Noncommutative geometry and quantum statistical mechanical systems

Abstract

In their paper, "Type III sigma-spectral triples and quantum statistical mechanical systems", M. Greenfield, M. Marcolli and K. Teh used the Bost-Connes system to construct a type III sigma-spectral triple where the twisting automorphism is a gauge automorphism. In this way they explored the link between quantum statistical mechanical systems and type III sigma-spectral triples. This thesis is dedicated to presenting their work pertaining to the Bost-Connes system, and to explore generalisations of their procedure. The text explores the possibilities for constructing type III sigma-spectral triples from lattice ordered semigroups and relating them to quantum statistical mechanical systems whose algebra of observables are semigroup crossed products.