Strict deformation quantization and local spin interactions

We define a strict deformation quantization which is compatible with any Hamiltonian with local spin interaction (e.g. the Heisenberg Hamiltonian) for a spin chain. This is a generalization of previous results known for mean-field theories. The main idea is to study the asymptotic properties of a suitably defined algebra of sequences invariant under the group generated by a cyclic permutation. Our point of view is similar to the one adopted by Landsman, Moretti and van de Ven, who considered a strict deformation quantization for the case of mean-field theories. However, the methods for a local spin interaction are considerably more involved, due to the presence of a strictly smaller symmetry group.


Introduction
In this paper we provide a rigorous C * -algebraic framework for the study of the semi-classical properties of any Hamiltonian with local spin interaction for a spin chain.This covers, for example, the Heisenberg Hamiltonian.This result is achieved by means of a suitable strict deformation quantization, whose construction is the main result of this paper -cf.Theorem 24.
Strict deformation quantization originates with Berezin [3] and Bayen et al. [1,2] and it is based on the idea of "deforming" a given commutative Poisson algebra representing a classical system into a given non-commutative algebra modelling the associated quantized system.In Rieffel's approach [17] the deformed algebras are C * -algebras.
Notably, the "classical-to-quantum" interpretation of a strict deformation quantization is not the unique point of view which can be taken.In Landsman's approach [10,11] the starting point of a strict deformation quantization is often taken to be a continuous field of C * -algebras.The latter models an increasingly larger sequence of quantum physical systems, whose limit defines a macroscopic classical theory.The advantage of this point of view is that it leads to a rigorous notion of the classical limit of quantum theories [11].This in turn yields a mathematically sound description of several physically interesting emergent phenomena, e.g.symmetry breaking [14,18,19].This paper is considering this "micro-to-macro" point of view on strict deformation quantization.
From a technical point of view a strict deformation quantization is defined by the following data: 1.A commutative Poisson C * -algebra A ∞ , namely a commutative C * -algebra A ∞ equipped with a Poisson structure { , } : A ∞ × A ∞ → A ∞ defined on a dense * -subalgebra A ∞ ⊆ A ∞ -cf.Section 3.3.

