Non-commutative geometry and new stable structures

Boris Zilber University of Oxford November 7, 2005

This paper grew out of an observation that some new stable structures discovered in the 1990’ as counterexamples to well-known conjectures in pure model theory might be related to non-commutative geometry.

The general meaning of the conjectures was that “very good”, or more technically, very stable structures must be in a certain way reducible to algebraic geometry over algebraically closed fields or to linear structures (Trichotomy conjecture and Algebraicity conjecture for groups, see [Z0]). This proved to be true to some extent (see [HZ]) but still two types of counterexamples signal the necessity to reconsider the connection between model theoretic classification principles and classical mathematics.

The first class of counterexamples shows that nonlinear one-dimensional Zariski geometries are not necessarily algebraic curves. Given a smooth algebraic curve C with big enough group of regular automorphisms one can produce a “smooth” Zariski curve C̃ along with a finite cover p : C̃ → C. C̃ can not be identified with any algebraic curve because the construction produces an unusual subgroup of the group of regular automorphisms of C̃ ([HZ, section 10). The main theorem of [HZ] states that it is the biggest deviation from an algebraic curve that can happen to a Zariski curve. Typical example of an unusual subgroup of a such C̃ is the nilpotent group of two generators U and V with the central commutator ϵ = [U, V] of finite order N. So, the defining relations are

UV = ϵ VU, ϵ^{N} = 1.

This, of course, hints towards the known structure of non-commutative geometry, the non-commutative (quantum) torus at the Nth root of unity. We call this example T_{N}.

The other example is of a different nature. B.Poizat constructed in [P] a multiplicative subgroup G of an algebraically closed field (we may assume this to be the field ℂ of complex numbers) such that (ℂ, +, ·, G) has ωstable theory of rank half of that of ℂ (so called “bad field”, related to the Algebraicity conjecture). The present author has shown in [Z2] that, assuming Schanuel’s conjecture, one can construct G by means of real analytic geometry. More specifically one can consider G of the form G = exp(αℤ) · exp(βℝ), α and β linearly independent over ℝ, β ∉ ℝ ∪ iℝ, and see that (ℂ, +, ·, G) is superstable of dimension half of that of ℂ. We then note that the structure on the quotient ℂ^{∗}/G is geometrically the same as what one gets in the quotient

T_{h}^{2} = (S × S)/L_{h}

of the square of the unit circle S ⊆ ℂ^{∗} by a Kronecker foliation L_{h} (setwise this is the same as the group ℝ/⟨1, h⟩). This is a basic example and motivation of A.Connes [C] for introducing non-commutative geometry.

Of course, one of the biggest challenges in relating non-commutative geometry to model theory comes from the difference in the way objects are represented in each of the approaches. Geometers tend to replace a structure M by the dual object, the algebra ℂ[M] of functions on M, or even more abstract non-commutative algebra of “observables” which take the role of the algebra of functions. Generally, non-commutative geometry does not assume that one has a reverse procedure of getting a structure back from the algebra of observables. Yet it is desirable to have a manifold-kind structure underlying a given algebra of observables. Yu.Manin makes this point in [Man] I.1.4 as a foundational problem.

In the present paper we undertake a thorough study of both classes of examples. We try to give answers to the following questions:

1. What are the “algebras of functions” for T_{N} and T_{h}^{2}? Can these structures be identified as objects of non-commutative geometry?

2. What is the structure that non-commutative geometry “sees” on T_{N} and T_{h}^{2}?

3. Is there a uniform representation of both types of structures?

By virtue of construction the algebra of Zariski continuous (regular) functions T_{N} → ℂ is the same as that of ℂ^{∗}, that is ℂ[t, t^{−1}], so does not reflect enough of the structure T_{N}. We show that specifically to the structures under question one can introduce the algebra of semi-definable functions. These are not uniquely defined but the commutative algebra H they generate is determined uniquely up to isomorphism. Moreover, uniquely determined is the action on H of certain linear operators related to the “hidden” structure of T_{N}. Algebra of these linear operators is the same as that of non-commutative torus at root of unity known to geometers.

One of the semi-definable functions plays a special role in the construction of U and V, this is the angular function

ang_{N} : ℂ^{∗} → ℂ[N], N-roots of 1,

satisfying certain conditions. Answering the second question above we show that T_{N} can be identified with a space of linear functionals H → ℂ of a positive orientation. We introduce the orientation in terms of the angular function. Alternatively but equivalently T_{N} can be identified with the space of N-dimensional irreducible modules of positive orientation over the coordinate algebra.

Then we look for a similar construction that can play a role of the limit structure T_{N} as N tends to ∞. The usual model-theoretic limit (the ultraproduct) does not quite work here, for the same reasons as the universal cover exp : ℂ → ℂ^{∗} can not be obtained as the ultraproduct of finite covers x ↦ x^{N}, ℂ^{∗} → ℂ^{∗}. We find a natural construction in terms of the structure of real and complex numbers, dependent on a real parameter h, the analytic Zariski structure T_{h}, which we show to behave as the limit structure in many respects. In particular the irreducible modules are of countable dimension with U-eigenvalues of the form q^{n}μ, m ∈ ℤ, q = exp 2πih, μ ∈ ℂ^{∗}, one μ for each module. The corresponding space H of semi-definable functions on T_{h} together with the action of U and V on it turns out to be a close analogue of the space with an action corresponding to Connes’ quantum torus T_{h}^{2}. The correspondent angular function gets the form of a function

(ℂ, +) → exp(2πihℤ),

behaving similarly to the function z ↦ exp(i Re z).

At this point we don’t have a full analogy yet, since setwise the space of our irreducible positively oriented modules is ℂ/⟨1, h⟩ rather than ℝ/⟨1, h⟩. Connes specifies, using his ℂ^{∗}-algebras language, that U and V must be unitary operators. This immediately translates into the fact that the eigenvalues q^{n}μ above must lie on the unit circle and so he gets S/⟨q⟩ while we have ℂ^{∗}/⟨q⟩. Instead of using the (unstable) ℂ^{∗}-algebras language we note that the group of regular automorphisms of T_{h} (commuting with U and V) is exactly the above group G = exp(2πihℤ + βℝ). This implies that the action of U and V is well-defined on the quotient ℂ^{∗}/G which is definable in our T_{h} and is representing Connes’ T_{h}^{2}.

We hence found a way to represent uniformly our T_{N}’s together with Connes’ T_{h}. Moreover, we can see that there exists a universal object U in this uniform representation. Namely, for each N ∈ ℕ ∪ {h} there is a surjective map

e_{N} : U → T_{N}

which also gives an interpretation of T_{N} in terms of U.

