Projective tensor products and Apq spaces

The aim of this paper is to extend the notion of Apq space from its historical context in the work of Herz and to recognise such spaces as preduals of spaces of intertwining operators of induced representations as suggested by the work of Rieffel. This generalisation of Apq spaces involves considering projective tensor products of Lp spaces of Banach space-valued functions (the spaces of induced representations) and constructing a convolution of functions of such spaces. Sufficient conditions for the existence of the integral of the convolution are established. Most of this analysis depends upon an identity we derive of Radon-Nikodym derivatives of measures on homogeneous spaces involved. The elements of the generalised Apq space are shown to be cross-sections of a Banach semi-bundle over the double coset space corresponding to the groups from which the representations are induced, and their properties are duly discussed. In particular, the generalised form of the classical result Lp*Lq is a subset of Lr; where 1/r = 1/p + 1/q - 1; is shown to be true in this situation. The result that the Apq space is the predual of the space of intertwining operators is then established, under the condition that the intertwining operators can be approximated, in the ultraweak operator topology, by integral operators.