A boundary quotient diagram for right LCM semigroups

Abstract

We propose a boundary quotient diagram for right LCM semigroups with property (AR) that generalizes the boundary quotient diagram for the $ax+b$-semigroup over the natural numbers. Our approach focuses on two important subsemigroups: the core subsemigroup and the semigroup of core irreducible elements. The diagram is then employed to unify several case studies on KMS-states, and we end with a discussion on $K$-theoretical aspects of the diagram motivated by recent findings for integral dynamics

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 ).…”

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

“…One of the keys to establishing our general theory is the insight that it pays off to work with the boundary quotient diagram for right LCM semigroups proposed by the fourth-named author in [Sta17]. The motivating example for this diagram was first considered in [BaHLR12], where it was shown to give extra insight into the structure of KMS-states on T (N ⋊ N × ).…”

“…If r, s, t ∈ S with sS ∩ tS = rS, then we call r a right LCM of s and t. In this work, all semigroups will admit an identity element (and so are monoids) and will be countable. We let S * denote the subgroup of invertible elements in S. Two subsets of a right LCM semigroup S are used in [Sta17]: the core subsemigroup…”

“…Questions 10.4. As indicated in [Sta17], the boundary quotient diagram ought to admit a reasonable description in terms of C * -algebras associated to groupoids. Assuming that this gap can be bridged, we arrive at the following questions:…”