2.
A continuous bundle of C * -algebras [7] N ∈N A N , where N = N ∪ {∞}; (Thorough the whole paper we will stick to the case of continuous bundle of C * -algebras over N, see [11] for the generic case.) 3. A family of linear maps, called quantization maps, ) for all a ∈ A ∞ .Moreover, the assignment defines a continuous section of the bundle N ∈N A N .
(b) For all a, a ′ ∈ A ∞ it holds lim The algebra A ∞ represents the classical (macroscopic) observables of the physical system.Likewise, the fibers A N , N ∈ N, of the bundle N ∈N A N recollect the quantum observables of the (increasingly larger) quantum system.A relevant example is the strict deformation quantization described in [12,16] for the C *algebra [B] ∞ π of (equivalence classes of) symmetric sequences -cf.Section 2 for further details.In this scenario the role of the commutative C * -algebra A ∞ is played by [B] ∞ π = C(S(B)) -here B = M κ (C), κ ∈ N, while S(B) denotes the states space over B. The continuous bundle of C * -algebras N ∈N B N π is such that, for N ∈ N, B N π ⊆ B N is the N -th symmetric tensor product of B.
From a physical point of view the ensuing quantization maps Q N are of particular interest as they relate to mean-field theories like the Curie-Weiss model [9, §2], for which the interaction between N spin sites is described by where σ 3 , σ 1 ∈ M 2 (C) denote the Pauli's matrices while I ∈ M 2 (C) is the identity matrix and I j := I ⊗j .Here J ∈ R represents the strength of the spin interaction whereas h ∈ R models an external magnetic field acting on the system.As observed in [12] one may recognize that where h cw ∈ C(S(B)) while R N ∈ B N π is such that R N N = O(1/N ).The physical interpretation is that C(S(B)) is the algebra containing macroscopic observables, i.e. observables of an infinite quantum system describing classical thermodynamics as a limit of quantum statistical mechanics.This has furthermore led to a significant contribution in the study of the classical limit of ground states [12,13,14,18].More precisely, in such works a mathematically rigorous description of the limit of ground states ω N of H cw,N in the regime of large particles N → ∞ is given.In particular, a classical counterpart ω ∞ (i.e. a probability measure) of the quantum ground state ω N ∈ S(B N π ) is constructed with the property that ω ∞ (a) := lim N →∞ ω N (Q N (a)) for all a ∈ C(S(B)).Additionally, this algebraic approach has revealed the existence of several physical emergent phenomena, see [19] for an overview.These results are consistent with the point of view of [11] -which is also the one considered in this paper-for which a quantum theory is pre-existing and the classical limit is computed in a second step, not vice versa.
As characteristic for mean-field models, the Curie-Weiss Hamiltonian describes the energy of a system of N spin sites under the assumption that the interaction is non-local, namely that every spin site interacts with all other spin site.This leads to interesting results, but it is ultimately an approximation as one would rather expect each spin site to interact with finitely many neighbouring spin sites.An exemplary model based on such a local interaction is the celebrated quantum Heisenberg Hamiltonian (for a spin chain) [6, §6.2] where J pq is the symmetric matrix describing the spin interaction while h p are the components of an external magnetic field -here for j = N − 1 the contribution in the first sum reads J pq σ q ⊗ I N −2 ⊗ σ p .For this model the interaction is restricted to two neighbouring sites.
Similarly to what happens with mean-field models one may wonder whether there exists a strict deformation quantization of a suitable C * -algebra such that The purpose of this paper is to prove that this is in fact the case, cf.Theorem 24.In [15] a different (though similar in spirit) point of view is taken, and a strict deformation is considered such that H He,N = Q κ (h He,N ) + O(1/κ) where the semi-classical parameter κ corresponds to the increasing dimension of the single site algebra B = M κ (C) for a fixed number N of lattice sites.In contrast, this paper deals with an arbitrary but fixed dimension κ ∈ N considering instead the increasing number N of spin sites as the semi-classical parameter.
Our result is particularly relevant because it provides an excellent basis for studying the classical limit of local quantum spin systems.Similarly to the case of mean-field theories [12,14,18,19], one may now consider a rigorous C * -algebraic formalization of the limit of ground states or Gibbs states [8,9].The latter can be used for the study of spontaneous symmetry breaking and phase transitions in realistic models such as the Heisenberg model.
From a technical point of view, the methods of this paper profit of those of [12,16] for mean-field models.Nevertheless the results obtained therein do not apply straight away to our case.As a matter of fact the strict deformation quantization for mean-field models (like the Curie-Weiss Hamiltonian) profits of: 1.A large symmetry group, that is, mean-field models are symmetric under the permutation group S N of all N spin sites.This leads to a high symmetry property which can be exploited in several steps of the construction, cf.[12,16].

A fairly explicit description of the classical algebra [B] ∞ π = C(S(B)). One may define [B] ∞
π in terms of equivalence classes of "symmetric sequences" -cf.Remark 3-but the description in terms of C(S(B)) simplifies the discussion, e.g. it allows to identify a Poisson structure in a rather direct way.
Contrary to this case, local quantum spin Hamiltonians (e.g. the Heisenberg model defined in (3)) are invariant under the strictly smaller subgroup generated by a fixed cyclic permutation of N objects.This spoils the possibility of applying the arguments of [12,16].The latter have to be reconsidered to take into account the smaller symmetry group.Moreover, the classical algebra [B] ∞ γ for such models does not have a "simple" explicit description.As a matter of fact, [B] ∞ γ is defined as the C * -algebra generated by (equivalence classes of) "γ-sequences" -cf.Definition 5. Nevertheless it is still possible to prove all properties of [B] ∞ γ relevant for the discussion of its strict deformation quantization.
The paper is structured as follows.In Section 2 we introduce the notion of "γ-sequences" -cf.Definition 2-and discuss their properties.The main result in this section is the proof that the C * -algebra [B] ∞ γ generated by (equivalence classes of) γ-sequences is a commutative C * -algebra.The latter will play the role of the classical algebra A ∞ for which we will present a strict deformation quantization.
In Section 3 we state and prove the main theorem of this paper, which provides a strict deformation quantization of the commutative C * -algebra [B] ∞ γ .To this avail, Section 3.1 is devoted to prove Proposition 12 which provides the continuous bundle of C * -algebra [B] γ needed in the formulation of Theorem 24.The main technical hurdle of this section is to prove that, given a γ-sequence (a N ) N , the sequence of the norms ( a N N ) N is convergent.While this is straightforward for symmetric sequences (i.e.those used when dealing with mean-field models) for γ-sequences this is non-trivial and has to be discussed carefully.Sections 3.2-3.3discuss further relevant properties of (equivalence classes of) γ-sequences as well as the Poisson structure on the C * -algebra [B] ∞ γ .Eventually Theorem 24 is proved by recollecting all results from the previous sections.
For the sake of clarity the following theorem recollects in a concise fashion the content of the main Theorem 24 together with the other relevant results of the paper.Finally, there exists a family of quantization maps The algebra of γ-sequences

