  A new method of analysing positive bistochastic maps on the algebra of
complex matrices $M_{3}$ has been proposed. By identifying the set of such maps
with a convex set of linear operators on $\mathbb{R}^{8}$, one can employ
techniques from the theory of compact affine semigroups to obtain results
concerning asymptotic properties of positive maps. It turns out that the
idempotent elements play a crucial role in classifying the convex set into
subsets, in which representations of extremal positive maps are to be found. It
has been show that all positive bistochastic maps, extremal in the set of all
positive maps of $M_{3}$, that are not Jordan isomorphisms of $M_3$ are
represented by matrices that fall into two possible categories, determined by
the simplest idempotent matrices: one by the zero matrix, and the other by a
one dimensional orthogonal projection. Some norm conditions for matrices
representing possible extremal maps have been specified and examples of maps
from both categories have been brought up, based on the results published
previously.