It is important to mention that the above description of the structures can not be complete without giving a detailed description of the languages involved. In fact there are at least two levels of languages. The basic language is the language of the example in [HZ], and we prove that T_{h} is superstable in this language (probably is analytic Zariski of dimension 1 see [PZ] and [Z1]).

We also discuss the language which allows the angular function ang. The conditions defining ang do not constitute a complete theory, so it is natural to choose a complete extension which axiomatises the existentially closed structures. In fact such a choice amounts to choosing ang in a uniformly random way. We conjecture that under this choice the theory is supersimple. This has been proven by D.Evans in a basic case. It seems both feasible and mathematically meaningful to undertake a detailed analysis of the structure of definable sets in the theory, and develop a probabilisitc measure theory on the sets.

Acknowledgement Most of the paper was written while I was a member of Model Theory and Applications to Algebra and Analysis programme at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. I am grateful to the organisers of the program, the staff of the Institute and to the participants.

1 Non-algebraic Zariski geometries

1.1 Recall the following theorem C of [HZ].

Theorem There exist irreducible pre-smooth Zariski structures (in particular of dimension 1) which are not interpretable in an algebraically closed field.

The construction

Let M be an irreducible pre-smooth Zariski structure, G ≤ ZAut M acting freely on M and for some G̃ with finite H :

1 → H → G̃ →^{p_{0}} G → 1.

Consider a set S ⊆ M of representatives of G-orbits: for each a ∈ M, G · a ∩ S is a singleton.

Consider the formal set

M(G̃) = M̃ = G̃ × S

and the projection map

p : (g, s) ↦ p_{0}(g) · s.

Consider also, for each f ∈ G̃ the function

f : (g, s) ↦ (fg, s).

Claim 1.The structure

(M̃, {f}_{f∈G̃}, p^{−1}(Zariski relations on M))

is an irreducible pre-smooth Zariski structure, its isomorphism type is determined by M and G̃ only and dim M̃ = dim M.

Proof. One can use obvious automorphisms of the structure to prove quantifier elimination. The statement of the claim then follows by checking the definitions. The detailed proof is given in [HZ] Proposition 10.1.

Claim 2. Suppose H does not split, for every proper G_{0} < G̃

G_{0} · H ≠ G̃.

Then, every equidimensional Zariski expansion M̃′ of M̃ is irreducible.

Indeed. Let C = M̃′ is an |H|-cover of the variety M, so dim C = dim M and C has at most |H| distinct irreducible components, say C_{i}, 1 ≤ i ≤ n. For generic y ∈ M the fiber p^{−1}(y) intersects every C_{i} (otherwise p^{−1}(M) is not equal to C).

Hence H acts transitively on the set of irreducible components. So, G̃ acts transitively on the set of irreducible components, so the setwise stabiliser G^{0} of C_{1} in G̃ is of index n in G̃ and also H ∩ G̃^{0} is of index n in H. Hence,

G̃ = G^{0} · H, with H ⊈ G^{0}

contradicting our assumptions. Claim proved.

Claim 3. G̃ ≤ ZAut M̃, that is G̃ is a subgroup of the group ZAut M of Zariski-continuous bijecions of M.

Immediate by the construction.

Lemma. Suppose M is a rational or elliptic curve (over an algebraically closed field F of characteristic zero), H does not split, G̃ is nilpotent and for some big enough integer μ there is a non-abelian subgroup G_{0}

|G̃ : G_{0}| ≥ μ.

Then M̃ is not interpretable in an algebraically closed field.

Proof First we show.

Claim 4. Without loss of generality we may assume that G̃ is infinite. Recall that G is a subgroup of the group ZAut M of rational (Zariski)

automorphisms of M. Every algebraic curve is birationally equivalent to a smooth one, so G embeds into the group of birational transformations of a smooth rational curve or an elliptic curve. Now remember that any birational transformation of a smooth algebraic curve is biregular. If M is rational then the group ZAut M is PGL(2, F). Choose a semisimple (diagonal) s ∈ PGL(2, F) be an automorphism of infinite order such that ⟨s⟩∩G = 1 and G commutes with s. Then we can replace G by G′ = ⟨G, s⟩ and G̃ by G̃′ = ⟨G̃, s⟩ with the trivial action of s on H. One can easily see from the construction that the M̃′ corresponding to G̃′ is the same as M̃, except for the new definable bijection corresponding to s.

We can use the same argument when M is an elliptic curve, in which case the group of automorphisms of the curve is given as a semidirect product of a finitely generated abelian group (complex multiplication) acting on the group on the elliptic curve E(F).

Now, assuming that M̃ is definable in an algebraically closed field F′ we will have that F is definable in F′. It is known to imply that F′ is definably isomorphic to F, so we may assume that F′ = F.

Also, since dim M̃ = dim M = 1, it follows that M̃ up to finitely many points is in a bijective definable correspondence with a smooth algebraic curve, say C = C(F).

G̃ then by the argument above is embedded into the group of rational automorphisms of C.

The automorphism group is finite if genus of the curve is 2 or higher, so by Claim 4 we can have only rational or elliptic curve for C.

Consider first the case when C is rational. The automorphism group then is PGL(2, F). Since G̃ is nilpotent its Zariski closure in PGL(2, F) is an infinite nilpotent group U. Let U^{0} be the connected component of U, which is a normal subgroup of finite index. By Malcev’s Theorem (see [Merzliakov], 45.1) there is a number μ (dependent only on the size of the matrix group in question but not on U) such that some normal subgroup U^{0} of U of index at most μ is a subgroup of the unipotent group

( \begin{matrix}1 z \\ 0 1 \end{matrix} )

this is Abelian, contradicting the assumption that G̃ has no abelian subgroups of index less than μ.

In case C is an elliptic curve the group of automorphisms is a semidirect product of a finitely generated abelian group (complex multiplication) acting freely on the abelian group of the elliptic curve. This group has no nilpotent non-abelian subgroups. This finishes the proof of the Lemma and of the theorem. □

In general it is harder to analyse the situation when dim M > 1 since the group of birational automorphisms is not so immediately reducible to the group of biregular automorphisms of a smooth variety in higher dimensions. But nevertheless the same method can prove the useful fact that the construction produces examples essentially of non algebro-geometric nature.

Proposition (i) Suppose M is an abelian variety, H does not split and G̃ is nilpotent not abelian. Then M̃ can not be an algebraic variety with p : M̃ → M a regular map.

(ii) Suppose M is the (semi-abelian) variety (F^{×})^{n}. Suppose also that G̃ is nilpotent and for some big enough integer μ = μ(n) has no abelian subgroup G_{0} of index bigger than μ. Then M̃ can not be an algebraic variety with p : M̃ → M a regular map.

