**ABSTRACT:**

Trace class operators play an important role not only in functional analysis but also in related theories such as quantum mechanics, where one identifies "mixed states" with positive trace class operators having trace equal to one. As is "well-known" it is the positivity of such operators which poses a difficult mathematical problem: we are still lacking complete conditions on the Weyl symbol of a trace class operators allowing to decided whether this operator is positive. In this talk, we review the so-called Kastler-Loupias-MiracleSole (KLM) conditions which have been known since the 1980s and complement them with new results having been obtained in collaboration with E. Cordero, F. Nicola, and which are still in progress. We briefly mention possible, but controversial, cosmological consequences for the early universe just after the Big Bang.