Bruno Iochum

Thierry Masson

Thomas Schucker

Andrzej Sitarz

T Schücker

κ-Deformation and Spectral Triples

Keywords: Pacs numbers: 11.10.Nx, 02.30.Sa, 11.15.Kc

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Lukierski, Ruegg, Nowicki & Tolstoy discovered a Hopf algebraic deformation of the Poincaré Lie algebra and called κ the deformation parameter [START_REF] Lukierski | q-deformation of Poincaré algebra[END_REF][START_REF] Lukierski | New quantum Poincaré algebra and κ-deformed field theory[END_REF]. Since this pioneering work, the subject became very active: the Hopf algebra was represented on the κ-deformation of Minkowski space [START_REF] Majid | Bicrossproduct structure of κ-Poincaré group and non-commutative geometry[END_REF][START_REF] Zakrzewski | Quantum Poincaré group related to the κ-Poincaré algebra[END_REF]. This has been used to generalize the notion of a quantum particle [START_REF] Lukierski | Classical quantum mechanics of free κ relativistic systems[END_REF] or in quantum fields [START_REF] Daszkiewicz | κ-deformed oscillators, the choice of star product and free κ-deformed quantum fields[END_REF]. Algebraic properties like differential calculi on the κ-Minkowski space were investigated [START_REF] Sitarz | Noncommutative differential calculus on the κ-Minkowski space[END_REF] as well as the Noether theorem [START_REF] Amelino-Camelia | On the 5D differential calculus and translation transformations in 4D κ-Minkowski noncommutative spacetime[END_REF][START_REF] Amelino-Camelia | A no-pure-boost uncertainty principle from spacetime noncommutativity[END_REF]. The κ-Minkowski space has also been popularized as 'double special relativity' [START_REF] Amelino-Camelia | Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale[END_REF] and appears in spin foam models [START_REF] Freidel | Effective 3d quantum gravity and noncommutative quantum field theory[END_REF].

It is natural to ask whether κ-geometry is a noncommutative one in the sense of Connes [START_REF] Connes | Gravity coupled with matter and the foundation of noncommutative geometry[END_REF][START_REF] Connes | On the spectral characterization of manifolds[END_REF] (see [START_REF]Spectral geometry of κ-Minkowski space[END_REF] for a first attempt). While the algebraic † Talk presented by B. I. at the conference "Geometry and Physics in Cracow", September 21-25, 2010, iochum@cpt.univ-mrs.fr ‡ The Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, [START_REF] Freidel | Effective 3d quantum gravity and noncommutative quantum field theory[END_REF]

setting is quite clear, the main difficulty is to overcome the analysis which is an essential part in the definition of spectral triples. The κ-deformation of n-dimensional Minkowski space is based on the Lie-algebraic relations [x 0 , x j ] := i κ x j , [x j , x k ] = 0, j, k = 1, . . . , n -1.

(

) 1
Here we assume κ > 0. As in [42, eq. (2.6)], one gets e icµx µ = e ic 0 x 0 e ic ′ j x j where c ′ j := κ c 0 (1 -e -c 0 /κ )c j .

Assuming that the x µ 's are selfadjoint operators on some Hilbert space, we define unitaries U ω := e iωx 0 and V k := e -i n-1 j=1 k j x j with ω, k j ∈ R, which generate the κ-Minkowski group considered in [START_REF] Agostini | κ-Minkowski representations on Hilbert spaces[END_REF].

If W ( k, ω) := V k U ω , one gets as in [1, eq. ( 13)]

W ( k, ω) W ( k ′ , ω ′ ) = W (e -ω/κ k ′ + k, ω + ω ′ ). (2)
The group law (2) is, for n = 2, nothing else but the crossed product G κ := R ⋊ α R with group isomorphism α(ω)k := e -ω/κ k, k ∈ R. [START_REF] Amelino-Camelia | On the 5D differential calculus and translation transformations in 4D κ-Minkowski noncommutative spacetime[END_REF] Note that G κ ≃ R ⋊ R * + is the affine group on the real line which is solvable and nonunimodular. The irreducible unitary representations are either onedimensional, or fall into two nonequivalent classes [START_REF] Agostini | κ-Minkowski representations on Hilbert spaces[END_REF]. When κ → ∞, the usual plane R 2 is recovered but with an unpleasant pathology at the origin [START_REF] Dabrowski | The κ-Minkowski spacetime: Trace, classical limit and uncertainty relations[END_REF].

For a given ω, a particular case occurs when m := e -ω/κ ∈ N * , since

U ω V k = (V k ) m U ω , (4)
m being independent of k j . This means that for chosen ω and k j , the presentation of the group is given by two generators and one relation.

Here, we investigate different spectral triples (A, H, D) associated to the group C * -algebra of G κ . To avoid technicalities due to a continuous spectrum of D, we want a unital algebra so that we consider a periodic time x 0 which induces a discrete version G a of G κ where a is a real parameter depending on κ. This is done in section 2.

In section 3, we give the main properties of the algebra U a = C * (G a ) and its representations. For C * -algebras of groups, the left regular representation is the natural one to consider. But due to the structure of G a (solvable with exponential growth, see Theorem 3.2), there is a known obstruction to construct finite-summable spectral triples on U a based on this representation. In order to bypass this obstruction, we need to refine our understanding of the structure of G a in terms of an underlying dynamical system. This structure permits to define a particular representation of U a using only the periodic points of the dynamical system. Following Brenken and Jørgensen [START_REF] Brenken | A family of dilation crossed product algebras[END_REF], the topological entropy of this dynamics is considered.

It is worthwhile to notice that the elementary building blocks G a given by a = m ∈ N * as in (4) are some of the amenable Baumslag-Solitar groups, already encountered in wavelet theory [START_REF] Jørgensen | Ruelle operators: Functions which are harmonic with respect to a transfer operator[END_REF]. The power of harmonic analysis on groups also justifies a reminder of their main properties in section 4.