Definition of γ-sequences
In this section we will introduce γ-sequences and discuss their properties.
To fix some notations, let κ ∈ N and set B := M κ (C).For the sake of simplicity we shall denote by B N := B ⊗N , where N ∈ N, with the convention that B 0 = C.The state space over B N will be denoted by S(B N ): Given η ∈ S(B) we set η N := η ⊗N ∈ S(B N ).Whenever needed we will denote N = N ∪ {∞}.
Following [12] we denote by I, b 1 , . . ., b κ 2 −1 , I ∈ B being the identity matrix, a basis of B (as a R-vector space) abiding by the requirements where c m jℓ denotes the structure constants of su(κ).In the particular case κ = 2 we may choose b j = σ j /2 while c m jℓ = ε jℓs δ sm , ε ijk being the Levi-Civita symbol.We will denote by B the vector space generated by {b j } κ 2 −1 j=1 .The latter corresponds to the ker τ , being τ : B → C the normalized trace defined by τ (a) := tr(a)/κ.
We then consider the linear operator (left-shift operator) γ N : B N → B N uniquely defined by continuous and linear extension of the following map defined on elementary tensors a (1) , . . ., a (N ) ∈ B . ( The operator γ N is an algebra endomorphism, moreover, γ N N = Id B , Id B : B → B being the identity operator.We denote by γ N : B N → B N the averaged γ N operator, defined by Clearly the C * -subalgebra of B N made by γ N -invariant elements.
Through this paper we will mostly consider sequences (a where I ∈ B denotes the identity of B and I N = I ⊗N .♦ Remark 3: γ is a linear operator with operator norm smaller than 1.This implies that, (γ M N a M ) N is bounded with where M denotes the norm on B M .
(ii) It is worth comparing our construction with the one presented in the literature [11,12,16], based on symmetric sequences.We stress that the latter are exploited to deal with the Curie-Weiss Hamiltonian -or more generally with mean-field theories [11, §10]-which prescribe a non-local interaction between spin sites.On the other hand we are interested in models compatible with Hamiltonian describing a local interaction between spin sites -e.g. the Heisenberg Hamiltonian, cf.Remark 4. To describe the non-local interaction algebraically one considers the symmetrization operator S N : B N → B N defined by continuous and linear extension of where S N is the set of permutation of N objects [11,16].Considering the C * -subalgebra B N π := S N B N ⊂ B N one then defines a symmetric-sequence (shortly, π-sequence) to be a sequence (a N ) N such that there exists M ∈ N and a M ∈ B M π fulfilling One immediately sees the relation with Definition 2: Actually a γ-sequence is defined in a way similar to π-sequences but averaging over a strictly smaller subgroup of S N .In fact γ-sequences and π-sequences share many similar properties, although π-sequences are generally speaking better behaved.♦

Asymptotic properties of γ-sequences
In what follows we will be mainly interested in the asymptotic behaviour as N → ∞ of the sequences under investigations.For this reason, following [16], we introduce the ∼-equivalence relation For a given sequence (a N ) N we will denoted by [a N ] N := [(a N ) N ] the corresponding equivalence class with respect to (9).The ∼-equivalence relation ( 9) has a nice interplay with the full C * -product N ∈N B N defined by As it is well-known [4] N ∈N B N is a C * -algebra with respect to sup norm (a N ) N ∞ := sup is a closed two-sided ideal in N ∈N B N and thus we may consider the quotient which is nothing but the space of ∼-equivalence classes [a N ] N for bounded sequences (a Remark 4: (i) Since both γ-and π-sequences are bounded -cf.Remark 3-i-they lead to well-defined elements This is in fact possible, but we postpone this discussion to Section 3.2 where we will prove that, for a given equivalence class

and only if the canonical representative vanishes -cf. Proposition 18. (ii) With reference to Equation (3) we have (considering
showing the relation between γ-sequences and the Heisenberg Hamiltonian.Similarly, as discussed in [12], Equation (2) leads to At this stage it is worth observing that γ-sequences model an arbitrary Hamiltonian with local spin interaction.We say that

