  In this paper, we follow two main goals. In the first attempt, we give some
functorial properties of the $p$-analog of the Fourier-Stieltjes algebras in
which we generalize some previously existed definitions and theorems in Arsac
and Cowling's works, to utilize them to prove $p$-complete boundedness of some
well-known maps on these algebras. In the second part, as an application of
these generalizations, we prove $p$-completely boundedness of homomorphisms
which are induced by continuous and proper piecewise affine maps that is a
generalization of Ilie's work on Fig\`a-Talamanca-Herz algebras.

Functional Analysis