Partition Functions as C*-Dynamical Invariants and Actions of Congruence Monoids
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Communications in
Mathematical Physics
Partition Functions as C*-Dynamical Invariants and Actions of Congruence Monoids Chris Bruce1,2 , Marcelo Laca3 , Takuya Takeishi4 1 School of Mathematical Sciences, Queen Mary University of London, Mile End Road, E1 4NS London,
UK.
2 School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK.
E-mail: [email protected]
3 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada.
E-mail: [email protected]
4 Faculty of Arts and Sciences, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, Japan.
E-mail: [email protected] Received: 20 January 2020 / Accepted: 11 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We study 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. We realize the extremal low-temperature KMS states as generalized Gibbs states in concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the type of extremal KMS states and we determine explicit partition functions for those of type I. The resulting collection of partition functions is an invariant for equivariant isomorphism classes of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. We use this to characterize several features of the underlying number field and congruence monoid. 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 type I KMS states. This discussion of partition functions for 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. 1. Introduction C*-dynamical systems of number-theoretic origin have attracted sustained interest ever since Bost and Connes [2] introduced their quantum statistical mechanical system based on a noncommutative Hecke C*-algebra associated to the inclusion of rings Z ⊆ Q. Their system has two remarkable features: it exhibits spontaneous symmetry breaking of a Galois symmetry at low-temperature and its partition function is the Riemann zeta Research partially supported by the Natural Sciences and Engineering Research Council of Canada through an Alexander Graham Bell CGS-D award (Bruce) and a Discovery Grant (Laca), and by JSPS KAKENHI Grant Number JP19K14551 (Takeishi).
C. Bruce, M. Laca, T. Takeishi
function. This second feature had been foreshadowed by the systems proposed independently by Jul
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