a (translation invariant) Hamiltonian with local spin interaction if and only if
were σ p (i)σ q (j) is a short notation for The parameter ℓ ∈ N determines the number of spin sites which interact with a fixed spin site i -e.g. for the Heisenberg Hamiltonian ℓ = 1.The strength of the interaction and of the external magnetic field is determined by J pq , h p .Notice that the latter do not depend on the spin site: This entails that we are considering translation invariant local spin interactions.
Any Hamiltonian H N as per Equation ( 14) leads to a γ-sequences as per Definition (2).Indeed, we have (iii) The ∼-equivalence relation (9) provides a first example showing the different behaviour of γ-sequences with respect to π-sequences.To this avail, let a M ∈ B M π and let us consider the π-sequence (π M N a M ) N .By direct inspection one immediately sees that, for all

The same property does not apply for γ-sequences, but it holds only asymptotically. Indeed for a
It then follows that where we used the γ-invariance while This shows that, although As we shall see, naively speaking most the results obtained for π-sequences holds true also for γ-sequences but only asymptotically -in the sense of relation (9).♦

The algebra generated by γ-sequences
In what follows we will consider the * -algebra As we will see, the latter algebra enjoys remarkable properties, in particular, it can be completed to a commutative Definition 5: Let Ḃ∞ γ be the * -algebra generated by γ-sequences -cf.Definition 2. We denote by where

algebra generated by equivalence classes of γ-sequences. To wit, an equivalence class [a
where denotes "total weighted symmetrization" over the factor a We will prove that, for N large enough, (18) where (9) this implies Equation (16).
We proceed by induction over ℓ ∈ N.For ℓ = 1 the right-hand side of Equation ( 18) reduces to γ M N a M + R N so that we may choose R N = 0.For ℓ = 2 we find, for N large enough (say, where in the last line we used the symmetry of j 1 , j 2 as well as the γ N -invariance of the whole term.The remainder R N coincides with where Roughly speaking, we removed the values of j for which a M 1 and a M 2 have "overlapping positions".This happens in M 1 + M 2 cases, whose fraction vanishes as N → ∞.This proves Equation ( 18) for ℓ = 2. Proceeding by induction on ℓ, we now assume that Equation ( 18) holds for all ℓ ′ < ℓ and prove it for ℓ.To this avail we consider, for where ).Thus, we focus on We now proceed as in the case ℓ = 2 by considering only those values j ℓ for which the position of a M ℓ "overlaps" with the ones of I j 1 , . . ., I j ℓ and not with those of a M 1 , . . ., a M ℓ−1 .Notice that, in focusing only on these j ℓ 's we are neglecting a contribution where R ′′ N N = O(1/N ) while the sum over the h p is empty if j p < M ℓ -notice that at least one of these sums is not empty if N is large enough.Notice that each of the ℓ − 1 sets of ℓ indexes is such that its elements sum to N − |M | ℓ .Considering now the summation over j 1 , . . ., j ℓ−1 and using the γ N -invariance each subset of indexes provides the same contribution.We are lead to where in the last line we used that for all ς ∈ S ℓ there are ℓ permutations which are equivalent to ς up to a cyclic permutation.Indeed, for any permutation of a M 1 , . . ., a M ℓ we may use the γ N -invariance to write the corresponding contribution fixing the position of the factor a M ℓ .This boils down to a permutation of a M 1 , . . ., a M ℓ−1 which is repeated ℓ times.By induction this proves Equation ( 18) for all ℓ ∈ N and thus Equation ( 16).
Remark 7: (i) The appearance of the total weighted symmetrization (17) ensures that, when a M j = I M j for all j ∈ {1, . . ., ℓ}, the right-hand side of Equation ( 16) coincides with [I N ] N .This is related to the fact that 1 (ii) A closer inspection to the remainder term R N of Equation (18) reveals that for all ℓ ′ , M ′ 1 , . . ., M ′ ℓ ′ ∈ N, and a M ∈ B M .Roughly speaking, the reason for this is due to the estimate R N N = O(1/N ) together with the fact that both (R N ) N and (γ ) N are sequences with "an increasing number of identities".In more details, Equation (18) where

The first contribution is estimated by
while for the second contribution we have where C M 1 ,...,M ℓ ′ > 0 does not depend on N .In the last inequality we used the estimate R N N = O(1/N ) and that, on account of the structure of R N -cf. the proof of Proposition 6-and of , the sum over p is non-vanishing for finitely many values, say L, where L is N -independent.This proves Equation (20).[16,Def. II.1].Moreover, as shown in [11,12,16], for any