Proof (i) If M is an abelian variety and M̃ were algebraic, the map p : M̃ → M has to be unramified since all its fibers are of the same order (equal to |H|). Hence M̃ being a finite unramified cover must have the same unversal cover as M has. So, M̃ must be an abelian variety as well. The group of automorphisms of an abelian variety A without complex multiplication is the abelian group A(F). The contradiction.

(ii) Same argument as in (i) proves that M̃ has to be isomorphic to (F^{×})^{n}. The Malcev theorem cited above finishes the proof.□

Proposition. Suppose M is an F-variety and, in the construction of M̃, the group G is finite. Then M̃ is definable in any expansion of the field F by a total linear order.

In particular, if M is a complex variety, M̃ is definable in the reals.

Proof Extend the ordering of F to a linear order of M and define

S := {s ∈ M : s = min G · s}.

The rest of the construction of M̃ is definable.□

Remark In other known examples of non-algebraic M̃ (with G infinite) M̃ is still definable in any expansion of the field F by a total linear order.

Problem (i) Classify Zariski structures definable in the reals. (ii) Classify Zariski structures definable in the reals as a smooth real

manifold.

(iii) Find new Zariski structures definable in ℝ_{an} as a smooth real manifold.

2 A non-algebraic Zariski curve and its coordinate algebra

2.1 Let F be an algebraically closed field of characteristic 0 and N a positive integer. Consider the groups given by generators and defining relations,

G = ⟨u, v : uv = vu⟩,
G̃ = ⟨U, V : [U, [U, V]] = [V, [U, V]] = 1 = [U, V]^{N}⟩.

Let a, b ∈ F^{∗} multiplicatively independent.

G acts on F^{×} :

u · x = ax, v · x = bx.

Taking M to be F^{×} this determines, by 1.1, a presmooth non-algebraic Zariski curve M̃ which from now on we denote T_{N}.

Since [U, V] is a central element, in every representation of G̃ one can replace [U, V] by an ϵ ∈ F, a primitive root of unity of order N. So, the defining relation for G̃ becomes just

VU = ϵUV,

or

VUV^{−1}U^{−1} = ϵ.

The correspondent definition for the covering map p : M̃ → M then gives us

p(Ut) = ap(t), p(Vt) = bp(t).  (1)

2.2 Semi-definable functions.

Lemma Given α, β such that α^{N} = a, β^{N} = b, one can define bijections

x_{k} : T_{N} → F^{∗} k = 0, …, N − 1

so that for any t ∈ T_{N} the following functional equations are satisfied,

x_{k}(t)^{N} = p(t)  (2)
x_{k}(Ut) = αϵ^{k}x_{k}(t),  (3)

x_{k}(Vt) = βx_{k+1}(t), where x_{N} = x_{0},  (4)
\frac{x_{k+1}(t)}{x_{k}(t)} = \frac{x_{k}(t)}{x_{k−1}(t)}.  (5)

Proof First, notice that (3),(4) imply

x_{k}([U, V]^{−1}t) = ϵx_{k}(t),  (6)

where [U, V]^{−1} = U^{−1}V^{−1}UV.

To construct the x_{k} choose randomly an injection ^{N} √ : F^{×} → F^{×} with the property

( \sqrt[N]{w})^{N} = w.

For any s ∈ S and t ∈ G̃ · s of the form t = U^{m}V^{n}[U, V]^{l} · s, set

x_{k}(U^{m}V^{n}[U, V]^{l} · s) := α^{m}β^{n}ϵ^{mk−l} \sqrt[N]{s}.

This satisfies (2)-(5).

To see that each x_{k} is injective consider t, t′ ∈ T_{N} such that x_{k}(t) = x_{k}(t′). We then have, by (2), that p(t) = p(t′). Hence t′ = ht for some h ∈ H, that is for h = [U, V]^{j}, some j ∈ {0, …, N − 1}. By (6) this is possible only if j = 0, that is t = t′.

In order to prove that x_{k} is surjective we need to solve the equation

x_{k}(t) = μ

for any given μ ∈ F^{×}. Since p is surjective we can find t′ ∈ T such that p(t′) = μ^{N}, and so by (2) we have x_{k}(t′) = ϵ^{l}μ, for some l ∈ ℤ. Take now t = [U, V]^{l}t′ and by (6) this solves the equation. □

2.3 Define the angular function on F^{∗} as a function ang : F^{×} → F[N], roots of unity of order N.

Set for λ ∈ F^{∗},

ang(λ) = \frac{x_{1}(t)}{x_{0}(t)}, if λ = x_{0}(t).

This is well-defined since x_{0} is a bijection.

Acting by U on t and using (3) we have

ang αλ = ϵang λ  (7)

We also have

ang ϵλ = ang λ.  (8)

since by (6)

x_{0}([U, V]^{−1}t) = ϵx_{0}(t) = ϵλ,

and at the same time

ang(ϵλ) = \frac{x_{1}([U, V]^{−1}t)}{x_{0}([U, V]^{−1}t)} = \frac{x_{1}(t)}{x_{0}(t)} = ang λ.

Finally, suppose x_{1}(t) = λ. Then x_{0}(Vt) = βλ, by (4), and x_{1}(Vt) = βx_{2}(t) = βλ · ang λ, by (5). Since ang βλ = x_{1}(Vt) : x_{0}(Vt), we have

ang βλ = ang λ.  (9)

Now we consider the structure

(F, +, ·, ang).

It is clear that F is partitioned into N ’sectors’ using the angular function:

P_{δ} = {μ ∈ F^{∗} : ang μ = δ}.

Proposition T_{N} is definable in (F, +, ·, ang) using parameters α and β. Moreover, x_{0}, …, x_{N−1} are definable in the structure as well.

Proof Define T = F^{×} as a set, and for any t ∈ F^{×} set

p(t) = t^{N}, Ut = α t, Vt = β ang(t) t.

We then have

t →^{U} α t →^{V} αβ ang(α t)t = αβ ang(t) ϵ t →^{U^{−1}} β ang(t) ϵ t →^{V^{−1}} ϵt.

That is

V^{−1}U^{−1}VUt = ϵ t

so, the group G̃ acts on the T freely.

It is also clear that

p(Ut) = ap(t), p(Vt) = bp(t), p^{−1}(p(t)) = {[U, V]^{−l}t : l = 0, …, N − 1} as required by the description of T_{N}.

Finally, set x_{k}(t) := (ang t)^{k} · t. □

From now on we use notation

Ť_{N} := (F, +, ·, ang).

