The Bost-Connes system

Thus, thenoncommutativealgebra of coordinates of the space of commensu- rability classes of 1-dimensional Q-lattices up to scaling can be identified with theC^∗-algebra crossed product

A1≡C^∗(Q/Z)αN^×

by the semi-groupN^×, where there is aC^∗-algebra isomorphismas C^∗(Q/Z)∼=C((Q/Z)^∧)∼=C(Z^p),

where Z^p is identified with the Pontrjagin dual (Q/Z)^∧= Hom(Q/Z,T) of the abelian group Q/Z. In fact, the C^∗-algebra A1 is Morita equivalent to the C^∗-algebraB1 (cf. [95]).

As shown in Laca and Raeburn [96] (added), the semi-group C^∗-algebra crossed product A1 is isomorphic to the Bost-Connes Hecke C^∗-algebra H1

considered in [13], which is obtained as a Hecke algebra for an inclusion Γ0⊂Γ of a pair of groups, with (Γ0,Γ) = (S_Z, S_Q), where SC denotes the solvable ax+b group with coefficients inC. The pair in this case satisfies that the left Γ0-orbits for any element of Γ/Γ0are finite, and the same holds for right orbits on the left coset. The ration of the lengths of left and right Γ0-orbits determines a canonical time evolution on theC^∗-algebra (cf. [13]).

Remark. (Added). Recall from Laca [95] the following. The group C^∗- dynamical system(C0(Af),Q^∗+, β) is theminimal automorphicdilationof the (Ore) semi-groupC^∗-dynamical system(C(Z^p),N^×, α), so that

A1=C(Z^p)αN^× ∼=χ_Z^p(C0(Af)βQ^∗+)χ_Z^p=χ_Z^pB1χ_Z^p,

where βr(f)(a) = f(r⁻¹a) for a ∈ Af and r ∈ Q^∗+ = (N^×)⁻¹N^×, and the characteristic function χ_Z^p on Z^p that is compact and open inAf, belongs to C0(A^f) the C^∗-algebra of all continuous functions on A^f vanishing at infinity, whereA^f is the locally compact ring of finite adeles:

ΠpQp⊃Af ={(ap)|ap∈Zp for all but finitelymany primes}. Moreover,

H1∼=q_Z^pC^∗(Af Q^∗+)q_Z^q ∼=A1,

where C^∗(Af Q^∗+) is the groupC^∗-algebra of the semi-direct product group Af Q^∗+, and q_Z^q is identified with the projection of the group C^∗-algebra C^∗(Af) corresponding to χ_Z^q under the Fourier transform, where A^∧f ∼=Af as a self-duality.

Such an actionβ dilatesthe action αifβn◦i=i◦αn forn∈N^×, where i:C(Z^p)→C0(Af) is the inclusionmap. Such a groupC^∗-dynamical systemby βis said to beminimalif the union∪n∈N^×β_n⁻¹(i(C(Z^p))) is dense inC0(Af)×β

Q^∗+. If these conditions are satisfied, the systembyβ is said to be theminimal automorphic dilation of the systembyα.

A semi-groupSis embeddable into a groupG=S⁻¹S if and only ifSis an Oresemigroup, that is, ifS is a cancellative semigroup such that Sk∩Sl=∅ for anyk, l∈S (cf. [95]).

Remark. (Added). As in Laca [94], the group inclusion of the ax+b-groups with coefficients inZandQ(as well asR) is

S_Z= 1 Z

0 1

⊂S_Q=

Q^∗+ Q

0 1 ⊂S_R=

R^∗+ R

0 1

act on

x 1

. Note thatS_Z∼=Z 1 andS_Q ∼=Q Q^∗+ as a semi-direct product group, so that the quotient group is also isomorphic to

S_Q/S_Z∼= (Q/Z) Q^∗+.

It then follows that the group C^∗-algebra of S_Q/S_Z is isomorphic to the C^∗- algebra crossed products by the groupQ^∗+ (extending the actionαofN^×):

C^∗(S_Q/S_Z)∼=C^∗(Q/Z) Q^∗+∼=C(Z^p) Q^∗+

⊃H1∼=C(Z^p)αN^× =A1.

The Morita equivalence between twoC^∗-algebras is just like the equivalence between two projections (corresponding to respective units). Masaka!

TheC^∗-algebraH1of the Bost-Connes systemhas an explicit presentation in terms of certain operators of two types. The first type consists of phase (unitary) operatorse(r), parameterized by elementsr∈Q/Z. These phase operators can be represented on the Fock space generated by occupation numbers |nas the operators defined ase(r)|n=α(ζ_rⁿ)|n, where we denote byζab=ζ_b^a(corrected) the abstract roots of unity generatingQ^cyc and byα:Q^cyc→Can embedding that identifies Q^cyc with the subfield of C generated by the concrete roots of unity.

Recall from[74, 3.1.2] thebracket(bra-ket) notation by Dirac as follows.

