  The main purpose of this paper is to study the validity of the
Hausdorff-Young inequality for vector-valued functions defined on a
non-commutative compact group. The natural context for this research is that of
operator spaces. This leads us to formulate a whole new theory of Fourier type
and cotype for the category of operator spaces. The present paper is the first
step in this program, where the basic theory is presented, the main examples
are analyzed and some questions are posed.
