Equilibrium states of twisted C*-algebras associated to number-theoretic semigroups

Abstract

Interest in studying equilibrium states of C*-dynamical systems arising from number theory began in 1995 when Jean-Benoît Bost and Alain Connes computed the KMS states of the so-called Bost-Connes system. Further interest was ignited in 2009 when Marcelo Laca and Iain Raeburn computed the KMS states of the Toeplitz algebra of the ax+b semigroup over the natural numbers.  Since then, much work has been done on computing KMS states for untwisted semigroup C*-algebras, but little has been done for twisted ones. The example of the free abelian monoid of rank $k>1$ suggests that twisted C*-algebras of  ‘number-theoretic’ semigroups can exhibit interesting KMS behaviour related to the ideal structure of the boundary quotient.  In this talk, I will first go over some relevant background, including KMS states for C*-dynamical systems and twisted C*-algebras associated to semigroups and groupoids. I will then mention some results in the literature related to my thesis and briefly state the main result I have proved so far in my PhD