  We describe (infinite-dimensional) irreducible representations of the crossed
product C$^*$-algebra associated with a topological dynamical system (based on
$Z$) and we show that their restrictions to the underling $\ell^1$-Banach
$*$-algebra are not algebraically irreducible under mild conditions on the
dynamical system. The above description of irreducible representations has two
ingredients, ergodic measures on the space and ergodic extensions for the
tensor product with type I factors; the latter which may not have been
explicitly taken up before will be explored by examples. A new class of ergodic
measures is also constructed for irrational rotations on the circle.