The question of the finite summability of these triples is carefully considered with results by Connes [START_REF] Connes | Compact metric spaces, Fredholm modules, and hyperfiniteness[END_REF] and Voiculescu [START_REF] Voiculescu | Some results of norm-ideal perturbations of Hilbert space operators[END_REF][START_REF] Voiculescu | On the existence of quasicentral approximate units relative to normed ideals. Part I[END_REF]: using the regular representation of C * (G a ), an obstruction to finite-summability appears and allows only θ-summability. However, faithful representations of C * (G a ), not quasi-equivalent to the left regular one and based on the existence of periodic points for dynamical systems, can give rise to arbitrary finite-summable spectral triples. These results are summarized in Theorems 5.2 and 5.3. All proofs will appear elsewhere.

Motivations and models

Let us consider the example given by (1) for two hermitian generators x 0 and x 1 . For any (k, ω) ∈ R 2 , one defines W (k, ω) := V k U ω = e -ikx 1 e iωx 0 . Then one has (2) which is a representation of the group G κ defined in [START_REF] Amelino-Camelia | On the 5D differential calculus and translation transformations in 4D κ-Minkowski noncommutative spacetime[END_REF]. The bounded operators W (f ) := Gκ f (k, ω) W (k, ω) e ω/κ dk dω for any f ∈ L 1 (G κ , e ω/κ dk dω) (here e ω/κ dk dω is the left Haar measure on G κ ) generate a representation of

C * red (G κ ). The product of f, g ∈ L 1 (G κ , e ω/κ dk dω) takes the form (f * κ g)(k, ω) = Gκ f (k ′ , ω ′ )g(e ω ′ /κ (k -k ′ ), ω -ω ′ ) e ω ′ /κ dk ′ dω ′ .
The advantage of considering the theory of group C * -algebras is twofold. Many structural properties on groups will turn out to be useful in studying some properties of the corresponding C * -algebras. Moreover, this allows us to construct in a natural way compact versions of noncommutative spaces as we now explain.

For an abelian topological group G, C * red (G) is isomorphic to C 0 ( G) where G is the Pontryagin dual of G. Both algebras are defined as spaces of functions. By duality, a discrete subgroup Γ ⊂ G produces the C * -algebra C * red (Γ) ≃ C( Γ) where C( Γ) is the C * -algebra of continuous functions on the compact space Γ. Notice that there is a natural dual map G → Γ. For the example of the plane, consider the discrete subgroup Γ

= Z 2 ⊂ R 2 . Then the resulting C * -algebra is C * (Z 2 ) ≃ C(T 2 ) because Z = T 1 . The dual map R ≃ R → Z = T 1
is explicitly given by x → e 2πix . The choice of the subgroup Γ = Z 2 ⊂ R 2 corresponds then to the choice of the compact version T 2 of the (dual) space R 2 ≃ R 2 . The compactification takes place in the space of the variables (x, y).

In order to get a compact version of this κ-deformed Minkowski space, one has to choose a discrete subgroup H κ ⊂ G κ . Since H κ is discrete and non-abelian, the associated algebra C * (H κ ) is unital and noncommutative, so it can be interpreted as a compact noncommutative space. This point is motivated in section 2.1 and is done in section 2.2.

As a final preliminary remark, let us mention that the groups we will encounter will be decomposed as crossed products with Z, so both the (related) theories of discrete dynamical systems and crossed products of C * -algebras will be intensively used in many parts of this work.

Spectral triples

The goal is to study the existence of spectral triples for the κ-deformed space. A spectral triple (or unbounded Fredholm module) (A, H, D) [START_REF] Connes | Noncommutative geometry and reality[END_REF][START_REF] Connes | Gravity coupled with matter and the foundation of noncommutative geometry[END_REF][START_REF] Connes | Noncommutative Geometry, Quantum Fields and Motives[END_REF] is given by a unital C * -algebra A with a faithful representation π on a Hilbert space H and an unbounded self-adjoint operator D on H such that -the set

A = { a ∈ A : [D, π(a)] is bounded } is norm dense in A, -(1 + D 2
) -1 has a compact resolvent. (A is always a * -subalgebra of A.) Of course, the natural choice of the algebra A is to take the C * -algebra of the group G κ , but since A = C * (G κ ) has no unit, we need to replace the second axiom by:

-π(a)(1 + D 2 ) -1 has a compact resolvent for any a ∈ A. This technical new axiom generates a lot of analytical complexities but is necessary to capture the metric dimension associated to D. For instance, if a Riemannian spin manifold M is non-compact, the usual Dirac operator D has a continuous spectrum on H = L 2 (S) where S is the spinor bundle on M . Nevertheless, the spectral triple C ∞ (M ), L 2 (S), D has a metric dimension which is equal to the dimension of M . A noncommutative example (the Moyal plane) of that kind has been studied in [START_REF] Gayral | Moyal planes are spectral triples[END_REF]. We try to avoid these difficulties here by using a unital algebra A.

The compact version model as choice of a discrete subgroup

We consider only dimension n = 2 but the results can be extended to higher dimensions thanks to (4). To get a unit, we choose a discrete subgroup H κ of G κ such that 1 ∈ C * (H κ ). Since we want also to keep separate the role of the variables x 0 and x 1 , we consider the subgroup of the form H κ = H ⋊ α Z: we first replace the second R of G κ in (3) by the lattice Z which corresponds to unitary periodic functions of a chosen frequency ω 0 (the time x 0 is now periodic). So, given κ > 0 and ω 0 ∈ R, with

a := e -ω 0 /κ ∈ R + , (5)
the group R ⋊ αa Z is a subgroup of G κ where α a (n) is the multiplication by a n . This subgroup is a non-discrete "ax + b" group. Then, we want a group H to be a discrete (now, not necessarily topological) subgroup of the first R in R ⋊ αa Z, which is invariant by the action α a . Given k 0 ∈ R, a natural building block candidate for a discrete H is given by H = B a • k 0 ≃ B a where

B a := { i, finite m i a n i : m i , n i ∈ Z },
and more generally, one can take

H ≃ ⊕ k 0 B a .
The search for a discrete subgroup

H κ of G κ such that 1 ∈ C * (H κ ) leads to H κ,a := B a ⋊ αa Z which is isomorphic to a subgroup of G κ once k 0 is fixed.
This procedure drives us to the analysis of the algebraic nature of a. For instance, when a = m ∈ N * is an integer, this group H κ,m is well known since it is the solvable Baumslag

κ,m = Z[ 1 m ]⋊ αm Z of G κ,m
and which can be seen as generated by two unitaries U, V such that

U = U ω 0 , V = V k 0 and U V = V m U .
Here, κ := -ω 0 log -1 (m) > 0 for some given integer m > 1 and some ω 0 ∈ R -, k 0 ∈ R.