The interpretation of T_{N} in the proof of the above proposition we will consider canonical, with respect to α and β.

Remark 1 The isomorphism type of T_{N} defined by means of Ť_{N} depends on the isomorphism type (so of the cardinality) of the field F with parameters α, β, ϵ only, and not on the choice of the angular function (equivalently P_{δ}) since by the construction in 1.1 any two structures M̃ with the same G̃ are isomorphic over M.

Corollary Assuming that F = ℂ and a, b ∈ ϵ · ℝ_{>0}, ϵ = exp 2πi/N, we have that T_{N} is definable in the reals using parameters α, β ∈ ℝ and ϵ such that α^{N} = a, β^{N} = b.

Proof It is enough to define an angular function with respect to the chosen parameters. Consider

P = \{z ∈ ℂ^{×} : \frac{2π}{N} > arg z ≥ 0\}.

Define

P_{ϵ^{k}} := ϵ^{k}P, k = 0, …, N − 1

and

ang λ := ϵ^{k} iff λ^{N} ∈ ϵ^{k}P.

This satisfies (7)-(9) by our assumptions.□

2.4 Question Consider a structure Ť_{N} which is existentially closed in the class of structures satisfying (7) - (9). What is the model-theoretic status of the theory of this structure? Is it supersimple?

Remark Before this paper has been finished D.Evans answered this question in positive.

The fact that Ť_{N} is supersimple has certain methodological significance. There is a common, albeit informal, understanding that simple structures (theories) come basically from stable structures by introducing a ’random noise’. So, one may think of Ť_{N} as an algebraic curve with a random angular function.

Problem Study the structure of definable subsets on Ť_{N}. Is there a good probabilistic measure theory on Ť_{N}?

2.5 Systems of semi-definable functions.

Denote ϵ̇ = [U, V] and let Γ = Γ_{N} := ⟨ϵ̇⟩ be the subgroup of G̃. We denote ϕ : G̃ → G = ⟨u, v⟩, the canonical embedding of the subgroup into F^{×}.

Lemma Given α, β such that α^{N} = a, β^{N} = b, one can define functions

x : G × T_{N} → F^{×}

so that for any g ∈ G, x(g, ·) : T_{N} → F^{×} is a bijection and for any t ∈ T_{N} the following functional equations are satisfied,

x(g, t)^{N} = p(t)  (10)
x(g, Ut) = αϕ(g)x(g, t),  (11)
x(g, Vt) = βx(gv, t),  (12)
x(gf, t)x(gf^{−1}, t) = x(g, t)^{2} for any f ∈ G̃.  (13)

Proof First, notice that (11),(12) imply

x(γϵ̇^{−1}t) = ϵx(γ, t).  (14)

To construct x choose randomly an injection ^{N} √ : F^{×} → F^{×} with the property

( \sqrt[N]{w})^{N} = w.

For any s ∈ S and t ∈ G̃ · s of the form t = U^{m}V^{n}[U, V]^{l} · s, set

x(γ, t) := α^{m}β^{n}ϕ(γ^{m})ϵ^{−l} \sqrt[N]{s}.

This satisfies (10)-(13).

To see that, for each γ, x(γ, ·) is injective consider t, t′ ∈ T_{N} such that x(γ, t) = x(γ, t′). We then have, by (10), that p(t) = p(t′). Hence t′ = for some δ ∈ Γ, that is for δ = ϵ̇^{j}, some j ∈ {0, …, N − 1}. By (14) this is possible only if j = 0, that is t = t′.

In order to prove that x(γ, ·) is surjective we need to solve the equation

x(γ, t) = μ

for any given μ ∈ F^{×}. Since p is surjective we can find t′ ∈ T such that p(t′) = μ^{N}, and so by (10) we have x(γ, t′) = ϵ^{l}μ, for some l ∈ ℤ. Take now t = ϵ̇^{l}t′ and by (14) this solves the equation. □

Define

ξ(t) = \frac{x(γϵ̇, t)}{x(γ, t)}.

By (13) this indeed does not depend on γ.

2.6 The space of semi-definable functions.

Let H be the F-algebra of semi-definable functions on T_{N} generated by x_{0}, …, x_{N−1}, x_{0}^{−1}, …, x_{N−1}^{−1}.

Remark H is determined as a commutative F-algebra uniquely up to isomorphism by its generators x_{0}, …, x_{N−1} satisfying the relations (2).

We may also regard it as an F-vector space with some linear operators on them.

We define linear operators U^{∗} and V^{∗} on H :

\begin{matrix}U^{∗} : ψ(t) ↦ ψ(Ut), \\ V^{∗} : ψ(t) ↦ ψ(Vt). \end{matrix}  (15)

Obviously these operators are invertible, so U^{∗}^{−1}, V^{∗}^{−1} are the inverses. Denote G̃^{∗} the group generated by the operators U^{∗}, V^{∗}, U^{∗}^{−1}, V^{∗}^{−1}.

H with the action of G̃^{∗} on it is determined uniquely up to isomorphism by the defining relation (2)-(6) and so is independent on the arbitrariness in the choices of x_{0}, …, x_{N−1}.

Finally we notice

Lemma The correspondence U ↦ U^{∗}, V ↦ V^{∗} generates the antiisomorphism G̃ → G̃^{∗} satisfying the property

(g_{1}g_{2})^{∗} = g_{2}^{∗}g_{1}^{∗}, for any g_{1}, g_{2} ∈ G̃.

Proof It can easily be seen if we define the pairing

H × T → F, (ψ, t) ↦ ψ(t).

This allows to consider the adjoint action of any g ∈ G̃ on H setting g^{∗}ψ as the unique element of H such that

(g^{∗}ψ, t) = (ψ, gt), for all t ∈ T.

We can immediately identify that this definition extends (15). The desired formula follows.□

2.7 Let Max(H) be the space of maximal ideals of the commutative algebra H.

Lemma 1 Max(H) consists of ideals Iμ̄, μ̄ = ⟨μ_{0}, …, μ_{N−1}⟩, μ_{0}^{N} = ⋯ = μ_{N−1}^{N},

Iμ̄ = ⟨(x_{0} − μ_{0}), …, (x_{N−1} − μ_{N−1})⟩.

Proof This is a standard fact of commutative algebra.□

Assuming F is endowed with an angular function ang : F^{×} → F[N] we call μ̄ as above oriented positively if μ_{k} = ang(μ_{0})^{k} · μ_{0}. Correspondingly, we call an ideal Iμ̄, oriented positively if μ̄ is.

Max^{+}(H) will denote the subspace of Max(H) consisting of positively oriented ideals I.