ξm, Aξn ≡ m|A|n,

where the vectorsξm, ξn(as states) in the inner product withAan (observable) operator on a (Fock) Hilbert space may be identified with the bra ξm = m and the ket ξn=n, respectively, and as well, Aξn =A|n. The Fock space is defined to be a Hilbert space obtained as the infinite direct sum of tensor products ⊗ⁿH forn∈Nof a certain Hilbert space H, with the vacuum state generating the orthogonal complement of the direct sum(cf. [150]).

Those phase operators are used in the theory of quantumoptics and optical coherence tomodel the phase quantum-mechanically (cf. [100], [102] both not at hand), as well as to model the phasors in quantum computing. They are based on the choice of a certain scale N at which the phase is discretized (No figure). Namely, the quantized optical phase is defined as a state

|θm,N=e m

N+ 1

vN,

vN = 1

√N+ 1 N n=0

|n

= 1

√N+ 1 N n=0

α(ζ^mn

N+1)|n,

ζ^mn

N+1 =e^2πiN+1mn

where vN is a superposition of occupation states as above (cf. [96, 2.4]). It is then necessary to ensure that the results are consistent over changes of scale.

The removed figure is a picture of phasors with Z⁶-discretization in the plane, given and parameterized as

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

1 1 1 · · · 1 1 · · · 1 e^2πi16 e^2πi26 · · · e^2πi56 1 · · · 1 e^2πi26 e^2πi46 · · · e^2πi106 1 · · · ... ... ... . .. ... ... . ..

1 e^2πik6 e^2πi2k6 · · · e^2πi5k6 1 · · · ... ... ... . .. ... ... . ..

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

The other operators that generate the Bost-Connes Hecke C^∗-algebra H1

can be thought of as implementing the changes of scales in the optical phases in a consistent way. These operators are isometriesμn, parameterized by positive integersn∈N^× =Z>0. The changes of scale are described by the action of μn

one(r) as

Ad(μn)e(r) =μne(r)μ^∗_n= 1 n

ns=r

e(s), n∈N^×, r∈Q/Z.

For instance, check that for [x]∈Q/Z, with 0≤x <1 assumed, [1

2] = 2[1 4] = 2[1

2+ 1

2²] = 2[3 4], [1

3] = 3[1

3²] = 3[1 3+ 1

3²] = 3[2 3 + 1

3²].

In addition to that compatibility condition, the phase operators e(r) and isometriesμnsatisfy other relations as below. For anyn, k∈N^×andr, s∈Q/Z,

μ^∗_nμn= 1, μkμn=μkn, (kn=k∨nas a lattice or L.C.M.) e(0) = 1, e(r)^∗=e(−r), e(r)e(s) =e(r+s).

These and that relations give such a presentation of the Hecke C^∗-algebra H1

of the BC system ([13], [94]). It then follows that the Bost-Connes HeckeC^∗- algebraH1 is isomorphic to the semi-groupC^∗-crossed productA1.

Those relations say that the mapN^×!n→μn is viewed as an isometric representation of the semi-group N^× and that the map Q/Z ! r → e(r) is viewed as a unitary representation of the groupQ/Z, and these representations become a covariant representation of both, to extend to define such a semi-group C^∗-algebra crossed product by taking aC^∗-algebra (norm) completion.

In terms of that explicit presentation, the timeevolution is defined as the form

σt(μn) =n^itμn=e^itlognμn and σt(e(r)) =e(r).

The space Sm^nc(GL1,{±1}) =GL1(Q)\A^◦/R^∗+ can be compactified by re- placingA^◦ byA, as in [29], to give the quotient

Sm^nc(GL1,{±1}) =GL1(Q)\A/R^∗+.

This compactification consists of adding the trivial lattice (with a possibly non- trivialQ-structure).

The dual space to Sm^nc(GL1,{±1}), under the duality of type II and type III factors, introduced by Connes (thesis), is a principalR^∗+-bundle over Sm^nc(GL1,{±1}), whose noncommutative algebra of coordinates is obtained as the C^∗-algebra crossed product of the C^∗-algebra corresponding to Sm^nc(GL1,{±1}) by the time evolutionσt as anR-action.

Namely, that is, and contains

(C(A)βQ^∗+)σR⊃(C0(Af)βQ^∗+)σR=B1σR as a closed ideal.

The space obtained in this way is the space of ad`ele classes as the principal R^∗+-bundle:

Lt1=GL1(Q)\A (bundle) ←−−−− R^∗+ (fiber)

⏐⏐

Lt1/R^∗+=GL1(Q)\A/R^∗+ (base space)

which gives the spectral realization of zeros of the Riemann zeta functionζ, as in [29]. The passing to this dual space corresponds to considering commensura- bility classes of 1-dimensionalQ-lattices (without up to scaling).

Consequently, wemay denote as

C^∗(Lt1/R^∗+) =C^∗(GL1(Q)\A/R^∗+) =C(A)βQ^∗+=C^∗(Sm^nc(GL1,{±1})) and C^∗(Lt1) =C^∗(GL1(Q)\A) = (C(A)βQ^∗+)×σR.

