  The Caldeira-Leggett master equation as an example of Markovian master
equation without Lindblad form is investigated for mathematical consistency. We
explore situations both analytically and numerically where the positivity
violations of the density operator occur. We reinforce some known knowledge
about this problem but also find new surprising cases. Our analytical results
are based on the full solution of the Caldeira-Leggett master equation obtained
via the method of characteristics. The preservation of positivity is mainly
investigated with the help of the density operator's purity and we give also
some numerical results about the violation of the Robertson-Schr\"odinger
uncertainty relation.