Lemma 2 μ̄ is positively oriented if and only if

⟨μ_{0}, …, μ_{N−1}⟩ = ⟨x_{0}(t), …, x_{N−1}(t)⟩,

for some t ∈ T.

Proof Indeed, since x_{0} is a bijection, there is t ∈ T such that x_{0}(t) = μ_{0}. Now apply the definition of natural angular function of 2.3.□

2.8 Lemma

(i) There is a bijective correspondence Ξ : Max^{+}(H) → T_{N} between the space of positively oriented maximal ideals and T_{N}.

(ii) The action (15) of G̃^{∗} on H induces an action on Max(H) and leaves Max^{+}(H) setwise invariant.

(iii) The action of g^{∗} ∈ G̃^{∗} on Max(H)(and so on T_{N}) can be identified as

g^{∗} : I_{⟨x_{0}(t),...,x_{N−1}(t)⟩} ↦ I_{⟨x_{0}(g^{−1}t),...,x_{N−1}(g^{−1}t)⟩}.

Proof (i). We set

Ξ(t) := Iμ̄, for μ̄ = ⟨x_{0}(t), …, x_{N−1}(t)⟩.

Then Ξ(t) is positively oriented by Remark 2 in 2.7.

Notice that by definition μ̄ is determined uniquely by μ_{0}. But x_{0} : T_{N} → F^{×} is bijective, so Ξ is bijective.

(ii)-(iii). For a given g ∈ G̃, the map ψ → g^{∗}ψ is an automorphism of the commutative F-algebra H, since g^{∗}ψ(t) = ψ(gt). So, it sends maximal ideals to maximal ideals, namely

g : ⟨(x_{0} − μ_{0}), …, (x_{N−1} − μ_{N−1})⟩ ↦ ⟨(x_{0}^{g} − μ_{0}), …, (x_{N−1}^{g} − μ_{N−1})⟩.

Notice that, for the unique t_{μ} ∈ T_{N} such that x_{0}(t_{μ}) = μ_{0}, …, x_{N−1}(t_{μ}) = μ_{N−1}

⟨x_{0}(U^{−1}t_{μ}), …, x_{N−1}(U^{−1}t_{μ})⟩ = ⟨α^{−1}μ_{0}, …, α^{−1}ϵ^{1−N}μ_{N−1}⟩,

by (3). Analogously, by (4)

⟨x_{0}(V^{−1}t_{μ}), …, x_{N−1}(V^{−1}t_{μ})⟩ = ⟨β^{−1}μ_{N−1}, β^{−1}μ_{0}, …, β^{−1}μ_{N−2}⟩.

So, by Lemma 2.7.2 both tuples on the right-hand side are positively oriented. Now notice that by (3) and (4)

U : ⟨(x_{0}−μ_{0}), …, (x_{N−1}−μ_{N−1})⟩ ↦ ⟨(αx_{0}−μ_{0}), …, (αϵ^{N−1}x_{N−1}−μ_{N−1})⟩ =

⟨(x_{0} − α^{−1}μ_{0}), …, (x_{N−1} − α^{−1}ϵ^{1−N}μ_{N−1})⟩ =
= ⟨(x_{0} − x_{0}(U^{−1}t)), …, (x_{N−1} − x_{N−1}(U^{−1}t))⟩

and

V : ⟨(x_{0} − μ_{0}), …, (x_{N−1} − μ_{N−1})⟩ ↦ ⟨(βx_{1} − μ_{0}), …, (βx_{0} − μ_{N−1})⟩ =

⟨(x_{0} − β^{−1}μ_{N−1}), …, (x_{N−1} − β^{−1}μ_{N−2})⟩ =
= ⟨(x_{0} − x_{0}(V^{−1}t)), …, (x_{N−1} − x_{N−1}(V^{−1}t))⟩.

This proves that the image of positive Iμ̄ under U and V is positive. Hence the image under the action of any g ∈ G̃ is positive, and we have (ii). The above also shows that the action induced by Ξ is anti-isomorphic to the original action and so proves (iii).□

2.9 We may also treat T as the space of F-linear functionals H → F defined by the pairing of 2.6,

H_{T}^{∗} = {F_{t} : ψ ↦ (ψ, t), t ∈ T }.

Obviously, the kernel of a nonzero functional is a maximal ideal. Moreover,

ker F_{t} = {ϕ ∈ H : (ϕ, t) = 0} = I_{⟨x_{0}(t),...,x_{N−1}(t)⟩}.

We also denote ker F_{t} := I^{t}.

We call a linear functional F on H positive if ker F is a positive maximal ideal.

Proposition

(i) The correspondence

t ↦ F_{t}

between T and the space H_{+}^{∗} of positive linear functionals on H is bijective.

(ii) The correspondence transfers isomorphically the natural action of G̃ on T to a natural action of G̃ on H_{+}^{∗}.

(iii) Consider also the commutative algebra H_{0} generated by p(t) and, for each linear functional F_{t} its restriction F_{t}^{0} on H_{0}. Then, for any t_{1}, t_{2} ∈ T,

F_{t_{1}}^{0} = F_{t_{2}}^{0} iff p(t_{1}) = p(t_{2}) iff F_{t_{1}} = ϵ^{j}F_{t_{2}}, for some j ∈ {0, …, N − 1},

and the correspondence

F_{t}^{0} ↦ p(t)

is a bijection between the space H_{0}^{∗} of all linear functionals of the form F_{t}^{0} and F^{×}.

Proof Let I ∈ Max(H). To any such I canonically corresponds the functional

F^{I} : ψ ↦ λ ∈ F, such that (ψ − λ) ∈ I.

We write

F (ψ) := {F, ψ}.

Now, in case I = I^{t} = I_{⟨x_{0}(t),...,x_{N−1}(t)⟩} we see that

{F^{I}, ψ} = ψ(t) = (ψ, t).  (16)

The latter establishes the required bijection between H_{T}^{∗} and T_{N}. On the other hand, since functionals of H_{T}^{∗} are in bijective correspondens with positive ideals, by Lemma 2.7.2, H_{T}^{∗} = H_{+}^{∗}, the set of all positive functionals. This proves (i).

(ii). Given F ∈ H^{∗} and f ∈ G̃^{∗} define f^{∗}F as the unique functional such that

{f^{∗}F, ψ} = (F, fψ).

Then by dualities we have the isomorphism of group with actions on T and H^{+} correspondingly

g ∈ G̃ ↦ g^{∗∗} ∈ G̃^{∗∗},
(ψ, gt) = (g^{∗}ψ, t) = {F^{t}, g^{∗}ψ} = {g^{∗∗}F^{t}, ψ}.

