  The structure of the set of positivity-preserving maps between matrix
algebras is notoriously difficult to describe. The notable exceptions are the
results by St{\o}rmer and Woronowicz from 1960s and 1970s settling the low
dimensional cases. By duality, these results are equivalent to the
Peres-Horodecki positive partial transpose criterion being able to
unambiguously establish whether a state in a 2 x 2 or 2 x 3 quantum system is
entangled or separable. However, even in these low dimensional cases, the
existing arguments (known to the authors) were based on long and seemingly ad
hoc computations. We present a simple proof, based on Brouwer's fixed point
theorem, for the 2 x 2 case (St{\o}rmer's theorem). For completeness, we also
include another argument (following the classical outline, but highly
streamlined) based on a characterization of extreme self-maps of the Lorentz
cone and on a link - noticed by R. Hildebrand - to the S-lemma, a well-known
fact from control theory and quadratic/semi-definite programming.
