  An alternative, geometrical proof of a known theorem concerning the
decomposition of positive maps of the matrix algebra $M_{2}(\mathbb{C})$ has
been presented. The premise of the proof is the identification of positive maps
with operators preserving the Lorentz cone in four dimensions, and it allows to
decompose the positive maps with respect to those preserving the boundary of
the cone. In addition, useful conditions implying complete positivity of a map
of $M_{2}(\mathbb{C})$ have been given, together with a sufficient condition
for complete positivity of an extremal Schwarz map of $M_{n}(\mathbb{C})$.
Lastly, following the same geometrical approach, a description in topological
terms of maps that are simultaneously completely positive and completely
copositive has been presented.
