  In this paper, some known results will be generalized. Firstly, the
idempotent theorem on the Fourier-Stieltjes algebra will be promoted and linked
to the $p$-analog of such an algebra. Next, the $p$-analog of the $\pi$-Fourier
space introduced by Arsac will be given, and by taking advantage of the theory
of ultra filters, the connection between the dual space of the algebra of
$p$-pseudofunctions and the $p$-analog of the $\pi$-Fourier space, will be
fully investigated.
  As the main result, one of the significant and applicable functorial
properties of the $p$-analog of the Fourier-Stieltjes algebras will be
achieved.