(iii). It is immediate from definitions that if F^{t} evaluates x_{0} as μ ∈ F^{×}, then the function p (as an element of H) is evaluated as μ^{N}. The statement follows. □

2.10 Comments

1.The space H is an analogue of the space S(ℝ^{2}, ℂ) of all Schwartz functions ℝ^{2} → ℂ decaying at infinity along with all its derivatives faster than \frac{1}{|x|^{n}} , any n (see A.Connes),.

2. In mathematical physics linear functionals on certain Hilbert spaces are called states.

Assume for a moment that H is an inner product space. Then any F ∈ H^{∗} can be identified with the orthogonal complement I^{⊥} of the maximal ideal corresponding to F. This is a one-dimensional subspace of H. This provides another version of the notion of states.

3. Even though the present definition of H considers it a finitely generated commutative ring, it can not treat it as the coordinate ring of an algebraic variety since we consider positively oriented ideals only.

We used in 2.9 the natural pairing H×T → F and the existence of enough functionals on the linear space H.

4. Despite the fact that T is in a bijective correspondence with a subset H_{+}^{∗} of the space of functionals we can not induce the additive structure on T since H_{+}^{∗} is not closed under addition.

3 The limit case

We introduce and study here a structure Ť_{∞} which can be seen as the limit version of Ť_{N}. It would be important in our view to formulate (and prove) the exact meaning of the transition N → ∞ but we only draw here parts of the possible picture towards this aim.

3.1 Let α, β ∈ ℂ^{×}, αℝ + βℝ = ℂ. Set, for w ∈ ℂ, the α-β - decomposition to be the uniquely determined decomposition

w = w_{a}α + w_{b}β, w_{a}, w_{b} ∈ ℝ.

Let i_{a}, i_{b} ∈ ℝ be the coordinates of the decomposition

i = i_{a}α + i_{b}β, here and below i^{2} = −1.

We also choose a real number h and assume that 1, 2πi_{a} and 2πi_{a}h are linearly independent over ℚ.

We define an additive α-β-version of the angular function, which we call band

bd_{h} : ℂ → 2πihℤ, fixed h ∈ ℝ \ ℚ

as follows.

First we define the function r ↦ [r]_{h} from ℝ to ℤ, the pseudo-integer part of r with the properties, for all r ∈ ℝ,

[0]_{h} = 0, [r + 1]_{h} = [r]_{h} + 1,  (17)
[r + 2πi_{a}]_{h} = [r]_{h},  (18)
[r + 2πi_{a}h]_{h} = [r]_{h}  (19)

Example Consider a direct sum decomposition

ℝ = ℝ′ ˙+2πi_{a}ℚ ˙+2πi_{a}hℚ, some subgroup ℚ < ℝ′ < ℝ,

and set, for all r′ ∈ ℝ′, c ∈ ℚ,

[r′ + c_{1} · 2πi_{a} + c_{2} · 2πi_{a}h]_{h} := [r′ + (c_{1} − [c_{1}]) · 2πi_{a} + (c_{2} − [c_{2}]) · 2πi_{a}h],

[ · ] the usual integer part of a real number. This satisfies (17)-(19).

Set

bd_{h} w := 2πih [w_{a}]_{h}.

We have then, by definition,

bd_{h}(rβ + w) = bd_{h} w, for every r ∈ ℝ;  (20)
bd_{h}(w + 2πi) = bd_{h}(w);  (21)
bd_{h}(w + 2πih) = bd_{h} w.  (22)

By (17),

bd_{h}(α + w) = 2πih + bd_{h} w.  (23)

Set,

Ũ : w ↦ α + w,
Ṽ : w ↦ β + w + bd_{h} w.

We have

w ↦^{U} α + w ↦^{V} α + β + w + bd_{h}(α + w) = α + β + w + bd_{h} w + 2πih ↦^{U^{−1}}

↦^{U^{−1}} β + w + 2πih + bd_{h} w ↦^{V^{−1}} 2πih + w.

That is

Ṽ^{−1}Ũ^{−1}ṼŨ w = w + 2πih,  (24)

3.2 Define the additive subgroup of ℂ

A_{h} = βℝ + 2πihℤ + 2πiℤ.

Proposition (i) A_{h} is the subgroup of all periods of bd_{h}, that is a ∈ ℂ such that bd_{h}(a + w) = bd_{h}w.

(ii) A_{h} is exactly the subgroup of shifts w ↦ a + w of ℂ which are automorphisms of (ℂ, Ũ, Ṽ).

(iii) A_{h} is definable in (ℂ, +, bd_{h}).

Proof (i). Immediate from (20)- (22). For (ii) notice that Ũ(a + w) = a + Ũw, for all a ∈ ℂ and

Ṽ(a + w) = a + Ṽw iff a ∈ A_{h}.

(iv) Immediate by definitions.□

3.3 We consider here the two-sorted structures

((ℂ, +, bd_{h}), exp, ℂ^{×}) and ((ℂ, +, A_{h}), exp, ℂ^{×})

where the second sort ℂ^{×} on the nonzero complex numbers comes with the usual language of all Zariski closed relations.

Obviously the functions Ũ and Ṽ are definable in (ℂ, +, bd_{h}). Conversely, bd_{h} is definable in (ℂ, +, Ṽ) using parameter β.

Proposition 1 The theory of ((ℂ, +, A_{h}), exp, ℂ^{×}) is superstable, provided the Schanuel conjecture is true.

Proof It is easy to see that the statement follows if the expansion of ℂ^{×} with the unary predicate for the subgroup G_{h} = exp(A_{h}) = exp(2πihℤ + βℝ) is superstable. A stronger theorem, stating ω-stability of the theory, for G = exp(βℝ + δℚ), β ∈ ℂ \ (ℝ ∪ iℝ), δ ∈ ℝ \ 2πiℚ, was proved in [Z2]. The same proof describes the elementary theory of the structure and yields superstability for the present theory. See also [Z3]. □

Notation G_{h} will stand for the subgroup exp(A_{h}) of ℂ^{×}.

On the other hand ((ℂ, +, bd_{h}), exp, ℂ^{×}) defines the following unstable structure on the sort ℂ^{×}.

Denote, for t = exp w,

ang_{h}t := exp bd_{h}w.

By (21) this is well-defined, and by (20),(22) we have analogues of (7)-(9), where q = exp 2πih,

ang_{h}qt = ang_{h}t,
ang_{h}e^{β}t = ang_{h}t,
ang_{h}e^{α}t = q · ang_{h}t.

Hence, defining

U : t ↦ e^{α} · t, V : t ↦ e^{β} · t · ang_{h}t,

we get

VUt = qUVt, for all t ∈ ℂ^{×}.

