Statistical Mechanics of Quantum Spin Systems. Ill

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous function P defined on a Banach space ~3 of translationally invariant interactions. We prove that each tangent functional to the graph of P defines a set of translationally invariant thermodynamic expectation values, l~Iore precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra 92 of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and i t is shown that they form a dense set in the set of all invariant states over 9A. With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Sehwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under timetranslations.