Classical and quantum statistical mechanics in a common Liouville space

It is shown that the extremal phase-space representations of quantum mechanics can be expressed in terms of wave-functions on L2-spaces which are embedded in L2(Γ). In L2(Γ) all these representations are restrictions of a globally defined representation of the canonical commutation relations. The master Liouville space B2(Γ) over L2(Γ) can accommodate representations of both classical and quantum statistical mechanics, and serves as a medium for their comparison. As a specific example, a Boltzmann-type equation on B2(Γ) is considered in the classical as well as quantum context.