(iii) In complete analogy with Definition 5 one may introduce the C
which shows that also [B] ∞ π is a commutative C * -algebra.In fact, the product of π-sequences is (asymptotically as N → ∞) a π-sequence.Additionally, one may prove that the system {B N π } N ∈N , {π M N } N ≥M is a generalized inductive system [4,5].This streamlines the identification of a bundle of C * -algebras N ∈N B N π out of which a strict deformation quantization can be constructed [12,16].
The situation for γ-sequences is slightly different.Indeed, Equation (16) shows that the product of γ-sequences is not a γ-sequence, even if its ∼-equivalence class is considered.Nevertheless, Equation (16) shows that the product of equivalence classes of γ-sequences is commutative.As we shall see in Section 3.1 this will be enough to identify a continuous bundle of C * -algebras N ∈N B N γ out of which a strict deformation quantization is obtained.Finally it is worth observing that, for all a ∈ B, one finds γ 1 N a = π 1 N a so that, given the results of [12,16] (iv) By standard Gelfand duality [4, § II.2] we find which identifies a sequence {ϕ M } M ∈N of states with ϕ M ∈ S(B M ).These states are "asymptotically equivalent" because of Equation (15).Indeed considering γ M N : [16,Lem. IV.4].This identification may also be seen as a consequence of the prominent quantum De Finetti Theorem [11,Thm. 8.9].As shown in [12], S(B) is a stratified manifold which carries a Poisson structure.

♦ 3 Strict deformation quantization of γ-sequences
The goal of this section is to construct a strict deformation quantization of the commutative algebra [B] ∞ γ .To this avail in Section 3.1 we will identity a suitable continuous bundle of C *algebras [B] γ by means of a standard construction [10,11].In Section 3.2 we will introduce the notion of "canonical representative" for an element (ii) For all α = (α N ) N ∈N ∈ C(N) and a ∈ A there exists a αa ∈ A with the property that A continuous section of A is an element a ∈ N ∈N A N such that there exists a ′ ∈ A fulfilling a N = ψ N (a ′ ) for all N ∈ N. Clearly A can be identified with its continuous sections, therefore, in the forthcoming discussion we shall always regard a ∈ A as an element a ∈ N ∈N A N .For this reason from now on we will implicitly identify ψ N , N ∈ N, with the projection N ∈N A N → A N .
Remark 8: In applications, it is often difficult to identity a continuous bundle of C * -algebras by assigning the triple A, {A N } N ∈N , {ψ N } N ∈N directly.However, a useful result -cf.[10,Prop. 1.2.3], [11,Prop. C.124]-shows that it is in fact sufficient to identify a dense set of (a posteriori) continuous sections of A. Actually, let A ⊆ N ∈N A N be such that: A is a * -algebra; 3. For all a ∈ A, it holds lim Then defining A by one may prove that A is a continuous bundle of C * -algebras over N [10,11].In fact, A is the smallest continuous bundle of C * -algebras over N which contains A. ♦ We will now prove that N ∈N [B] N γ identifies a continuous bundle of C * -algebras where To this avail we will identity a subset A ⊂ N ∈N [B] N γ fulfilling conditions 1-2-3 of Remark 8. From a technical point of view, condition 3 will require to prove that, for all [a N ] N ∈ [B] ∞ γ , the sequence ( a N N ) N has a limit as N → ∞: This is proved in Proposition 10.To this avail, the following Lemma comes in handy.
Lemma 9: Let (α N ) N ∈N be a bounded sequence of real numbers such that Proof.Let (α N j ) j∈N be a convergent subsequence of (α N ) N ∈N .Then for all K ≥ N 0 we find Since this holds true for all convergent subsequences we conclude that lim inf Considering again a convergent subsequence (α K j ) j∈N of (α N ) N ∈N the above inequality implies Since this holds for all convergent subsequences we conclude that lim sup α N exists and it is finite.
In particular we have To begin with we prove the claim for where η p ℓ ∈ S(B) for all ℓ = 1, . . ., K. We then consider where N = r + qK, q ∈ N and r ∈ {0, . . ., K − 1} while τ ∈ S(B) is normalized the trace state.We consider By direct inspection we have that, for all ℓ ∈ {0, . . ., K − 1}, The same contribution arises if j ≤ N − r − M = qK − M and j = ℓ mod K.The number of such j's is roughly where the O(1) contribution is bounded both in N and in K.The net result is where we observed that the sum over j ∈ terms each of which is bounded by a M M .Overall we find where C > 0 depends on M but not on N or K.The arbitrariness of ω K ∈ S(B K ) leads to Thus, Lemma 9 applies to the sequence ( γ M N a M N ) N proving that lim We now consider an arbitrary element [a N ] N .Although our proof works for an arbitrary element of [a N ] N ∈ [ Ḃ] ∞ γ , for the sake of (notational) simplicity we restrict ourself to the case where the sum over k 1 , k 2 is finite.To prove that ( a N N ) N has a limit as N → ∞ we rely on Equation ( 16) together with an argument similar in spirit to the one used for the case of a single γ-sequence.In fact, Proposition 6 implies that so that we may restrict to the first factor on the right-hand side.
As for the case of single γ-sequence let N, K ∈ N be such that N ≥ K ≥ max } where the maximum is taken over all pairs k 1 , k 2 ∈ N appearing in the sum defining [a N ] N .We consider ω K ∈ S(B K ) and, as above, we set where N = r + qK, q ∈ N and r ∈ {0, . . ., K − 1} while is an arbitrary finite convex decomposition of ω K into product states.We then evaluate To this avail we fix k 1 , k 2 and split the sum over j 2 in two cases: (a) Let consider the sum for 0 we find, with the same argument used for a single γ-sequence, The number of j 2 's such that 0 ≤ j 2 ≤ N − M (k 1 ) − M (k 2 ) − r and j 2 = ℓ mod K is roughly q = N/K + O(1), therefore, summing over such j 2 's leads to a contribution of It remains to discuss the sum over 0 where we observed that, for each of the Loosely speaking these contributions arise when j 2 is such that "a (k 2 ) overlaps with the (translated) position of a (k 1 ) ".This does not allow to reconstruct ω K , therefore, these cases are estimated by Recollecting our result we have where C 1 , C 2 > 0 do not depend neither on N nor on K.The arbitrariness of ω K ∈ S(B K ) leads to Thus, Lemma 9 applies and the limit lim exists and it is finite.
Then, for all ε > 0 there exists For N, M ≥ max{N ε , N ′ ε } we then have Remark 11: The result of Proposition 10 applies also for π-sequences.For this latter case the proof streamlines because γ is twofold.On the one hand, for γ-sequences γ M N a M N is not decreasing, although it fulfils a similar properties asymptotically.On the other hand, the product of γ-sequences is not a γ-sequence, even when equivalence classes are considered.This requires a different strategy to ensure the existence of the limit lim The following proposition proves the existence of the continuous bundle of C * -algebras of interest.
γ be the subset defined by Then [ Ḃ] γ fulfils conditions 1-2-3 and thus it leads to a continuous bundle of C * -algebras (25)