The algebra U a and its representations

We now compute the C * -algebra U a which is our model for a compact version of the 2-dimensional κ-Minkowski space. The structure of this algebra is described through a semi-direct product of two abelian groups, one of which depends explicitly on the real parameter a > 0. This semi-direct structure gives rise to a dynamical system which is heavily used in the following. The classification of the algebras U a is performed: the K-groups are not complete invariants, and we use the entropy defined on the underlying dynamical system to complete this classification. Then some representations of U a are considered. They strongly depend on the algebraic or transcendental character of a. In the algebraic case, some particular finite dimensional representations are introduced based on periodic points of the dynamical system. This construction will be used in section 5.

Let a = e -ω 0 /κ ∈ R * + with a = 1, and let us recall general facts from [START_REF] Brenken | A family of dilation crossed product algebras[END_REF]:

Define B a := { i m i a n i for finitely many m i , n i ∈ Z }.
This discrete group is torsion-free so its Pontryagin dual B a is connected and compact. Let α a be the action of Z on b ∈ B a defined by α a (n)b := a n b, let α a be the associate automorphism on B a and

G a := B a ⋊ αa Z, U a := C * (G a ) = C * (B a ) ⋊ αa Z = C( B a ) ⋊ αa Z.
This kind of C * -algebras also appeared in [START_REF] Carey | Families of type III KMS states on a class of C * -algebras containing O n and Q N[END_REF] for totally different purposes.

The group G a is generated by u := (0, 1) and v := (1, 0).

Lemma 3.1. Let a ∈ R * + , then B a = B 1/a and G a ≃ G 1/a . Thus the C * -algebra U a and U 1/a are isomorphic.
The symmetry point a = 1/a corresponds to the commutative case in (4) with a = 1 or the undeformed relation ( 1) with κ = ∞. In this spirit U a can be viewed as a deformation of the two-torus.

The dynamical system U a ≃ C( B a ) ⋊ α Z has an ergodic action (if a = 1) [START_REF] Brenken | The local product structure of expansive automorphisms of solenoids and their associated C * -algebras[END_REF][START_REF] Brenken | A family of dilation crossed product algebras[END_REF] and if the set of q-periodic points is Per q ( B a ) := { χ ∈ B a : α k (χ) = χ, ∀k < q, α q (χ) = χ } then the growth rate lim q→∞ q -1 log #Per q ( B a ) of this sets is an invariant of U a which coincides with the topological entropy h( α a ):

h( α a ) = lim q→∞ q -1 log #Per q ( B a ) . (6)
This entropy can be finite or infinite, dividing the algebraic properties of a into two cases: a can be a an algebraic or a transcendental number.

Transcendental case

If a is a transcendental number, then

B a ≃ Z[a, a -1 ]. Thus, B a ≃ ⊕ Z Z, B a ≃ S a := { z = (z k ) ∞ k=-∞ ∈ T Z } and α(z) k = z k+1 for z ∈ S a , k ∈ Z so that α is just the shift σ on T Z . Thus U a ≃ C(T Z ) ⋊ σ Z and h( α) = ∞.
Note that the wreath product ≀ appears with its known presentation:

G a = B a ⋊ α Z ≃ Z ≀ Z ≃ u, v : [u i vu -i , v] = 1 for all i ≥ 1 .
This group is amenable (solvable), torsion-free, finitely generated (but not finitely presented), residually finite with exponential growth. S a contains a lot of q-periodic points: they are obtained by repeating any sequence (z k ) q-1 k=0 of arbitrary elements in T. Aperiodic points are easily constructed also. Moreover, S a is the Bohr compactification b Ba R of R.

Algebraic case

Assume now that a is algebraic. Let P ∈ Q[x] be the monic irreducible polynomial such that P (a) = 0 and let

P = c Q a where Q a ∈ Z[x]. If d is the degree of Q a , we get the ring isomorphism B a ≃ Z[x, x -1 ]/(Q a ) [11]. Moreover, B a has a torsion-free rank d. If Q a (x) = d j=0 q j x j (so Q a has leading coefficient q d ∈ N * ), let A a ∈ M d×d (Z) be the d × d-matrix defined by (A a ) i,j := q d δ i,j-1 for 1 ≤ j ≤ d and (A a ) d,j = -q j-1 . Then q d a j = d k=1 (A a ) j,k a k-1 . For instance, if a = 1/m for m ∈ N * then P (x) = x -1/m, Q a (x) = mx -1, d = 1, so B a = Z[ 1 m
] and A a is just the number 1. Let σ be the shift on the group (T d )

Z and consider its σ-invariant sub-

group K a := { z = (z k ) ∞ k=-∞ ∈ (T d ) Z : q d z k+1 = A a z k } (use T ≃ R/Z).
If S a is the connected component of the identity of K a , then there exists a topological group isomorphism ψ :

B a → S a such that σ |Sa • ψ = ψ • α [41, Theorem 19]. For any χ ∈ B a , the associated z = (z k ) ∞ k=-∞ is given by z (i) k = χ(a k+i-1 ) where z k = (z (i) k ) d i=1 ∈ T d .
In particular z 0 is given by χ(1), χ(a), . . . , χ(a d-1 ) . This map is only surjective on the connected component of the identity. When a = 1/m with m ∈ N * , we will recover S 1/m = S m in [START_REF] Carey | Families of type III KMS states on a class of C * -algebras containing O n and Q N[END_REF].

