  Given a contractive tuple of Hilbert space operators satisfying certain
$A$-relations we show that there exists a unique minimal dilation to generators
of Cuntz-Krieger algebras or its extension by compact operators. This
Cuntz-Krieger dilation can be obtained from the classical minimal isometric
dilation as a certain maximal $A$-relation piece. We define a maximal piece
more generally for a finite set of polynomials in $n$ noncommuting variables.
We classify all representations of Cuntz-Krieger algebras ${\mathcal O}_A$
obtained from dilations of commuting tuples satisfying $A$-relations. The
universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed
algebra generated by it is studied in terms of invariant subspaces.
