Positive definite maps, representations and frames

We present a unitary approach to the construction of representations and intertwining operators. We apply it to the $C^{\ast }$-algebras, groups, Gabor type unitary systems and wavelets. We give an application of our method to the theory of frames, and we prove a general dilation theorem which is in turn applied to specific cases, and we obtain in this way a dilation theorem for wavelets.