There is a morphism of groups ι : R d → S a defined as follows: to any

φ = (φ (i) ) i=1,...,d ∈ R d , one associates ι(φ) = z = (z k ) ∞ k=-∞ ∈ (T d ) Z with z (i) k = exp 2iπq -k d d j=1 (A k a ) i,j φ (j) . (7)
This shows that S a is a Bohr compactification of

R d [10, Proposition 2.4]. Then α(φ) = q -1 d A a φ defines an action of Z on R d which satisfies ι• α = α•ι. If r i , i = 1, • • • d are the roots of P , then by [11, Proposition 3, Corollary 1], c q (a) := # Per q (S a ) = Π q k=1 |Q a (e i2πk/q )| = |q d | q Π d k=1 |1 -r k q |. (8)
Thus by [START_REF] Bekka | Some properties of C * -algebras associated to discrete linear groups[END_REF], the topological entropy is

h( α a ) = log |q d | + i, |r i |>1 log |r i |. (9)
In case a = m or a = 1/m, (9

) gives h( α m ) = h( α 1/m ) = log(m).
Aperiodic points in S a can be easily constructed using the map ι defined by [START_REF] Blackadar | Operator Algebras, Theory of C * -Algebras and von Neumann Algebras[END_REF]: any φ ∈ (R\Q) d defines an aperiodic point ι(φ) ∈ S a .

On the structure and classification of algebras U a

We can now give the main properties of the algebras U a :

Theorem 3.2. Let a ∈ R * + and a = 1. Then (i) The group G a = B a ⋊ α Z is a torsion-free discrete solvable group with exponential growth and B a is a compact set isomorphic to a solenoid S a .

(ii A main point of this theorem, crucial for the sequel is that the algebra U a is residually finite and its proof is based on properties of the underlying dynamical system:

) U a = C * red (B a ⋊ α Z) ≃ C(S a ) ⋊ α Z is a NGCR
Proposition 3.3. Let a ∈ R *
+ and a = 1. The subgroup of periodic points and the set of aperiodic points of S a under α are dense. The space of orbits of S a is not a T 0 -space.

The classification of algebras U a is also based on the dynamical system:

Theorem 3.4. Let ω 0 ∈ R and κ ∈ R * + defining a = 1 in (5). (i) U a ≃ U a ′ yields c q (a) = c q (a ′ ), ∀q ∈ N * . (ii) The entropy h( α) is also an isomorphism-invariant of U a .
This result has important physical consequences since a full Lebesgue measure dense set of different parameters a (namely the transcendental ones) generates the same algebra or κ-deformed space, while in the rational case, these spaces are different: Corollary 3.5. As already seen, U a ≃ U 1/a . Moreover, (i) All transcendental numbers a generate isomorphic algebras U a .

(ii) If U a ≃ U a ′ , then a and a ′ are both simultaneously algebraic or transcendental numbers.

(iii) If U a ≃ U a ′ , then a ′ = a or a ′ = a -1 in the following cases: a, a ′ or their inverses are in Q * or are quadratic algebraic numbers.

(iv

) If a = m/l ∈ Q * + , K 0 (U a ) ≃ Z and K 1 (U a ) ≃ Z ⊕ Z l-m .
1 A C * -algebra A is said to be CCR or liminal if π(A) is equal to the set of compact operators on the Hilbert space Hπ for every irreducible representation π. The algebra A is called NGCR if it has no nonzero CCR ideals.

An AF -algebra is a inductive limit of sequences of finite-dimensional C * -algebras. The algebra A is residually finite-dimensional if it has a separating family of finitedimensional representations.

Using [START_REF] Brenken | K-Groups of solenoidal algebras I[END_REF], we can show that the K-groups do not give a complete classification even in this algebraic case.

On some representations of U a for algebraic a

We will concentrate on a = 1 algebraic and follow the construction of finite dimensional representations [START_REF] Svensson | On the commutant of C(X) in C * -crossed products by Z and their representations[END_REF][START_REF] Yamashita | Circle correspondence C * -algebras[END_REF]: Let z q ∈ Per q (S a ) be a q-periodic point of α. Let ρ zq : C(S a ) → M q (C) be a representation of C(S a ) defined by

ρ zq (f ) := Diag f (z q ), • • • , f α q-1 (z q ) ∈ M q (C)
and for x ∈ T, let u x,zq := 0 x 1 q-1 0 ∈ M q (C). It is a unitary which satisfies the covariance relation u *

x,zq ρ zq (f )

u x,zq = ρ zq (f • α) thus π x,zq := ρ zq ⋊ u x,zq is a representation of U a on M q (C). Again, π x := ⊕ ∞ q=1 ⊕ zq∈Perq(Sa) is a representation of U a . So, for a dense family { x l } ∞ l=1 in T,
π := ⊕ ∞ l=1 π x l is a faithful representation of U a such that π(U a ) ⊂ ⊕ ∞ l=1 ⊕ ∞ q=1 ⊕ z∈Perq(Sa) M q (C).
The representation π x,χ where χ = z q ∈ B a = S a , can be extended to a representation π χ on the Hilbert space

H χ = L 2 (T) ⊗ C q by π χ = T π x,χ dx. ( 10
)
Denote by {e

(q)
s } s=1,...q the canonical basis of C q and by e n : θ → e inθ , for n ∈ Z, the basis of L 2 (T) ≃ ℓ 2 (Z). Then let, for f ∈ C * (B a ) = C( B a ):

π χ (f )(e n ⊗ e (q) s ) := f • α s-1 (χ) e n ⊗ e (q) s ,
U (e n ⊗ e (q) s ) := e n ⊗ e (q) s for 1 ≤ s < q, e n+1 ⊗ e (q) 1 for s = q, where U is the generator of Z. This representation is constructed from the representation of G a = B a ⋊ α Z given by π χ (b)(e n ⊗e

(q) s ) = χ•α s-1 (b) e n ⊗e (q) s
for any b ∈ B a (the generator U of the action of Z is the same).

