C^*-Algebras of algebraic dynamical systems and right LCM semigroups

Abstract

We introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group. To each algebraic dynamical system we associate a C * -algebra and describe it as a semigroup C * -algebra. As part of our analysis of these C * -algebras we prove results for right LCM semigroups. More precisely we discuss functoriality of the full semigroup C * -algebra and compute its K-theory for a large class of semigroups. We introduce the notion of a Nica-Toeplitz algebra of a product system over a right LCM semigroup, and show that it provides a useful alternative to study algebraic dynamical systems.

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

“…for s, t ∈ S. It follows that C * (S) admits a dense spanning set v s v * t with s, t ∈ S, see [Nor14,BLS18]. We let e sS denote the projection v s v * s in C * (S), for each s ∈ 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].…”

“…on G p . Each Σ p is an injective endomorphism of G whose image has index p. As Σ p commutes with f q , it is easy to check that (G, Σ × f q , N × × Z) forms an algebraic dynamical system in the sense of [BLS18]. Hence the induced semidirect product S := G ⋊ θ (N × × Z) is a cancellative right LCM monoid with S c = S * = G ⋊ fq Z.…”

The nature of generalized scales

“…This monoid has close ties to the well-known Ledrappier shift from [Led78], see [Sta, Example 2.9] and [ER07, Section 11]. It is also an example of a right LCM monoid arising from an algebraic dynamical system in the sense of [BLS18]. The two generators for the N 2 -action σ and id +σ are independent b injective group endomorphisms whose images have index two.…”

Amenability and functoriality of right-LCM semigroup C*-algebras

“…Since Nica amenability is a property of C*-algebras constructed from semigroups, it is not surprising that functoriality of the construction is at the heart of the question. Functoriality results specific to right-LCM semigroups have been obtained in [5], and include criteria to decide whether a homomorphism of right-LCM semigroups lifts to a C*-algebra homomorphism of their universal C*-algebras for covariant representations. The difficulty here is that, in contrast to what happens for groups, it is not clear whether the C*-algebra homomorphism resulting from the inclusion of semigroups is injective.…”