It is easy to see that also

U exp w = exp Ũw, V exp w = exp Ṽw.

We define

Ť_{h} := (ℂ, +, ·, ang_{h}).

This is an obvious analogue of Ť_{N} defined in 2.3.

Note that the group Γ_{h} = exp 2πihℤ = ang_{h}(ℂ^{×}) is definable in Ť_{h}.

The full analogy with Ť_{N} of 2.3 requires also a definition of p_{h}. We define

p_{h} : ℂ^{×} → ℂ^{×}/Γ_{h},

the canonical homomorphism. This agrees with 2.3, moreover in the finite case ℂ^{×}/⟨ϵ⟩ can be definably identified with ℂ^{×} in the full Zariski language, in particular the whole construction is a Zariski structure (obviously, of finite Morley rank).

We also define the maps u and v on ℂ^{×}/Γ_{h} by

u p_{h}(t) := p_{h}(Ut), v p_{h}(t) := p_{h}(Vt),

that is

u : t · Γ_{h} ↦ e^{α} · t · Γ_{h}, v : t · Γ_{h} ↦ e^{β} · t · Γ_{h}.

This is obviously well-defined.

Proposition 2 The group of shifts t ↦ gt on ℂ^{×} commuting with ang_{h} (and so with U and V) is G_{h}. This group is definable in Ť_{h}. The theory of the structure (ℂ, +, ·, G_{h}, Γ_{h}) is superstable.

Proof Essentially the same argument as for Proposition 1. The superstability of the weaker structure (ℂ, +, ·, Γ_{h}) is well-known and follows from the Lang property of Γ_{h}. □

Problems 1. Fix the theory T_{h}^{G} of structures of the form (F, +, ·, ang, e_{a}), (e_{a} a constant ) saying that

(F, +, ·, Aut(ang), ang(F^{×}), e_{a}) ≡ (ℂ, +, ·, G_{h}, Γ_{h}, e^{α})

(where Aut(ang) is the group of shifts of F^{×} commuting with ang, and ang(F^{×}) is the image under ang)

and

∀t ∈ F^{×} ang g · t = q · ang t iff g^{−1}e_{a} ∈ Aut(ang).

Consider the class Ṫ_{h}^{G} of existentially closed models of T_{h}^{G}. What is the stability status of completions of Ṫ_{h}^{G}. Are they supersimple?

2. Is Ť_{h} above based on the band function bd_{h} given in the Example in 3.1 existentially closed in T_{h}^{G}? Is it supersimple?

3.4 We notice here that in ((ℂ, +, bd_{h}, 2πi_{a}·, h· ), exp, ℂ^{×}) ( 2πi_{a}· and h· are unary operations here) one can definably construct an inverse to the usual exponentiation exp : ℂ → ℂ^{×}.

Define the function

ln_{0} : ℂ^{×} → ℂ

by setting, for t = exp w,

ln_{0} t = w − h^{−1}bd_{h}(w/2πi_{a}).

It is immediate that

exp(ln_{0} t) = t.

Claim ln_{0} t is well-defined and is injective.

Indeed, if also exp w′ = t, w′ = w + 2πik, some k ∈ ℤ, then

bd_{h}(w′/2πi_{a}) = bd_{h}(\frac{w + 2πik}{2πi_{a}}) = bd(\frac{w}{2πi_{a}}) + 2πihk, by (23).

Hence,

w′ − h^{−1}bd_{h}(w′/2πi_{a}) = w − h^{−1}bd_{h}(w/2πi_{a}),

as required.

In more detail,

ln_{0} t = w − 2πi[\frac{w_{a}}{2πi_{a}}]_{h}.  (25)

So,

ln_{0} t = ln_{0} t′ iff w − 2πi[\frac{w_{a}}{2πi_{a}}]_{h} = w′ − 2πi[\frac{w′_{a}}{2πi_{a}}]_{h},

whence w − w′ ∈ 2πiℤ and t = t′,

hence ln_{0} is injective.

Remark The logarithm constructed here resembles the random logarithm constructed (non-effectively) by T.Hyttinen [Hy].

3.5 Now we redefine Ť_{N} in a way compatible both with 2.3 and 3.3.

Define, for each positive N ∈ ℕ the map

e_{Nh} : ℂ → ℂ^{×}; e_{Nh}(w) = exp(N^{−1}h^{−1}w).

It is convenient to distinguish the copies of ℂ^{×} which are images of e_{Nh} for different N as T_{N}.

Set, for t = e_{Nh}(w) ∈ T_{N},

U_{N}t := e_{Nh}(Ũw), V_{N}t := e_{Nh}(Ṽw).

It follows,

U_{N}t := e_{Nh}(α) · t, V_{N}t := e_{Nh}(β) · t · exp \frac{2πi}{N}[w_{a}]_{h}.

Denote

ang_{N}(t) := exp \frac{2πi}{N}[w_{a}]_{h}.

This is well-defined. Indeed, any other representation of t would be of the form t = e_{Nh}(w + 2πihNk), k ∈ ℤ. But (w + 2πihNk)_{a} = w_{a} + 2πi_{a}hNk, and [w_{a} + hNk]_{h} = [w_{a}]_{h} by (19).

Similarly one checks that ang_{N} satisfies (7)-(9) with ϵ = exp \frac{2πi}{N} and corresponding parameters for α, β. So we get, by 2.3

V_{N}U_{N} t = ϵ U_{N}V_{N} t.  (26)

Define

Ť_{N} = (ℂ, +, ·, ang_{N})

This is the same definition as 2.3 except here we specified our choice of the angular function.

Proposition The group of periods of ang_{N}, that is g ∈ ℂ^{×} such that ang_{N}(g · t) = ang_{N}t is equal to

G_{N^{−1}h^{−1},αh^{−1}} · ℂ[N] = exp(2πiN^{−1}h^{−1} + αh^{−1}ℤ + βℝ) · ℂ[N].

In particular, this group is definable in the above Ť_{N} and the theory of

(ℂ, +, ·, G_{N^{−1}h^{−1},αh^{−1}})

is superstable.

Proof By calculation: for t = exp N^{−1}h^{−1}w and g = exp N^{−1}h^{−1}u, by definition,

ang_{N}(gt) = exp \frac{2πi}{N}[w_{a} + u_{a}]_{h},

so g is a period if and only if

∀r ∈ ℝ [r + u_{a}]_{h} ≡ [r]_{h} modNℤ,

iff u_{a} ∈ 2πi_{a}ℤ + 2πi_{a}hℤ + Nℤ iff

g ∈ exp(2πi_{a}h^{−1}N^{−1}+2πi_{a}αN^{−1}ℤ+αh^{−1}ℤ+βℝ) = exp(2πiN^{−1}h^{−1}ℤ+2πiN^{−1}ℤ+αh^{−1}ℤ+βℝ).

