Bost Connes System Research Articles

Reconstructing the Bost–Connes semigroup actions from K-theory

We complete the classification of Bost–Connes systems. We show that two Bost–Connes C*-algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissen–de Smit–Li–Marcolli–Smit, we conclude that two Bost–Connes C*-algebras are isomorphic if and only if the original number fields are isomorphic.

Q-Deformations of statistical mechanical systems and motives over finite fields

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F_1 and to categorifications of Bost-Connes systems.

Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers

In this article we develop a broad generalization of the classical Bost–Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck’s category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost–Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

Bost-Connes Systems for Local Fields of Characteristic Zero

The Bost-Connes system is a $C^*$-dynamical system which has a relation with the class field theory. The purpose of this paper is to generalize the notion of Bost-Connes systems to local fields of characteristic zero. The notable phenomenon in the case of local fields is the absence of the phase transition. We also investigate the relation between the global and local ones.

Quantum statistical mechanics in arithmetic topology

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli crossed product. We describe the main properties of the associated quantum statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.

Primitive ideals and K-theoretic approach to Bost–Connes systems

By KMS-classification theorem, the Dedekind zeta function is an invariant of Bost–Connes systems. In this paper, we show that it is in fact an invariant of Bost–Connes C⁎-algebras. We examine second maximal primitive ideals of Bost–Connes C⁎-algebras, and apply K-theory to some quotients.

Irreducible representations of Bost–Connes systems

The classification problem of Bost–Connes systems was studied by Cornelissen and Marcolli partially, but still remains unsolved. In this paper, we give a representation-theoretic approach to this problem. We generalize the result of Laca and Raeburn, which is concerned with the primitive ideal space of the Bost–Connes system for \mathbb Q . As a consequence, the Bost–Connes C^* -algebra for a number field K has h^1_K -dimensional irreducible representations and does not have finite-dimensional irreducible representations for the other dimensions, where h^1_K is the narrow class number of K . In particular, the narrow class number is an invariant of Bost–Connes C^* -algebras.

Bost–Connes systems associated with function fields

With a global function field K with constant field \mathbb{F}_q , a finite set S of primes in K and an abelian extension L of K , finite or infinite, we associate a C*-dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld modules and by Consani–Marcolli using commensurability of K -lattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMS _\beta -state for every 0<\beta\le1 gives rise to an ITPFI-factor (ITPFI stands for “infinite tensor product of finite type I factors”) of type III _{q^{-\beta n}} , where n is the degree of the algebraic closure of \mathbb{F}_q in L . Therefore for n=+\infty we get a factor of type III _0 . Its flow of weights is a scaled suspension flow of the translation by the Frobenius element on Gal (\bar{\mathbb{F}}_q/\mathbb{F}_q) .

Bost–Connes systems, Hecke algebras, and induction

We consider a Hecke algebra naturally associated with the affine group with totally positive multiplicative part over an algebraic number field K and we show that the C*-algebra of the Bost–Connes system for K can be obtained from our Hecke algebra by induction, from the group of totally positive principal ideals to the whole group of ideals. Our Hecke algebra is therefore a full corner, corresponding to the narrow Hilbert class field, in the Bost–Connes C*-algebra of K ; in particular, the two algebras coincide if and only if K has narrow class number one. Passing the known results for the Bost–Connes system for K to this corner, we obtain a phase transition theorem for our Hecke algebra. In another application of induction we consider an extension L/K of number fields and we show that the Bost–Connes system for L embeds into the system obtained from the Bost–Connes system for K by induction from the group of ideals in K to the group of ideals in L . This gives a C*-algebraic correspondence from the Bost–Connes system for K to that for L . Therefore the construction of Bost–Connes systems can be extended to a functor from number fields to C*-dynamical systems with equivariant correspondences as morphisms. We use this correspondence to induce KMS-states and we show that for \beta>1 certain extremal KMS _\beta -states for L can be obtained, via induction and rescaling, from KMS _{[L: K]\beta} -states for K . On the other hand, for 0<\beta\le1 every KMS _{[L: K]\beta} -state for K induces to an infinite weight.

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes (Sel. Math. 1(3):411–457, 1995). In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Consani and Marcolli (Adv. Math. 214(2):761–831, 2007), and using a classification result of Borger and de Smit (arXiv:1105.4662) for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreover the uniqueness of the arithmetic model is shown by Sergey Neshveyev in an appendix. In the second part of the paper we introduce a base-change functor for a class of algebraic endomotives and construct in this way an algebraic refinement of a functor from the category of number fields to the category of Bost-Connes systems, constructed recently by Laca, Neshveyev and Trifkovic (arXiv:1010.4766).

Type III 1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T ( N ⋊ N × ) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 ⩽ β ⩽ 2 , the unique KMS β -state is of type III 1. We prove this by reducing the type classification from T ( N ⋊ N × ) to that of the symmetric part of the Bost–Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of N ⋊ N × in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of N ⋊ N × on the Nica spectrum, we can also recover all the KMS-states of T ( N ⋊ N × ) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMS β -states for β > 2 that does not ostensibly come from an action of T on the C ⁎-algebra.

Bost–Connes type systems for function fields

We describe a construction which associates to any function field k and any place ∞ of k a \mathrm C^* -dynamical system (C_{k,∞},σ_t) that is analogous to the Bost–Connes system associated to ℚ and its archimedean place. Our construction relies on Hayes’ explicit class field theory in terms of sign-normalized rank one Drinfel’d modules. We show that C_{k,∞} has a faithful continuous action of \mathrm{Gal} (K/k) , where K is a certain field constructed by Hayes such that k^{\mathrm{ab},∞} ⊂ K ⊂ k^{\mathrm{ab}} . Here k^{\mathrm{ab},∞} is the maximal abelian extension of k that is totally split at ∞ . We classify the extremal KMS {}_β states of (C_{k,∞},σ_t) at any temperature 0 < 1/β < ∞ and show that a phase transition with spontaneous symmetry breaking occurs at temperature 1/β = 1 . At high temperature 1/β ≥ 1 , there is a unique KMS {}_β state, of type III {}_{q^{−β}} , where q is the cardinal of the constant subfield of k . At low temperature 1/β < 1 , the space of extremal KMS {}_β states is principal homogeneous under \mathrm{Gal} (K/k) . Each such state is of type I _∞ and the partition function is the Dedekind zeta function ζ_{k,∞} . Moreover, we construct a \ast -subalgebra \mathcal H , we give a presentation of \mathcal H and of C_{k,∞} , and we show that the values of the low-temperature extremal KMS _β states at certain elements of \mathcal H are related to special values of partial zeta functions.

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ℓ-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost–Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.

KMS states and complex multiplication

We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the Dedekind zeta function as partition function and the idele class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. The geometric notion underlying the construction is that of commensurability of K-lattices.

Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem

We study relatively invariant measures with the multiplicators Q + ∗ ∋ q ↦ q − β {\mathbb Q}^*_+\ni q\mapsto q^{-\beta } on the space A f \mathcal A_f of finite adeles. We prove that for β ∈ ( 0 , 1 ] \beta \in (0,1] such measures are ergodic, and then deduce from this the uniqueness of KMS β _\beta -states for the Bost-Connes system. Combining this with a result of Blackadar and Boca-Zaharescu, we also obtain ergodicity of the action of Q ∗ \mathbb Q^* on the full adeles.