♦
Proof.We will prove conditions 1-2-3 of Remark 8.The space [ Ḃ] γ is a * -algebra, therefore, condition 2 is fulfilled.Concerning condition 1, we have to prove that where the existence of lim N →∞ a N N is ensured by Proposition 10.

Canonical representative of [a
To proceed further in the construction of the deformation quantization of [B] ∞ γ we have to discuss the possibility of identifying a canonical representative of an element [a N ] N ∈ [ Ḃ] ∞ γ .This is required for both endowing [B] ∞ γ with a Poisson structure as well as for defining the quantization maps Theorem 24.To begin with, we address the following problem: A positive answer in this direction would imply that, given an equivalence class [γ M N a M ] N , one is able to determine uniquely the γ-sequence (γ M N a M ) N .Unfortunately, the answer to this question is negative because although the associated sequences are not the same.Indeed This counterexample suggests to focus on the C * -subalgebra B M where B = ker τ , τ ∈ S(B) being the trace state -cf.Section 2. In fact, therein the situation is slightly better as shown by the following Lemma.
) and let τ ∈ S(B) be the normalized trace state τ (a) := tr(a)/κ.Let N ≥ M + 1, q ∈ N and r ∈ {0, . . ., M } be such that N = r + q(M + 1).We consider the state By direct inspection we find that Indeed, for j = 0 the resulting contribution is ω M ( a M )/N .The same contribution appears when j = 0 mod M +1: Since j ∈ {0, . . ., N −1} this happens q times, moreover, q = N/(M +1)+O(1) leading to the right-hand side of Equation (26).Whenever j = 0 mod M + 1 the resulting contribution is 0, on account of the fact that τ vanishes on B. Equation (26) implies that, for all ω M ∈ S(B M ), The arbitrariness of ω M leads to a M M = 0, that is, a M = 0.
Thus, although the equivalence class [γ M N a M ] N does not identify a unique sequence (γ M N a M ) N , Lemma 13 suggests that a (a posteriori unique) canonical representative may be extracted by working with the " B-irreducible components" of the γ-sequence.To this avail, we introduce the notion of B-irreducibility.This identifies those elements in B M which cannot be written as For the sake of completeness, Appendix A provides a complete characterization of B M irr for all M ∈ N.
♦ The notion of B-irreducible elements leads to a proper definition of "canonical representative" for a γ-sequence -cf.Definition 16.Indeed, let consider an arbitrary a M ∈ B M .By considering a basis I, b 1 , . . ., b κ 2 −1 of B fulfilling (4) we may decompose a M as where a 0 , c k 1 ...k ℓ j 1 ...j ℓ+1 ∈ C and the sum over k 1 , . . ., k ℓ is finite.At this stage we observe that We stress that some of the a ′ j 's may vanish in the process.However, it is important to observe that, moving from a M to a ′ M , the γ-sequence (and thus its equivalence class) does not change.Notice that, if we replace a M with I K ⊗ a M or a M ⊗ I K , the B-irreducible elements {a ′ j } M j=0 do not change.