The superstability follows by the same argument as in 3.3. □

Problem Is the theory of Ť_{N} as given by the present construction, supersimple?

3.6 Denote

U = (ℂ, +, bd_{h}, h·).

By the construction in 3.3 and 3.5 Ť_{N} is definable in (U, exp, ℂ^{×}) , for all N ∈ ℕ ∪ {h}.

The resulting picture is as follows, with the arrows showing definable surjections.

[figure: U; Ť_{h}; ℂ^{×} = Ť_{1}; e_{1}; e_{h}; e_{2h}; e_{3h}; e_{Nh}; Ť_{3}; Ť_{2}; Ť_{N}; …; ·; ·; ·; p_{N}; p_{2}; p_{3}; p_{h}; ℂ^{×}/Γ_{h}]

where e_{1}(w) := exp w.

4 Quantum torus

Our aim here is to connect the construction of Ť_{h} to the well-known definition of the noncommutative (quantum) torus usually denoted T_{h}^{2}.

4.1 Following the pattern of 2.2 and 2.3 we introduce the algebra H generated by functions

x_{k} : ℂ^{×} → ℂ^{×}, k ∈ ℤ,

where x_{0} = x is the identity function and

x_{k} = ξ^{k} · x, ξ(t) = ang_{h}t.

We have by 3.3,

x_{k}(Ut) = e^{α}q^{k} · x_{k}(t),
x_{k}(Vt) = e^{β}x_{k+1}(w),
ξ(Ut) = q · ξ(t), ξ(Vt) = ξ(t).

As in ?? we normalise the operators U^{∗} and V^{∗} on functions by defining operators on H,

\begin{matrix}U̇ : ψ ↦ U^{∗}ψ, U^{∗}ψ(w) = ψ(Uw); \\ V̇ : ψ ↦ ξ · ψ. \end{matrix}

Using the identities above we get immediately the usual

U̇V̇ = qV̇U̇.

4.2 We can introduce an isomorphic space with operators in an alternative but closely connected way.

Let z and ζ be the functions ℂ → ℂ^{×} given by

z(w) = exp w, ζ(w) = exp bd_{h}w.

Denote Ḣ the commutative F-algebra generated by z and ζ, and denote z_{k} = ζ^{k}z.

We have, using identities for bd_{h},

z(Ũw) = e^{α} · z(w), ζ(Ũw) = q · ζ(w),
z(Ṽw) = e^{β}ζ(w)z(w), ζ(Vw) = ζ(w).

Again, we define operators on Ḣ :

\begin{matrix}U̇ : ψ ↦ Ũ^{∗}ψ, \\ V̇ : ψ ↦ ζ · ψ. \end{matrix}

The space Ḣ is an analogue of the space S(ℝ^{2}, ℂ) of all Schwartz functions ℝ^{2} → ℂ decaying at infinity along with all its derivatives faster than \frac{1}{|x|^{n}} , any n (see [C]), or S(ℤ^{2}, ℂ) the Hilbert space of Schwartz sequences, that is complex valued sequences (c_{m,n}) decaying faster than any polynomial of m, n.

In [C] with each leaf of the Kronecker foliation

L_{a} = {⟨r, s⟩ ∈ ℝ^{2} : s + θr = a}

one associates the ℂ[U̇, V̇, U̇^{−1}, V̇^{−1}]-module H_{a} obtained by restricting functions of S(ℝ^{2}, ℂ) to L_{a} and defining operators U̇ and V̇. Namely, the operator U̇ is defined by exactly the same formula as here and V̇ sends ψ(r, s) (function of two real variables r and s) to exp(is) · ψ(r, s) (notice that extra to these data there is a linear dependence between r and s). So, ξ is a good analogue of the function exp(is) taking values in the unit circle.

Notice that U̇ and V̇ are unitary operators if we see H_{a} as a Hilbert space. This makes the completion of ℂ[U̇, V̇, U̇^{−1}, V̇^{−1}] a ℂ^{∗}-algebra.

By A.Connes the quantum torus T_{θ}^{2} is the space of all the modules H_{a} on the correspondent L_{a}.

Remark Consider again the algebra of functions Ḣ and denote, for a ∈ ℂ, Ḣ_{a} the algebra obtained by restricting functions from Ḣ to the coset a + A_{h}. It follows from Proposition 3.2(ii) that the action of U̇ and V̇ on Ḣ induces a well-defined action on Ḣ_{a}, so this is a ℂ[U̇, V̇, U̇^{−1}, V̇^{−1}]-module for any a ∈ ℂ.

4.3 To understand further relations of Connes’ construction to our T_{h} we prove the following.

Claim 1. There is a natural bijective correspondence

ϕ : ℂ/A_{h} → T_{θ}^{2},

for θ = h, where T_{θ}^{2} is seen as the space of leaves of the Kronecker foliation.

Indeed, we have the decomposition of ℂ into two real lines

ℂ = iℝ + αℝ, for any z ∈ ℂ z = xi + yα, x, y ∈ ℝ.

Rescale the real coordinates

r := h^{−1}x, s := 2π(2πi_{a})^{−1}y

and consider the mapping onto the direct product of two unit circles

z ↦ ⟨x, y⟩ ↦ ⟨r, s⟩ ↦ ⟨exp ir, exp is⟩.

Under the map

2πihℤ + 2πi_{a}αℤ → ⟨2πhℤ, 2πi_{a}ℤ⟩ → ⟨2πℤ, 2πℤ⟩ → 1,

and since 2πi − 2πi_{a}α ∈ βℝ,

βℝ → ⟨2π, −2πi_{a}⟩ℝ → ⟨2πh^{−1}, −2π⟩ℝ → L_{0}.

This establishes the bijection between the cosets of A_{h} and the leaves L_{a} of the foliation.

Claim 2. There is a bijective correspondence

p̃_{h} : ℂ/A_{h} → ℂ^{×}/G_{h},

induced by p_{h}. Moreover, the action of Ũ and Ṽ on ℂ induces a well-defined action on ℂ/A_{h} and correspondingly the action on ℂ^{×}/G_{h}. The latter action coincides with the one induced by u and v on the cosets of G_{h}.

This is the direct consequence of Proposition 3.2(iii) and the definition of p_{h}.

Corollary p̃_{h} ∘ ϕ^{−1} identifies T_{h}^{2} with ℂ^{×}/G_{h}, with all the structure on the latter induced from Ť_{h}.

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