contractive. We produce a contractive module C_N^m over A (B^m) such that it is completely bounded with the complete bound equal to √m; that is, C_N^m is not completely contractive.