Semigroups of *-Endomorphisms, Dirichlet Series, and Phase Transitions

We study phase transitions of C*-dynamical systems (A, σ) in whichAis the crossed product of a C*-algebra by a lattice semigroup of *-endomorphisms, andσis a one-parameter subgroup of the dual action, determined by a real-valued scale on the semigroup. We show that the KMS equilibrium condition is equivalent to a Markov-type condition on the (predual) semigroup dynamical system, and give criteria for the existence of equilibrium states and of phase transitions in terms of the semigroup action. When the semigroup is an integer lattice, the partition function of the real scale is a Dirichlet series having an Euler product expansion; on their convergence region there is a phase transition whose geometry is fully determined by the ground states and is independent of the chosen scale, generalizing a model recently constructed by Bost and Connes motivated by earlier work of Julia. As applications we simplify part of the proof the Bost–Connes theorem on phase transition with spontaneous symmetry breaking and we discuss generalizations to other systems associated to subsets of primes and to number fields of class number 1.