Another natural representation to consider is the representation of U a obtained from the left regular representation of G a .

In [START_REF] Lim | A direct integral decomposition of the wavelet representation[END_REF] and more systematically in [START_REF] Dutkay | A duality approach to representations of Baumslag-Solitar groups[END_REF], the induced representations à la Mackey of G a for algebraic a have been investigated. The main results are the following.

For any χ ∈ B a , let the space of functions ϕ : B a ⋊ α Z → C such that ϕ(b, k) = χ(b)ϕ(0, k) for any b ∈ B a and k ∈ Z be endowed with the norm ϕ 2 χ = k∈Z |ϕ(0, k)| 2 . This defines a Hilbert space denoted by H Ind χ . The induced representation of G a on H Ind χ is given by (π Ind χ (g)ϕ)(h) = ϕ(hg) for any g, h ∈ G a .

This representation is unitarily equivalent to the following π ′ χ [29, Theorem 4.2]: the Hilbert space is ℓ 2 (Z) and for any ξ = (ξ k ) k∈Z , one takes

(π ′ χ (b)ξ) k := χ • α k (b)ξ k and the generator of Z is (U ξ) k = ξ k+1 . As a representation of U a , one has (π ′ χ (f )ξ) k = f • α k (χ)ξ k for any f ∈ C * (B a ).
Theorem 3.6. Assume a = 1 is algebraic. (i) There is a natural bijection between the set of orbits of α in S a and the set of all equivalence classes of induced representations of U a = C * (G a ). This bijection is realized by χ → π Ind χ . (ii) The representation π Ind χ is irreducible if and only if χ is aperiodic. (iii) The commutant of π Ind χ for a q-periodic point χ is the commutative algebra C(T).

(iv) The right regular representation R of

U a = C * (G a ) is unitarily equivalent to the representation ⊕ Ba π Ind χ dµ(χ).
For a q-periodic χ, the representation π Ind χ is reducible. Explicitly one has:

Proposition 3.7. If χ is q-periodic, π Ind χ is unitarily equivalent to the representation π χ on H χ , so that its continuous decomposition into irreducible finite dimensional representations on C q is realized by [START_REF] Brenken | The local product structure of expansive automorphisms of solenoids and their associated C * -algebras[END_REF] along T.

This proposition states that, while the finite dimensional representations of U a are not obtained as induced representations, they are nevertheless reductions of induced representations. The right regular representation R contains the infinite dimensional irreducible induced representations which are only accessible using aperiodic points. The representations R and π χ are not quasi-equivalent: this difference will play a crucial role in the construction of different spectral triples, see Remark 5.4.

While the π Ind χ 's yield a von Neumann factor of type I, R gives a type II 1 factor because of the integral. So the group G a is non-type I.

The particular case a = m ∈ N *

According to Lemma 3.1, the case a = m ∈ N * also covers the case a = 1/m. We do insist on this κ-deformed space since the algebra is then generated by two unitaries related by one relation (see [START_REF] Brenken | A family of dilation crossed product algebras[END_REF] below) in the spirit of the noncommutative two-torus: G m = BS(1, m) is the Baumslag-Solitar group which is generated by two elements and one-relator while, when a and a -1 are not integers, G a is not a finitely presented group (still with two generators). This simplifies the computations of section 3.2. Moreover, the results described now rely more on some properties of the Baumslag-Solitar group than on the dynamical system. Thus, these results (which for the most are already valid and exposed for generic values of a) are presented independently. These structures appear also naturally in wavelet theory, which could benefit from our analysis.

The algebra

U V U -1 = V m . ( 11
)
This universal C * -algebra U m is denoted by O(E 1,m ) in [START_REF] Katsura | A class of C * -algebras generalizing both graph algebras and homeomorphism C * -algebras IV, pure infiniteness[END_REF], and also O m,1 (T) in [START_REF] Yamashita | Circle correspondence C * -algebras[END_REF] (where only m ∈ N * is considered.) These algebras are topological graph C * -algebras which can be seen as transformation group C * -algebras on solenoid groups as already noticed in [START_REF] Brenken | The local product structure of expansive automorphisms of solenoids and their associated C * -algebras[END_REF][START_REF] Brenken | A family of dilation crossed product algebras[END_REF][START_REF] Jørgensen | Ruelle operators: Functions which are harmonic with respect to a transfer operator[END_REF]. They have been used in wavelets and coding theory [START_REF] Dutkay | Orthogonal dilations of Parseval wavelets[END_REF][START_REF] Dutkay | Unitary representations of wavelet groups and encoding of iterated function systems in solenoids[END_REF][START_REF] Dutkay | A duality approach to representations of Baumslag-Solitar groups[END_REF].

Relation [START_REF] Brenken | A family of dilation crossed product algebras[END_REF] also appeared in the Baumslag-Solitar group BS(1, m) introduced in [START_REF] Baumslag | Some two-generator one-relator non-Hopfian groups[END_REF] as the group generated by u, v with a one-relator:

BS(1, m) := u, v | uvu -1 = v m .
This group plays a role in combinatorial and geometric group theory. It is a finitely generated, meta-abelian, residually finite, Hopfian, torsion-free, amenable (solvable non-nilpotent) group. It has infinite conjugacy classes, a uniformly exponential growth (for m = 1) but is not Gromov hyperbolic [START_REF] De La Harpe | Topics in Geometric Group Theory[END_REF]. Note that BS(1, 1) is the free abelian group on two generators and BS(1, -1) is the Klein bottle group. As for the BS(1, m) groups, within the algebras U m , we remark that U 1 and U -1 play a particular role: U 1 = C(T 2 ) and U -1 ⊃ C(T 2 ) will not be considered here since we need a = m > 0.