Summing up, every equivalence class
∞ γ has a unique canonical representative obtained by decomposing a M into its B-irreducible components.
We shall now discuss the notion of canonical representative for a generic element [a N ] N ∈ [ Ḃ] ∞ γ .Proposition 6 and Remark 17-ii lead to the following definition. where for all k j , while the sum over ℓ, k 1 , . . ., k ℓ is finite and is called the canonical representative of [a N ] N .♦ Similarly to Proposition 18 we have the following result, showing that the canonical representative introduced in Definition 19 is unique.
where the sum over ℓ, k 1 , . . ., k ℓ ∈ N is finite and for all k j .♦ Proof.For the sake of clarity, we will discuss the proof for ℓ ≤ 2. This simplifies the construction without affecting the validity of the argument.We thus consider the sequence where the sum over k 1 , k 2 is finite and a (k) ∈ B M (k) irr for all k.Notice that, whenever M (k 1 ) = 0 or M (k 2 ) = 0 the corresponding contribution reduces to a single γ-sequence up to a remainder O(1/N ).We have to prove that lim N →∞ a N N = 0 implies a N = 0 for all N ∈ N.
We now analyse (34) with the help of the following parameters: Roughly speaking M is the maximal degree of the a (k) 's appearing in (34).The parameter M 1 ≤ M is the minimal "bigger length" among all pairs (k 1 , k 2 ) appearing in (34).Notice that M 1 > 0 on account of the hypothesis (M (k 1 ), M (k 2 )) = (0, 0).Finally M 2 ≤ M 1 is the minimal length of the a (k) 's appearing when considering only those pairs (k 1 , k 2 ) for which max{M where where in the second line we observed that This follows from the fact that, if On account of the symmetry in k 1 , k 2 in the sum, we may assume that M (k 1 ) = M 1 and M (k 2 ) = M 2 for all pairs (k 1 , k 2 ).Equation (44) reduces to Then Equation (45) leads to This shows that Equation (45) implies We now observe that Equation ( 46) is equivalent to (47) Indeed, by direct inspection Equation (46) implies that and thus it implies Equation (47).Conversely, if Equation (47) holds true then evaluation on the state where ℓ 1 , ℓ 2 are such that At this stage we may either argue that this is in contradiction with the definition of M 1 , M 2 -unless a N = 0-because min Alternatively we may consider the remaining contribution to a N and argue again as above identifying new values M 1 , M 2 , M .In either case we have a N = 0 for all N ∈ N as claimed.
Remark 21: The notion of canonical representative applies also for symmetric sequences.In- π , one obtain the following unique decomposition:

With this decomposition at hand the canonical representative of
This point of view is equivalent to the one adopted in [12].♦

