Phase transitions on C∗-algebras arising from number fields and the generalized Furstenberg conjecture

Laca, Marcelo; Warren, Jacqueline M.

doi:10.7900/jot.2018jul13.2201

Cited by 4 publications

(1 citation statement)

References 14 publications

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“…Certain almost minimal algebraic actions were studied by Berend [Ber83;Ber84] and by Laca and Warren [LW20]. We also refer the reader to Schmidt's book [Sch95, Section 29] for further discussions of examples from algebraic actions.…”

Section: Preliminariesmentioning

confidence: 99%

A tracial characterization of Furstenberg’s conjecture

Bruce,

Scarparo

2023

Can. Math. Bull.

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We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$ -algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$ . We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$ .

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Section: Preliminariesmentioning

confidence: 99%

A tracial characterization of Furstenberg’s conjecture

Bruce,

Scarparo

2023

Can. Math. Bull.

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Partition Functions as C*-Dynamical Invariants and Actions of Congruence Monoids

Bruce

Laca

Takeishi

2020

Commun. Math. Phys.

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C*-algebras from actions of congruence monoids on rings of algebraic integers

Bruce

2019

Trans. Amer. Math. Soc.

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Let K be a number field with ring of integers R. Given a modulus m for K and a group Γ of residues modulo m, we consider the semi-direct product R Rm,Γ obtained by restricting the multiplicative part of the full ax + b-semigroup over R to those algebraic integers whose residue modulo m lies in Γ, and we study the left regular C*-algebra of this semigroup. We give two presentations of this C*-algebra and realize it as a full corner in a crossed product C*-algebra. We also establish a faithfulness criterion for representations in terms of projections associated with ideal classes in a quotient of the ray class group modulo m, and we explicitly describe the primitive ideals using relations only involving the range projections of the generating isometries; this leads to an explicit description of the boundary quotient. Our results generalize and strengthen those of Cuntz, Deninger, and Laca and of Echterhoff and Laca for the C*-algebra of the full ax + b-semigroup. We conclude by showing that our construction is functorial in the appropriate sense; in particular, we prove that the left regular C*-algebra of R Rm,Γ embeds canonically into the left regular C*-algebra of the full ax + b-semigroup. Our methods rely heavily on Li's theory of semigroup C*-algebras.

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Phase transitions on C∗-algebras arising from number fields and the generalized Furstenberg conjecture

Cited by 4 publications

References 14 publications

A tracial characterization of Furstenberg’s conjecture

A tracial characterization of Furstenberg’s conjecture

Partition Functions as C*-Dynamical Invariants and Actions of Congruence Monoids

C*-algebras from actions of congruence monoids on rings of algebraic integers

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