Equilibrium States on Right LCM Semigroup $C^*$-Algebras

Abstract

We determine the structure of equilibrium states for a natural dynamics on the boundary quotient diagram of C * -algebras for a large class of right LCM semigroups. The approach is based on abstract properties of the semigroup and covers the previous case studies on N ⋊ N × , dilation matrices, self-similar actions, and Baumslag-Solitar monoids. At the same time, it provides new results for right LCM semigroups associated to algebraic dynamical systems.

𝐶*-algebras of right LCM monoids and their equilibrium states

“…The present work was triggered as we became aware of Spielberg's work [Spi12] on Baumslag-Solitar monoids that are not quasi-lattice ordered, but right LCM. This class went unnoticed until after a discussion with Jack Spielberg at a conference in Newcastle, Australia, where the authors of the present paper realised that it fell outside the scope of [ABLS17]. The current paper grew out of our attempts to gain insight into what happens for this special class of Baumslag-Solitar monoids.…”

“…The philosophy of unifying different case studies of C * -algebras in a broad framework is the leitmotiv in [ABLS17], where a first step was taken towards uncovering internal structure of semigroup C * -algebras via classification of KMS-states for what could be deemed as intrinsic one-parameter groups on C * (S). The main result of [ABLS17] provides general methods for analysing KMS-states for C * -algebras associated to admissible right LCM monoids.…”

“…The philosophy of unifying different case studies of C * -algebras in a broad framework is the leitmotiv in [ABLS17], where a first step was taken towards uncovering internal structure of semigroup C * -algebras via classification of KMS-states for what could be deemed as intrinsic one-parameter groups on C * (S). The main result of [ABLS17] provides general methods for analysing KMS-states for C * -algebras associated to admissible right LCM monoids. This covers new cases such as algebraic dynamical systems [BLS18], and also offers a unified perspective onto this problem for C * -algebras associated to the affine semigroup over the natural numbers [LR10], algebraic number fields with trivial class number [CDL13], integer dilation matrices [LRR11], self-similar group actions [LRRW14], and quasi-lattice ordered Baumslag-Solitar monoids [CaHR16].…”

Boundary Quotient -algebras of Products of Odometers

“…We should mention that F + θ here is not necessary to be right LCM, and so F + θ ⊲⊳ G is not right LCM in general. Therefore, one can not apply the results in the recent works on right LCM semigroups, such as [ABLS16,BOS15,BLS16,BRRW14,Stam16,Star15], to our cases.…”

“…Since then, there has been attracting a lot of attention to the study of semigroup C*-algebras. See, for example, [ABLS16,BOS15,BLS16,BRRW14,Stam16,Star15] and the references therein. In [BRRW14], Brownlowe-Ramagge-Robertson-Whittaker defined a quotient C*-algebra Q(P ) of C * (P ).…”