LCM Semigroups Research Articles

Couniversality and controlled maps on product systems over right LCM semigroups

Couniversality and controlled maps on product systems over right LCM semigroups

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify with the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

Dilation theory for right LCM semigroup dynamical systems

This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and prove a generalized Stinespring's dilation theorem showing that these representations can be dilated if and only if the map on the C⁎-algebra is unital and completely positive. This generalizes earlier results about dilations of right LCM semigroups of contractions. In addition, we also give sufficient conditions under which a contractive covariant representation of a right LCM system can be dilated to an isometric representation of the boundary quotient.

Regular Dilation and Nica-covariant Representation on Right LCM Semigroups

Regular dilation has recently been extended to graph product of $$\mathbb {N}$$ , where having a $$*$$ -regular dilation is equivalent to having a minimal isometric Nica-covariant dilation. In this paper, we first extend the result to right LCM semigroups, and establish a similar equivalence among $$*$$ -regular dilation, minimal isometric Nica-covariant dilation, and a Brehmer-type condition. This result can be applied to various semigroups to establish conditions for $$*$$ -regular dilation.

Nica–Toeplitz algebras associated with right-tensor C∗-precategories over right LCM semigroups

We introduce and analyze the full [Formula: see text] and the reduced [Formula: see text] Nica–Toeplitz algebra associated to an ideal [Formula: see text] in a right-tensor [Formula: see text]-precategory [Formula: see text] over a right LCM semigroup [Formula: see text]. These [Formula: see text]-algebras unify cross-sectional [Formula: see text]-algebras associated to Fell bundles over discrete groups and Nica–Toeplitz [Formula: see text]-algebras associated to product systems. They also allow a study of Doplicher–Roberts versions of the latter. A new phenomenon is that when [Formula: see text] is not right cancellative then the canonical conditional expectation takes values outside the ambient algebra. Our main result is a uniqueness theorem that gives sufficient conditions for a representation of [Formula: see text] to generate a [Formula: see text]-algebra naturally lying between [Formula: see text] and [Formula: see text]. We also characterize the situation when [Formula: see text]. Unlike previous results for quasi-lattice monoids, [Formula: see text] is allowed to contain nontrivial invertible elements, and we accommodate this by identifying an assumption of aperiodicity of an action of the group of invertible elements in [Formula: see text]. One prominent condition for uniqueness is a geometric condition of Coburn’s type, exploited in the work of Fowler, Laca and Raeburn. Here we shed new light on the role of this condition by relating it to a [Formula: see text]-algebra associated to [Formula: see text] itself.

Nica–Toeplitz algebras associated with product systems over right LCM semigroups

We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^*$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^*$-precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type $C^*$-algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz crossed products by actions with completely positive maps. This provides a unified framework for Nica-Toeplitz semigroup crossed products by endomorphisms and by transfer operators. We illustrate these two classes of examples with semigroup $C^*$-algebras of right and left semidirect products.

GRAPH PRODUCTS AND THE ABSENCE OF PROPERTY (AR)

We discuss the internal structure of graph products of right LCM semigroups and prove that there is an abundance of examples without property (AR). Thereby we provide the first examples of right LCM semigroups lacking this seemingly common feature. The results are particularly sharp for right-angled Artin monoids.

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

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 $\mathbb{N} \rtimes \mathbb{N}^\times$, dilation matrices, self-similar actions, and Baumslag-Solitar monoids. At the same time, it provides new results for large classes of right LCM semigroups, including those associated to algebraic dynamical systems.

The inner structure of boundary quotients of right LCM semigroups

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,\theta)$. Our work includes a complete solution to the problem of extending Bogolubov automorphisms from the Cuntz algebra in $2 \leq p<\infty$ generators to the $p$-adic ring $C^*$-algebra. For the case where $P$ is abelian and $C^*(G)$ is a maximal abelian subalgebra, we establish a picture for the automorphisms of the boundary quotient that fix $C^*(G)$ pointwise. This allows us to show that they form a maximal abelian subgroup of the entire automorphism group. The picture also leads to the surprising outcome that, for integral dynamics, every automorphism that fixes one of the natural Cuntz subalgebras pointwise is necessarily a gauge automorphism. Many of the automorphisms we consider are shown to be outer.

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

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.

A boundary quotient diagram for right LCM semigroups

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.

On $C^*$-algebras associated to right LCM semigroups

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.

The boundary quotient for algebraic dynamical systems

We introduce the notion of accurate foundation sets and the accurate refinement property for right LCM semigroups. For right LCM semigroups with this property, we derive a more explicit presentation of the boundary quotient. In the context of algebraic dynamical systems, we also analyse finiteness properties of foundation sets which lead us to a very concrete presentation. Based on Starling's recent work, we provide sharp conditions on certain algebraic dynamical systems for pure infiniteness and simplicity of their boundary quotient.

Boundary quotients of C*-algebras of right LCM semigroups

We study C*-algebras associated to right LCM semigroups, that is, semigroups which are left cancellative and for which any two principal right ideals are either disjoint or intersect in another principal right ideal. If P is such a semigroup, its C*-algebra admits a natural boundary quotient Q(P). We show that Q(P) is isomorphic to the tight C*-algebra of a certain inverse semigroup associated to P, and thus is isomorphic to the C*-algebra of an étale groupoid. We use this to give conditions on P which guarantee that Q(P) is simple and purely infinite, and give applications to self-similar groups and Zappa–Szép products of semigroups.