Partition functions as C*-dynamical invariants and actions of congruence monoids

TL;DR

This paper investigates the phase transitions of KMS states for C*-algebras associated with algebraic integers and congruence monoids, introducing invariants based on partition functions that encode number-theoretic information.

Contribution

It introduces a novel invariant based on partition functions for C*-dynamical systems, linking number theory and operator algebras, and characterizes features of number fields via these invariants.

Findings

Extremal low-temperature KMS states are realized as generalized Gibbs states.

Partition functions serve as invariants under C*-dynamical system isomorphisms.

Most systems exhibit infinitely many type I and at least one type II KMS states at the same inverse temperature.

Abstract

We study the phase transition of KMS states for the C*-algebras of $ax+b$-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz, Deninger, and Laca. Here we realize the extremal low-temperature KMS states as generalized Gibbs states by constructing concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the Murray--von Neumann type of extremal KMS states and we determine explicit partition functions for the type I factor states. The collection of partition functions that arise this way is an invariant under $\mathbb{R}$-equivariant isomorphism of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. As an application we characterize several features of the underlying number field and congruence monoid in terms of these invariants. In most cases our systems have infinitely many type I factor KMS states and at least one type II factor KMS state at the same inverse temperature and there are infinitely many partition functions. In order to deal with this multiplicity, we establish, in the context of general C*-dynamical systems, a precise way to associate partition functions to extremal KMS states that are of type I, and we then show that for our systems these partition functions depend only on connected components in the KMS simplex. The discussion of partition functions of general C*-dynamical systems may be of interest by itself and is likely to have applications in other contexts, so we include it in a self-contained initial section that is partly expository and is independent of the number-theoretic background and of the technical results about congruence monoids.