For m ≥ 2, a solenoid appears as in section 3.2, as well as a crossed product structure, a fact that we recall now in this particular context.

Assume 2 ≤ m ∈ N and let the subring of Q generated over Z by

1 m B m = B 1/m := Z[ 1 m ] := l∈N m -l Z ⊂ Q.
It is the additive subgroup of Q which is an inductive limit of the rankone groups m -l Z, for l = 0, 1, 2, . . . and B m has a natural automorphism α defined by α(b) := mb. Note that the abelian group B m is not finitely generated. When m → ∞, BS(1, m) → Z ≀ Z (in the space of marked groups on two generators) [START_REF] Stalder | Convergence of Baumslag-Solitar groups[END_REF]. This group also appears when m = e -ω 0 /κ is replaced by a transcendental number a ∈ R * + as seen in section 3.

B m can be identified with the subgroup of the affine group Aff 1 (Q) generated by the dilatation u : x → mx and the translation v : x → x+ 1. It is the subgroup normally generated in BS(1, m) by v, u -1 vu, u -2 vu 2 , ... . The Baumslag-Solitar group BS(1, m) is then isomorphic to the crossed product BS(1, m) ≃ B m ⋊ α Z so that one has the group extension

1 → Z[ 1 m ] → B(1, m) → Z → 1.
Using this crossed product decomposition, the group BS(1, m) has the following explicit law:

(b, l)(b ′ , l ′ ) = (b+α l (b ′ ), l+l ′ ) for l, l ′ ∈ Z and b, b ′ ∈ B m .
It is of course generated by the elements u := (0, 1) and

f b := (b, 0) with b ∈ B m . Thus for j, l ∈ Z, n ∈ N, uf b u -1 = f α(b) , and if f α -n j := (α -n j, 0) = u -n f j u n , then (( 1 m ) n j, l) = f ( 1 m ) n j u l . (12)
BS(1, m) is a subgroup of the "ax + b" group (endowed with the law (b, a)(b ′ , a ′ ) := (b + ab ′ , aa ′ )) and can be viewed as the following subgroup of two-by-two matrices { m l b 0 1

: l ∈ Z, b ∈ B m }. BS(1, m) ≃ B(1, m ′ ) is equivalent to m = m ′ [49].
Let B m be the Pontryagin dual of B m endowed with the discrete topology. It is isomorphic to the solenoid

S m = S 1/m ≃ { (z k ) ∞ k=0 ∈ ∞ i=0 T : z m k+1 = z k , ∀k ∈ N 0 } (13)
using

z k := χ ( 1 m ) k for any χ ∈ B m .
The group S m is compact connected and abelian. Notice that (see section 3.2)

S m ≃ { (z k ) ∞ k=-∞ ∈ T Z : z m k+1 = z k , k ∈ Z } defining z -k := z mk 0 for k > 0. The embedding ι : θ ∈ R → χ θ ∈ S m where χ θ (b) := e i2πθb ∈ T for b ∈ B m identifies S m as the Bohr compactification b Bm R of R.
S m is endowed with a natural group automorphism α given by

α(z 0 , z 1 , z 2 , . . . ) = (z m 0 , z 0 , z 1 , . . . ) α -1 (z 0 , z 1 , z 2 , . . . ) = (z 1 , z 2 , . . . ).
All q-periodic points in S m are of the following form: if z 0 is a solution of z m q -1 = 1, then (z 0 , z m q-1 0 , . . . , z m 0 , z 0 , . . . ) ∈ S m ; so there are only finitely many periodic points, namely c q (m) = m q -1 such points.

The C * -algebra C(S m ) ≃ C * (B m ) is precisely the algebra of almost periodic functions on R, with frequencies in B m and the isomorphism is the map

f → f • ι. Thus U m = C * (BS(1, m)) ≃ C * (B m ) ⋊ α Z ≃ C(S m ) ⋊ α Z.
The unitary element U of Definition 4.1 is precisely the generator of the action

α of Z on C * (B m ) while V is one of the generators { U -ℓ V U ℓ : ℓ ∈ Z } of the abelian algebra C * (B m ). As a continuous function on S m , U -ℓ V U ℓ is the function (z k ) ∞ k=0 → z ℓ and in particular, V : (z k ) ∞ k=0 → z 0 . The subgroup { z := (z k ) ∞
k=0 ∈ S m : α q (z) = z for some q ∈ N * } of periodic points is dense in S m and α is ergodic on S m for m ≥ 2 as previously seen [11, Proposition 1].

The representations

The knowledge of * -representations of U m is essential in the context of spectral triples (see Definition 5.1). According to [START_REF] Brown | C * -algebras and Finite Dimensional Approximations[END_REF], any unitary representation of BS(1, m) is given by a unitary operator U and a family of unitaries T k , k ∈ Z, with the constraint U T k U -1 = T mk , so there is a bijection between the * -representations of U m on some Hilbert space H and the corresponding unitary representations of BS (1, m). This is rephrased usefully in the following lemma [START_REF] Jørgensen | Ruelle operators: Functions which are harmonic with respect to a transfer operator[END_REF]: Lemma 4.2. The algebra U m is the C * -algebra generated by L ∞ (T) and a unitary symbol Ũ with commutation relations, where e n (z

) := z n Ũ f Ũ -1 = f • e m , ∀f ∈ L ∞ (T). (14)
U m contains a family of abelian subalgebras

A n := Ũ -n L ∞ (T) Ũ n for n ∈ N, which is increasing since Ũ -n f Ũ n = Ũ -(n+1) f • e m Ũ (n+1
) . If we choose the Hilbert space H := L 2 (R), the scaling and shift operators give rise to a representation π of U m on H by π(U ) :

ψ(x) → 1 √ m ψ( x m ) and π(V ) : ψ(x) → ψ(x -1).
The Haar measure ν on B m gives rise to a faithful trace on U m and, since there are many finite dimensional representations of U m (see proof of Theorem 3.2), there are many traces on it.