The Poisson structure of [B] ∞ γ
In this section we will endow [B] ∞ γ with a Poisson structure defined on [ Ḃ] ∞ γ .Eventually we will discuss the deformation quantization of [B] γ .
We recall that a Poisson structure over a C * -algebra A is given by a bilinear map { , } : A 0 × A 0 → A 0 defined on a dense * -subalgebra A 0 ⊂ A which fulfils: for all a, a ′ , a ′′ ∈ A 0 .
Indeed, this is due to the identity together with the fact that the result holds true for ℓ = ℓ ′ = 1.Moreover, we have ) so that it remains to discuss the term where the latter estimate is due to Remark 7-ii.Overall we have shown that where R ′′ N N = O(1/N ) and by direct inspection fulfils Equation (20).This implies in particular that so that { , } γ is well-defined.
(50) − (51) By proceeding in a completely analogous way one also proves conditions (50)-(51).Indeed, considering without loss of generality The first contribution fulfils (51).With an argument similar in spirit to Remark 7-ii, the second contribution can be estimated by Finally N [a can N , R N ] N = O(1/N ) because of Remark 7-ii so that with R ′′′ N N = O(1/N ).This proves condition 51 for { , } γ .By proceeding in a similar fashion we also have where R ′′′ N N = O(1/N ) while the first contribution fulfils (50).This proves condition (50) for { , } γ .
Remark 23: The proof of Proposition 22 shows that, if a M ∈ B M irr and a M ′ ∈ B M ′ irr then where a M +2M ′ ∈ B M +2M ′ is not B-irreducible in general.♦ At last, we can finally state and prove the main theorem of this paper.With reference to Section 1 we have We will now prove conditions 3a-3b-3c.Notice that no other term from Φ( a M ) provides a non-vanishing contribution because τ ( B) = {0}.The arbitrariness of η 1 , η 2 ∈ S(B) leads to ω 2 ( a 2 ) = 0 for all ω 2 ∈ S(B 2 ) and thus a 2 = 0.

3 . 1
. Definitions 16-19.Eventually, in Section 3.3 will show that [B] ∞ γ carries a Poisson structure and we will prove Theorem 24, which provides the strict deformation quantization of [B] ∞ γ .The continuous bundle of C * -algebras [B] γ associated with [B] ∞ γ In this section we will define a continuous bundle of C * -algebras [B] γ over N = N ∪ {∞} whose fibers are [B] N γ := B N γ for N ∈ N and [B] ∞ γ , defined as per Definition 5, for N = ∞.To this avail we briefly recall the main definitions and results we need -cf.[11, App.C.19], [4, §IV.1.6].We denote by C(N) the space of C-valued sequences (α N ) N ∈N such that α ∞ := lim N →∞ α N ∈ C exists.A continuous bundle (or field) of C * -algebras over N is a triple A, {A N } N ∈N , {ψ N } N ∈N made by C * -algebras A, A N , N ∈ N, and surjective homomorphisms ψ N : A → A N such that: (i) The norm of A is given by a A := sup N ∈N ψ N (a) A N ;

= 0
For example if a M = a 0 I M then the canonical representative is the constant sequence a N = a 0 I N , N ∈ N, which coincides with (γ M N a M ) N only for N ≥ M .(ii) On account of the previous discussion we observe that the algebra generated by γ-sequences of the form (γ M N a M ) N for a M ∈ B M irr , M ∈ N, exhaust the whole space Ḃ∞ γ .♦ The following proposition shows that the canonical representative introduced in Definition 16 is unique.Proposition 18: Let M ∈ N and a j ∈ B j irr for all j = 0, . . ., M .Then lim ⇐⇒ a 0 = 0 , . . ., a M = 0 .(31) ♦ Proof.The proof is similar to the one of Lemma 13.By direct inspection we have a j = |a 0 | .

Theorem 24 :
Let [B] γ ⊂ N ∈N [B] N γ be the continuous bundle of C * -algebras defined as per Proposition 12.For K∈ N let Q K : [ Ḃ] ∞ γ → [B]K γ be the linear map defined byQ K ([a N ] N ) := a can K K ∈ N [a N ] N K = ∞ (53)where (a can N ) N is the canonical representative of [a N ] N as per Definitions 16-19.Then the family of maps{Q N } N ∈N defines a strict deformation quantization of [B] ∞ γ .♦ Proof.Notice that Q N is well-defined for all N ∈ N on account of the uniqueness of the canonical representative -cf.Propositions 18-20.
for the canonical representative we only have equality of equivalence classes, i.e. [γ M N a M and ω M 2 ∈ S(B M 2 ) are arbitrary states.This implies Equation (45).By comparison with (34) we conclude that Equation (47) is nothing but the sum of the terms in a N whose pairs k 1

)
where (a can N ) N denotes the canonical representative of [a N ] N -cf.Definitions 16-19.Then { , } γ is a Poisson structure on [B] ∞ γ .♦ At this stage we observe that