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Reduction of Lie-Jordan Banach algebras and quantum states

A theory of reduction of Lie-Jordan Banach algebras with respect to either a Jordan ideal or a Lie-Jordan subalgebra is presented. This theory is compared with the standard reduction of C*-algebras of observables of a quantum system in the presence of quantum constraints. It is shown that the later corresponds to the particular instance of the reduction of Lie-Jordan Banach algebras with respect to a Lie-Jordan subalgebra as described in this paper. The space of states of the reduced Lie-Jordan Banach algebras is described in terms of equivalence classes of extensions to the full algebra and their GNS representations are characterized in the same way. A few simple examples are discussed that illustrates some of the main results.


Introduction
In this paper we present a theory of reduction of Lie-Jordan algebras that can be used as an alternative to deal with symmetries and local constraints in quantum physics and quantum field theories. R. Haag's algebraic approach to quantum systems [1] has had a profound influence in both the foundations and applications of quantum physics. The background for that approach is to consider a quantum system described by a C * -algebra A of observables, its quantum states ω are normalized positive complex functionals on it. The state space S becomes a convex w * -compact topological space and pure states are its extremal points. Moreover, given a state ω, the GNS construction allows to represent the C * -algebra A on a cyclic complex Hilbert space H ω .
A geometrical approach to Quantum Mechanics [4], [5] has been developed in the last twenty years (see also Cirelli and Lanzavecchia [2] and Ashtekar and Schilling [3]). After this, it has become obvious that the geometrical picture of Quantum Mechanics could be useful to gain insight into such properties of quantum systems like integrability [6], the intrinsic nature of different measures of entanglement [7], [8], etc. (see for instance [9] and references therein).
The geometrical description of dynamical systems provides a natural setting to describe systems with symmetry and/or constraints. If the system carries a symplectic or Poisson structure several procedures were introduced along the years to cope with the different cases, like Marsden-Weinstein reduction, symplectic reduction, Poisson reduction, reduction of contact structures, etc. Again, it was soon realized that the algebraic approach to reduction provided a convenient setting to deal with reduction of systems [10] [11].
Whenever constraints are imposed on a quantum system or symmetries are present, both dynamical or gauge, some reduction on the state space must be considered either because not all states are physical and/or because families of states are equivalent. In the standard approach to Quantum Mechanics, constraints are imposed on the system by selecting subspaces determined by the quantum operators corresponding to the constraints of the theory, Dirac states, and equivalence of quantum states was dealt with by using the representation theory of the corresponding group of symmetries. Many difficulties and ambiguities emerge when implementing this analysis as different classes of singular Lagrangian systems are considered or other singularities were found (quantization of singular level sets of momentum maps for instance).
Taking as a departing point for the analysis the algebraic approach to Quantum Mechanics and Quantum Field Theory the problem of reduction of the quantum system becomes the problem of reducing the C * -algebra of observables. Such programme was succesfully developed for some gauge theories and the proposed reduction of the algebra of observables was called T-reduction (see Grundling et al [12] and references therein) and more generally for quantum local constraints it was discussed at [13]. There is however a point that should be raised here, and it is that the state space S of the quantum system does not determine univocally the C * -algebra structure of the algebra of observables but only its Jordan Banach real algebra [14] [15]. In fact the real (or self-adjoint) part of a C * -algebra A, because of a theorem by Kadison is isometrically isomorphic to the space of all bounded affine functions on the state space [16]. Then the question of when a given Jordan Banach algebra is the real part of a C * -algebra raises. A. Connes on one side [17] and Alfsen and Shultz on the other [18], gave different answers to this question. In the present context the characterization obtained by Alfsen and Schultz in terms of the so-called dynamical correspondence on a Jordan Banach algebra amounts to state that the relevant structure to discuss properties of the state space of a quantum system is that of Lie-Jordan Banach algebra [19] [20]. In fact the topological properties of the state space are completely captured by the Jordan Banach-algebra structure and the Lie algebra structure allows to construct the C * -algebra setting for them, their GNS representations, etc.
Thus, as stated above, we will address the problem of the reduction of Quantum systems as the reduction of Lie-Jordan Banach algebras. Inspired by the general reduction theory of Jordan algebras (briefly reviewed in section 4) and combining it with the theory of reduction of (noncommutative) Poisson algebras, we will obtain a general reduction theorem for Lie-Jordan Banach algebras with respect to Jordan ideals, Sect. 4, and with respect to Lie-Jordan subalgebras, Sect. 5. These results will be compared to the theory of reduction of C * -algebra with respect to quantum constraints that will be shown to be a particular instance of the theory discussed in this paper, Sect. 6. An explicit description of the states of the reduced Lie-Jordan Banach algebras will be obtained and the corresponding GNS representations will be discussed in Sects. 7 and 8.