With

H := L 2 (S m , ν) ≃ ℓ 2 (B m ) and U : ψ ∈ H → ψ• α ∈ H, C(S m
) acts on H by left multiplication, we get a covariant representation for (S m , U ) of the dynamical system (C(S m ), α, Z), so a representation of U m on H. Since α is ergodic, this representation is irreducible and faithful [START_REF] Brenken | A family of dilation crossed product algebras[END_REF]Theorem 1].

If we choose H := ℓ 2 (Z), for each θ ∈ R we get an induced representation of U m by π θ (U )ψ(k) := ψ(k -1) and π θ (V )ψ(k) := χ θ (m -k ) ψ(k), k ∈ Z.

On the existence of spectral triples

Since we want to construct spectral triples on U a , it is worthwhile to know the heat decay of G a = B a ⋊ α Z via a random walk on the Cayley graph of G a with generators S = { x, x -1 , y, y -1 } where x = (0, 1) and y = (1, 0), and with a constant weight and standard Laplacian.

The decay of the heat kernel p t , with t ∈ N, has been computed on the diagonal in [51, Theorem 1.1], [START_REF] Coulhon | A geometric approach to ondiagonal heat kernel lower bounds on groups[END_REF]Theorem 5.2]: when t → ∞, we get p 2t ∼ e -t 1/3 (log t) 2/3 if a is transcendental while p 2t ∼ e -t 1/3 if a is algebraic. This is related to the fact that G a has exponential volume growth.

However, for a finite dimensional connected non-compact Lie group, the behaviour of the heat kernel p t depends on t ∈ R * + and can diverge for the short time behaviour when t → 0. Let us explain how this point is related to the dimension: In noncommutative geometry, a (regular simple) spectral triple (A, H, D) has a (spectral) dimension which is given by max{ n ∈ N : n is a pole of ζ D : s ∈ C → Tr(|D| -s ) } (here D is assumed invertible). In particular, when M is a n-dimensional compact Riemannian spin manifold, and A = C ∞ (M ), H = L 2 (S) where S is the spinor bundle and D is the canonical Dirac operator, the spectral dimension coincides with n. Via the Wodzicki residue, an integral X := Res s=0 Tr(X|D| -s ) is defined on (classical) pseudodifferential operators X acting on smooth sections of S. For instance, |D| -n coincides (up to a universal constant) with the Dixmier trace Tr Dix (|D|

-n ) = lim N →∞ log(N ) -1 N k=1 |λ k | -n
where the λ k are the singular values of D. The dimension of M appears in Tr(e -tD 2 ) ∼ N ≥0 1 t (n-N)/2 a N (D) when t → 0. In particular, when M = R n with Lebesgue measure and D 2 = -△ is the standard Laplacian (noncompactness is not a problem), the heat kernel is p t (x, x) = 1 (4πt) n/2 for all x ∈ M (see [START_REF] Connes | Noncommutative geometry[END_REF][START_REF] Connes | Noncommutative Geometry, Quantum Fields and Motives[END_REF][START_REF] Gracía-Bondía | Elements of Noncommutative Geometry[END_REF]). As a consequence, Tr(|D| -(n+ǫ) ) < ∞ for all ǫ > 0.

We will see that, depending on the chosen representation of U a , such an n does not always exist, meaning that the "dimension is infinite". Definition 5.1. A spectral triple (also called unbounded Fredholm module) (A, H, D) is given by a unital C * -algebra A with a faithful representation π on a Hilbert space H and an unbounded self-adjoint operator D such that -the set

A = { a ∈ A : [D, π(a)] is bounded } is norm dense in A, -(1 + D 2 ) -1 ∈ J where J is a symmetrically-normed ideal of the compact operators K(H) on H. The triple is p-summable if J = L p (H) for 1 ≤ p < ∞ which means Tr (1 + D 2 ) -p/2 < ∞. It is p + -summable if J = L p+ (H).
It is finitely summable if it is p-summable for some p.

It is θ-summable if there exists t 0 ≥ 0 such that Tr e -tD 2 < ∞ for all t > t 0 (thus J = K(H)).

Note that A is a * -subalgebra of A and p-summability implies θ-summability.

Connes proved in [START_REF] Connes | Compact metric spaces, Fredholm modules, and hyperfiniteness[END_REF] that, for an infinite, discrete, non-amenable group G, there exist no finitely summable spectral triples on A = C * red (G). How-ever, in this case, there always exist θ-summable spectral triples on A (even with D > 0). Using a computable obstruction to the existence of quasicentral approximate units relative to J for A, Voiculescu was able to derive, for solvable groups with exponential growth, the non-existence result for unbounded (generalized) Fredholm modules using the Macaev ideal J = L ∞,1 (H) [START_REF] Voiculescu | On the existence of quasicentral approximate units relative to normed ideals. Part I[END_REF]. We use these results in the following:

Theorem 5.2. Non-existence of finite-summable spectral triples.

Let A = U a , G a = B a ⋊ α Z and A = C[G a ].
(i) There is no finitely summable spectral triple π(A), H π , D when the representation π is quasiequivalent to the left regular one.

(ii) There exist θ-summable spectral triples π(A), H π , D with t 0 = 0 where the representation π is quasiequivalent to the left regular one.

Despite the previous result, we add a few explicit examples of spectral triples using the fact that the algebra U a is residually finite. Clearly, these triples deal with a restrictive part of the geometry of the κ-deformation based on U a , namely the dynamical system which is behind. The residually finite property is seen via the periodic points of this dynamics. In case (i), [D, π(x)] is not necessarily zero but the summability is not controlled while for case (ii), the condition [D, π(x)] = 0 enables us to control summability. In a sense, case (iii) is a mixed situation requiring that a be algebraic. In that situation, we have an explicit representation π so that formulae for Dirac operators can be proposed.

