C*-algebras from actions of congruence monoids
Date
2020-04-20
Authors
Bruce, Chris
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Abstract
We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic
considerations; namely, we generalize the construction of Cuntz, Deninger,
and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions
of congruence monoids on rings of algebraic integers in number fields. Our motivation
for considering actions of congruence monoids comes from class field theory and work
on Bost–Connes type systems. We give two presentations and a groupoid model for
these algebras, and establish a faithfulness criterion for their representations. We
then explicitly compute the primitive ideal space, give a semigroup crossed product
description of the boundary quotient, and prove that the construction is functorial
in the appropriate sense. These C*-algebras carry canonical time evolutions, so that
our construction also produces a new class of C*-dynamical systems. We classify the
KMS (equilibrium) states for this canonical time evolution, and show that there are
several phase transitions whose complexity depends on properties of a generalized
ideal class group. We compute the type of all high temperature KMS states, and
consider several related C*-dynamical systems.
Description
Keywords
C*-algebras, Semigroup C*-algebras, Operator algebras, Primitive ideals, KMS states, C*-dynamical system, Number fields, Rings of algebraic integers, Congruence monoids, von Neumann algebras, Noncommutative geometry, Faithful representations, Groupoid C*-algebras, Type III_1 factors