Quantum states and Lie-Jordan Banach algebras
Quantum states are positive linear functionals ω on a C * -algebra A of observables, thus ω(a * a) ≥ 0 ∀a ∈ A. We will assume in what follows that the C * -algebra is unital and states are normalized, i.e. ω(1) = 1, where 1 denotes the unit element of A. We will denote by A sa the real (or self-adjoint) part of A, i.e.
A sa = {a ∈ A | a * = a}, and we will denote by · the induced norm. The self-adjoint part of A inherits the structure of a real Lie-Jordan algebra. It is a Jordan algebra with the non-associative commutative Jordan product and by setting a • a = a 2 , the Jordan product • satisfies which is the usual replacement for associativity for Jordan algebras. The Lie product is obtained by taking a scaled commutator: The skew-symmetric bilinear form [·, ·] λ equips A sa with a Lie algebra structure since it verifies the Jacobi identity These two operations are compatible in the sense that the following Leibniz identity is satisfied: or, in other words, the linear map D a (·) ≡ [a, · ] λ is a derivation of the Jordan product •.
A vector space V with two operations • and [·, ·] satisfying the properties of the operations above, eqs. (1)-(5), is called a (pre-) Lie-Jordan algebra. The complete definition of a Lie-Jordan algebra requires that the associator of the structure product is related to the Lie bracket by: κ being a positive real number. When using the Lie bracket (3), we obtain: A Lie-Jordan algebra will be called unital if it contains the unit element with respect to the Jordan product.
Moreover if a Lie-Jordan algebra L carries a complete norm · such that ∀a, b ∈ L: and [·, ·] λ is continous, then L is a Lie-Jordan Banach algebra (LJB-algebra for short). A LJB-algebra (L, •, [·, ·] λ ) is in particular a JB-algebra (a Jordan-Banach algebra) when considered with the Jordan product alone. If L is a LJB-algebra, by taking combinations of the two products we can define an associative product on the complexification L C . Specifically, we define: Associativity follows from the Leibniz property (5) and the Jacobi identity (4). Such associative algebra equipped with the norm ||a + ib|| 2 = ||a|| 2 + ||b|| 2 becomes a C * -algebra whose real part is precisely L. When we consider a Lie-Jordan algebra to consist of all smooth functions on a classical carrier space, equipped with the pointwise product such that ρ(1) = 1 and ρ(a 2 ) ≥ 0 ∀a ∈ L. The state space of a JBalgebra is convex and compact with respect to the w * -topology. It was an important problem to determine which state spaces of Jordan algebras are the state spaces of a C * -algebra. In fact it follows from results of Kadison [16] that if S(A) denotes the state space of a C * -algebra A, then A sa is naturally isomorphic to the space of affine w * -continous real functions on S(A), hence the state space S(A) as a topological space is determined just by the self-adjoint part of A. The previous problem was solved by Alfsen and Shultz [18] and A. Connes [17] independently. We will describe the reconstruction of the C * -algebra in the following section.

