arXiv subjects

Nicolai Stammeier

Publications and source records attributed to Nicolai Stammeier.

13 recordsLinked to original sources

arXiv2025

The groupoid approach to equilibrium states on right LCM semigroup C*-algebras

Given a right LCM semigroup $S$ and a homomorphism $N\colon S\to[1,+\infty)$, we use the groupoid approach to study the KMS$_β$-states on $C^*(S)$ with respect to the dynamics induced by $N$. We establish necessary and sufficient conditions for the existence and uniqueness of KMS$_β$-states. As an application, we show that the sufficient condition for the uniqueness obtained for so-called generalized scales is necessary as well. Our most complete results are obtained for inverse temperatures $β$ at which the $ζ$-function of $N$ is finite. In this case we get an explicit bijective correspondence between the KMS$_β$-states on $C^*(S)$ and the tracial states on $C^*(\operatorname{ker} N)$.

math.OA↗

arXiv2019

The nature of generalized scales

The notion of a generalized scale emerged in recent joint work with Afsar-Brownlowe-Larsen on equilibrium states on C*-algebras of right LCM monoids, where it features as the key datum for the dynamics under investigation. This work provides the structure theory for such monoidal homomorphisms. We establish uniqueness of the generalized scale and characterize its existence in terms of a simplicial graph arising from a new notion of irreducibility inside right LCM monoids. In addition, the method yields an explicit construction of the generalized scale if existent. We discuss applications for graph products as well as algebraic dynamical systems, and reveal a striking connection to Saito's degree map.

math.GR↗

arXiv2019

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

We study the internal structure of $C^*$-algebras of right LCM monoids by means of isolating the core semigroup $C^*$-algebra as the coefficient algebra of a Fock-type module on which the full semigroup $C^*$-algebra admits a left action. If the semigroup has a generalised scale, we classify the KMS-states for the associated time evolution on the semigroup $C^*$-algebra, and provide sufficient conditions for uniqueness of the KMS$_β$-state at inverse temperature $β$ in a critical interval.

math.OA↗

arXiv2018

The inner structure of boundary quotients of right LCM semigroups

We study distinguished subalgebras and automorphisms of boundary quotients arising from algebraic dynamical systems $(G,P,θ)$. 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.

math.OA↗

arXiv2017

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.

math.OA↗

arXiv2017

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.

math.OA↗

arXiv2017

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.

math.OA↗

arXiv2017

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.

math.OA↗

arXiv2016

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.

math.OA↗

arXiv2016

On the K-theory of C*-algebras arising from integral dynamics

We investigate the $K$-theory of unital UCT Kirchberg algebras $\mathcal{Q}_S$ arising from families $S$ of relatively prime numbers. It is shown that $K_*(\mathcal{Q}_S)$ is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct $C^*$-algebra naturally associated to $S$. The $C^*$-algebra representing the torsion part is identified with a natural subalgebra $\mathcal{A}_S$ of $\mathcal{Q}_S$. For the $K$-theory of $\mathcal{Q}_S$, the cardinality of $S$ determines the free part and is also relevant for the torsion part, for which the greatest common divisor $g_S$ of $\{p-1 : p \in S\}$ plays a central role as well. In the case where $\lvert S \rvert \leq 2$ or $g_S=1$ we obtain a complete classification for $\mathcal{Q}_S$. Our results support the conjecture that $\mathcal{A}_S$ coincides with $\otimes_{p \in S} \mathcal{O}_p$. This would lead to a complete classification of $\mathcal{Q}_S$, and is related to a conjecture about $k$-graphs.

math.OA↗

arXiv2015

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.

math.OA↗

arXiv2014

Topological freeness for $*$-commuting covering maps

A countable family of $*$-commuting surjective, non-injective local homeomorphisms of a compact Hausdorff space $X$ gives rise to an action $θ$ of a countably generated, free abelian monoid $P$. For such a triple $(X,P,θ)$, which we call an irreversible $*$-commutative dynamical system, we construct a universal $C^*$-algebra $\mathcal{O}[X,P,θ]$. Within this setting we show that the following four conditions are equivalent: $(X,P,θ)$ is topologically free, $C(X) \subset \mathcal{O}[X,P,θ]$ has the ideal intersection property, the natural representation of $\mathcal{O}[X,P,θ]$ on $\ell^2(X)$ is faithful, and $C(X)$ is a masa in $\mathcal{O}[X,P,θ]$. As an application, we characterise simplicity of $\mathcal{O}[X,P,θ]$ by minimality of $(X,P,θ)$. We also show that $\mathcal{O}[X,P,θ]$ is isomorphic to the Cuntz-Nica-Pimsner algebra of a product system of Hilbert bimodules naturally associated to $(X,P,θ)$. Moreover, we find a close connection between $*$-commutativity and independence of group endomorphisms, a notion introduced by Cuntz and Vershik. This leads to the observation that, for commutative irreversible algebraic dynamical systems of finite type $(G,P,θ)$, the dual model $(\hat{G},P,\hatθ)$ is an irreversible $*$-commutative dynamical system and $\mathcal{O}[\hat{G},P,\hatθ]$ is canonically isomorphic to $\mathcal{O}[G,P,θ]$. This allows us to conclude that minimality of $(G,P,θ)$ is not only sufficient, but also necessary for simplicity of $\mathcal{O}[G,P,θ]$ if $(G,P,θ)$ is commutative and of finite type.

math.OA↗

arXiv2014

On C*-algebras of irreversible algebraic dynamical systems

Extending the work of Cuntz and Vershik, we develop a general notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as irreversible analogues of the dynamical systems considered by Schmidt. To each irreversible algebraic dynamical system, we associate a universal C*-algebra and show that it is a UCT Kirchberg algebra under natural assumptions. Moreover, we discuss the structure of the core subalgebra, which is closely related to generalised Bunce-Deddens algebras in the sense of Orfanos. We also construct discrete product systems of Hilbert bimodules for irreversible algebraic dynamical systems which allow us to view the associated C*-algebras as Cuntz-Nica-Pimsner algebras. Besides, we prove a decomposition theorem for semigroup crossed products of unital C*-algebras by semidirect products of discrete, left cancellative monoids.

math.OA↗