On K-theoretic invariants of semigroup C*-algebras attached to number fields

Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

International Mathematics Research Notices ◽

10.1093/imrn/rnaa056 ◽

2020 ◽

Author(s):

Chris Bruce

Keyword(s):

Phase Transitions ◽

Class Group ◽

Number Fields ◽

Algebraic Integers ◽

Ring Of Integers ◽

C Algebras ◽

Rings Of Algebraic Integers ◽

Beta 2 ◽

Factor State ◽

Type Classification

Abstract We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS$_\beta $ state, and we prove that it is a factor state of type III$_1$. There are phase transitions at $\beta =2$ and $\beta =\infty $ involving a quotient of a ray class group. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

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

Mathematische Annalen ◽

10.1007/s00208-012-0826-9 ◽

2012 ◽

Vol 355 (4) ◽

pp. 1383-1423 ◽

Cited By ~ 25

Author(s):

Joachim Cuntz ◽

Christopher Deninger ◽

Marcelo Laca

Keyword(s):

Algebraic Number ◽

Number Fields ◽

Algebraic Number Fields ◽

C Algebras

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Asymptotic Semicircular Laws Induced by p-Adic Number Fields Qp and C-Algebras over Primes p*

Symmetry ◽

10.3390/sym11060819 ◽

2019 ◽

Vol 11 (6) ◽

pp. 819

Author(s):

Ilwoo Cho

Keyword(s):

Number Fields ◽

C Algebras ◽

Probability Spaces

In this paper, we study asymptotic semicircular laws induced both by arbitrarily fixed C * -probability spaces, and p-adic number fields { Q p } p ∈ P , as p→∞ in the set P of all primes.

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