$$C^*$$ -algebras of Toeplitz type associated with algebraic number fields

Abstract

We associate with the ring R of algebraic integers in a number field a C*-algebra T [R]. It is an extension of the ring C*-algebra A[R] studied previously by the first named author in collaboration with X.Li. In contrast to A[R], it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the axThe algebra T[R] carries a natural one-parameter automorphism group (σ t ) t∈R . We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where R is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMSstates splits over the class group. The "partition functions" are partial Dedekind ζ-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial ζ-functions, which we then use to show uniqueness of the β-KMS state for each inverse temperature β ∈ (1, 2].

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

“…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].…”

Ground states of groupoid C∗-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

“…The paper [22] by Laca and Raeburn marked the first C * -algebra with a time evolution admitting an interesting phase transition at infinity for ground states, see also [20]. Further results of this type were achieved in [8].…”

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

“…Gradually, it has become apparent that the study of KMS-states for systems that do not necessarily have physical origins brings valuable insight into the structure of the underlying C * -algebra, and uncovers new directions of interplay between the theory of C * -algebras and other fields of mathematics. A rich supply of examples is by now present in the literature, see [BC95,LR10,LRRW14,CDL13,Nes13], to mention only some.…”