Remark 5.4.

There is no contradiction between Theorems 5.2 and 5.3 since the faithful quasidiagonal representation (or residually finite one) π of U a used above to construct D is not quasiequivalent to the left regular one: actually, as already mentioned, the von Neumann algebra generated by π(U a ) is a II 1 factor when π is the left regular representation, while it is of type I when π is the quasidiagonal or residually finite one [26, 5.4.3.].

A more direct way to confirm that the representation π used in the proof of point (iii) of Theorems 5.3 is not quasiequivalent to the left regular representation (or to the right regular representation which is unitarily equivalent to the left one) is to notice that π is the direct integral π = ⊕ Per π χ dµ(χ) of the finite dimensional representations π χ defined in [START_REF] Brenken | The local product structure of expansive automorphisms of solenoids and their associated C * -algebras[END_REF]. As such, this representation is strictly contained in the right regular representation R as can be checked using (iv) of Theorem 3.6. The part of R which is not in π is given by the induced infinite dimensional irreducible representations constructed on aperiodic χ's.

As noticed in [START_REF] Skalski | A note on spectral triples and quasidiagonality[END_REF], if (A, H π , D) is a spectral triple with [D, π(x)] = 0, ∀x ∈ A, then A is a residually finite C * -algebra.

Remark 5.5. Theorem 5.2 says that the 2-dimensional κ-deformed space reflected by the algebra U a with κ = -ω 0 log -1 (a) is in fact "infinite dimensional" as a metric noncommutative space. Theorem 5.3 is a tentative to restore a metric. For instance, the distances on the state space S(U a ) generated by Connes are infinite in case (ii) of Theorem 5.3, while in case (iii) some states can be at finite distances.

Remark 5.6. The operator D given in Theorem 5.3 (iii) is not directly related to the group structure of G a but rather connected to the underlying dynamical system associated to the algebraic nature of a: it depends explicitly of the isomorphism-invariant { c q (a) : q ∈ N * }.

Conclusion

We have shown that κ-Minkowski space defined by (1) can be reduced to a compact or discrete version. Depending on κ, or on a defined in [START_REF] Baumslag | Some two-generator one-relator non-Hopfian groups[END_REF], this involves discrete amenable groups G a , in particular the well-known Baumslag-Solitar ones. The associated C * -algebras U a can be viewed as a deformation of the two-torus. They are different when a varies within the rational numbers (of zero Lebesgue measure) because of the structure of the underlying dynamical system. For transcendental values of a, which are dense in R + and of full Lebesgue measure, all these algebras are isomorphic to each other.

Due to the exponential growth of G a , we have proved that the algebras U a are not only quasidiagonal but also residually finite dimensional. They admit different spectral triples: the ones which are quasi-equivalent to the left regular representation and are never p-summable but only θ-summable, i.e. they are of "infinite metric dimension". This situation reminds us of the passage from non-relativistic to relativistic quantum mechanics: in quantum field theory, the θ-summability (and not the p-summability) naturally occurs due to the behaviour of Tr(e -tH ) (when t → 0) where H is the Hamiltonian (or D 2 ), see for instance [START_REF] Carpi | Spectral triples and the super-Virasoro algebra[END_REF].

The other faithful representations can generate fancy spectral triples which can have arbitrary summability (or "dimension") depending on the algebraic properties of the real parameter a, but are in fact degenerate to some extent. It is also not entirely clear what the topological content of these unbounded Fredholm modules is (i.e. whether they correspond to nontrivial elements of K-homology). The dimension of these spectral triples is unrelated to the number of coordinates defining the κ-deformed Minkowski spaces.

The nonexistence theorem, though powerful, does not preclude the possible existence of a genuine, non-degenerate, nontrivial spectral geometry on the κ-deformation spaces presented here, they only restrict the possible algebra representations that could be used in the construction.

This shows how delicate the notion of spectral or metric dimensions of κ-Minkowski space is, and how subtle its analysis through noncommutative geometry.

Lemma 2 . 1 .

 21 -Solitar group BS(1, m) = Z[ 1 m ] ⋊ αm Z as shown in section 4 where we get B m = B 1/m = Z[1/m]. A broad family of noncommutative spaces appears: In two dimensions, there exists a unital subalgebra C * (H κ,m ) of the κ-deformation algebra, associated to the subgroup H

  using the canonical faithful conditional expectation C(S a ) ⋊ α Z → C(S a ) and the density of periodic points, one show that

Definition 4 . 1 .

 41 Let U m be the universal C * -algebra labelled by m ∈ Z * (restricted to N * later) and generated by two unitaries U and V such that

Theorem 5 . 3 .

 53 Existence of finite-summable spectral triples. LetA = U a and A = C[G a ].(i) There exist spectral triples π(A), H π , D which are compact, i.e. [D, π(x)] is compact for all x ∈ A.(ii) There exist spectral triples π(A), H π , D such that [D, π(x)] = 0, ∀x ∈ U a , and with arbitrary summability.(iii) When a is algebraic, there exist spectral triples π(A), H π , D such that [D, π(v)] = 0, [D, π(u)] = 0 and with an arbitrary summability p ≥ 2.

  ' formula d(ω, ω ′ ) := sup{ |ω(a) -ω ′ (a)| : a ∈ A, ||[D, a]|| ≤ 1 }, ω, ω ′ ∈ S(U a )

  1 , AF-embeddable, non-simple, residually finite dimensional C * -algebra and its generated von Neumann algebra for the left regular representation is a type II 1 -factor.

Acknowledgments

We thank Alain Connes, Michael Puschnigg, Adam Skalski and Shinji Yamashita for helpful discussions or correspondence. B. I. and T. S. acknowledge the warm hospitality of the Institute of Physics at the Jagiellonian University in Krakow where this work was started under the Transfer of Knowledge Program "Geometry in Mathematical Physics".