As well

C^∗(Sm^nc(GL1,{±1})) =C^∗(GL1(Q)\A^◦/R^∗+) =C0(Af) Q^∗+. Moreover, wemay denote as well that

C^∗(AfβQ^∗+) =C0(Af)βQ^∗+=B1⊂C^∗(Lt1/R^∗+), C^∗((Q/Z)αN^×) =C^∗(Q/Z)αN^×=A1⊂B1,

both of which are reduced from C^∗(Lt1/R^∗+), so may be denoted as B1 = C_r^∗(Lt1/R^∗+) orA1=C_r^∗(Lt1/R^∗+).

Structure of KMS states. The Bost-Connes Hecke C^∗-algebra H1∼=A1 the semigroup C^∗-algebra crossed product of Laca and Raeburn has irreducible representations on the Hilbert space H = l²(N^×). These are parameterized

by elements α ∈ Z^p,∧ = GL1(Z^p) = (Z^p)⁻¹. Any such element α defines an embedding α: Q^cyc → C of the subfieldQ^cyc of C generated by the roots of unity (by the same symbol), and the corresponding representation has the form

πα(e(r))ξk =α(ζ_r^k)ξk=e^2πirkξk and πα(μn)ξk=ξnk

forr= [r] =r+Z∈Q/Z andk∈N^×, where{ξk}k∈N^× is the canonical basis forl²(N^×), withξk=χk the characteristic function atk(cf. [96, 2.4]).

TheHamiltonianimplementing the time evolution by the actionαofN^× is given by Hξk = (logk)ξk. Thus, thepartitionfunction of the Bost-Connes system is the Riemann zeta function

Z(β) = tr(e^−βH) = ∞ k=1

1

k^β =ζ(β).

It is shown by Bost and Connes [13] that KMS states have the following structure, with a phase transition atβ= 1.

Theorem 3.6. (Edited). Consider on A1=H1 withσ the time evolution.

• In the range0< β ≤1, there is a unique KMSβ stateϕβ. Its restriction toQ[Q/Z]in C^∗(Q/Z)has the form

ϕβ(e(a

b)) =b^−βΠpprime,p|b

1−p^β−1 1−p⁻¹ ,

where the product corresponds to the prime factorization for b. Moreover, each ϕβ is associated to the hyper-finite factor of type III1, that is, R_∞ of Araki- Woods.

•For1< β ≤ ∞, the set Eβ of extremalKMSβ states can be identified with (Z^p)^∗, which has a free and transitive action by (itself) the group also induced by some automorphisms of A1. The extremal KMSβ stateϕβ,α corresponding toα∈(Z^p)^∗ has the form onC^∗(Q/Z)

ϕβ,α(x) = 1

ζ(β)tr(πα(x)e^−βH)

= 1

ζ(β) ∞ n=1

α(x)ⁿ n^β

. There states are associated to the typeI_∞ factor, that is,B(H).

•Atβ=∞, the Galois groupG= Gal(Q^cyc/Q)acts on the values ofKMS_∞ states ϕ ∈ E∞ on a certain arithmetic subalgebra A1,Q of A1. These states have a property thatϕ(A1,Q)⊂Q^cyc and that the class field theory isomorphism θ:G∼= (Z^p)^∗intertwines the Galois action on values with the action of(Z^p)^∗ by symmetries, namely,γϕ(x) =ϕ(θ(γ)x) for any ϕ∈E_∞,γ ∈G, andx∈A1,Q, where the arithemetic subalgebra A1,Q can be taken as an algebra over Q, generated by the operators e(r)andμn, μ^∗_n forr∈Q/Z andn∈N^×.

Remark. (Added). Asmentioned in [13], the critical temperature is atT = 1.

At lower temperature forβ >1, the phases of the BC systemare parameterized

by all possible embedding of Q^cyc into C. The Galois group G acts naturally as a group of automorphisms ofA1=H1 commuting with the time evolution σ and the spontaneous symmetry breaking occurs for β >1.

As shown in [41], thatQ-algebraA1,Q can also be obtained as the algebra generated by μn, μ^∗n forn∈N^× and by homogeneous functions of weight zero on 1-dimensional Q-lattices obtained as a normalization of the functions

ξk,a(Λ, ϕ) =

y∈Λ+ϕ(a)

1

y^k by covolume.

Namely, consider the functionsc^kξk,a, wherec(Λ) is proportional to the covol- ume|Λ|and satisfies 2π√

−1c(Z) = 1.

The choice of such an arithmetic subalgebra ofA1 corresponds to endowing the noncommutative space (asA1,Q) with an arithmetic structure. The subal- gebra corresonds to the rational functions and the values of KMS_∞ states at elements of this subalgebra should be thought of as values of rational functions at the classical points of the noncommutaitve space (cf. [48]).

3.3 Noncommutative geometry and the Hilbert 12th prob-

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