# Nicolai Stammeier: A boundary quotient diagram for right LCM semigroups I - construction & examples

Abstract: For the ax+b semigroup over the natural numbers, which is known to be part of a quasi-lattice ordered group, Laca and Raeburn considered its Nica-Toeplitz algebra and its Cuntz-Nica-Pimsner algebra, with a special appeal to nice presentations by generators and relations as well as the structure of KMS states for a natural dynamics. Shortly thereafter, Brownlowe-an Huef-Laca-Raeburn showed that there are two intermediate quotients between the Nica-Toeplitz algebra and the Cuntz-Nica-Pimsner algebra that exhibit interesting structural properties, especially with regards to KMS states. Since then, analogous quotients have been considered (partly in disguise) in a growing list of case studies on the KMS state structure, e.g. for dilation matrices, self-similar actions, and Baumslag-Solitair monoids. Somewhat surprisingly, all these case studies can be viewed from the perspective of semigroup C*-algebras of right LCM semigroups, and in this talk, I shall describe a unifying perspective on such boundary quotient diagrams. Thereby several questions concerning the general structure of right LCM semigroups are raised.