In this talk, we advertise a noncommutative analog of P-points, called quantum P-points.

There is a canonical correspondence between pure states on a $C^*$-algebra $C$, maximal quantum filters in $C$, and irreducible representations of $C$. By axiomatizing pure states, we show that the properties of a quantum filter have a measurable influence on their associated irreducible representation. In particular, we show that representations corresponding to P-points have countable transitivity properties when $C$ has a weak saturation property called countable degree-1 saturation, while those associated to idempotent ultrafilters do not.

This is work under the supervision of Ilijas Farah.