Dynamical correspondence and LJB-algebras
Following [18] we will define a derivation of a JB-algebra L by focusing first only on the order structure, ignoring the algebraic multiplicative aspect. All the proofs contained in [18] will be omitted. First note that a unital JB-algebra L is a complete order unit space with respect to the positive cone We denote the Jordan multiplier determined by an element b ∈ L by It can be shown that Jordan multipliers δ b are order derivations ∀ b ∈ L.
Definition 2. An order derivation δ on a unital JB-algebra L is selfadjoint if there exists a ∈ L such that δ = δ a and is skew-adjoint if δ(1) = 0.
Again, it can be shown that if δ is an order derivation, then δ is skew if and only if δ is a Jordan derivation, i.e., is a derivation with respect to the Jordan product, that is

Definition 3. A dynamical correspondence on a unital JB-algebra L is a linear map
It follows immediately from the definitions that: Definition 4. Let L be a unital JB-algebra. A C * -product compatible with L is an associative product on the complex linear space L ⊕ iL which induces the given Jordan product on L and makes L ⊕ iL into a C * -algebra with involution (a + ib) * = a − ib and whose norm induces the given norm in the JB-algebra L.
Note that if a JB-algebra L is the self-adjoint part of a C * -algebra A, then there is a natural product and a norm induced in L ⊕ iL by using the representation A = a+ib with A ∈ A, a, b ∈ L. Such product and norm organizes L ⊕ iL into a C * -algebra. It follows that a JBalgebra is the self-adjoint part of a C * -algebra if and only if there exists a C * -product compatible with L on L⊕iL. We can now state the main result in [18], relating the existence of a dynamical correspondence on a JB-algebra to the existence of a compatible C * -product. Theorem 1. A unital JB-algebra L is Jordan isomorphic to the selfadjoint part of a C * -algebra if and only if there exists a dynamical correspondence on L. Each dynamical correspondence ψ on L determines a unique associative C * -product compatible with L defined as and each C * -product compatible with L arises in this way from a unique dynamical correspondence ψ on L.
We will now show that the existence of a dynamical correspondence on L is equivalent to the existence of a Lie product organizing L into a LJB-algebra. First we need the following Lemmas: Then there exists an associative bilinear product on L × L defined as Proof: Bilinearity of the product follows directly from the bilinearity of the Jordan and Lie products. We have to prove the associativity, i.e.: The l.h.s. of the previous equation leads to: and the r.h.s.: where we have used (4), (5) and (6).
Note that the Jordan and Lie products can be also expressed in terms of the associative product as: Proof: We have to prove that By Hadamard's formula, the l.h.s. of the previous equation is: By using formula (14), the r.h.s. of (16) becomes:  Proof: From Definition 1, we have to prove that e t[a,·] L (L + ) ⊂ L + ∀ a ∈ L and ∀ t ∈ R. Since e t[a,·] L is a Jordan automorphism (Lemma 2), we have: Proof: First assume that L is a LJB-algebra. From Definition 1 we which is an easy computation once the Jordan and Lie products are expressed as in (15) and (14). From the antisymmetry of the Lie product it is also true that ψ a a = [a, a] L = 0 ∀ a ∈ L. Hence the linear map a −→ [a, ·] L from the LJB-algebra L to the skew-order derivations on L is a dynamical correspondence. Conversely, assume L is a JB-algebra with a dynamical correspondence ψ. Then from (10) ψ a b = [a, b] L is antisymmetric. By using the expression 11 it is an easy computation to check that the Jacobi property (4) is satisfied ψ a (ψ b c) + ψ b (ψ c a) + ψ c (ψ a b) = 0. The Leibniz identity (5) follows from (8) and also the compatibility condition (6) is easy to check with a simple computation. Hence a JB-algebra with a dynamical correspondence is a LJB-algebra.

Reduction of Lie-Jordan algebras by Jordan ideals
The algebraic reduction of a Lie-Jordan algebra should start from the consideration of both products, the Jordan • and the Lie product [·, ·] L . As usual to reduce a Jordan algebra L we have to identify a Jordan ideal, say J , and then form the quotient algebra L/J that carries the structure of a Jordan algebra, however such space will not inherit a Lie bracket in general. The space of all Jordan derivations of L carries a natural structure of Lie algebra. We consider the subalgebra of those derivations which map J into itself. By using the Lie product, i.e., the dynamical correspondece, it is possible to associate with any element a in L a Jordan derivation 1 as in eq. (17)  The normalizer is also a Jordan subalgebra, in fact due to the Leibniz identity (5), It is also a Lie-Jordan subalgebra since it verifies the compatibility conditions (5) by definition of Jordan derivation and (6) because it is a subalgebra of L. It also contains the Jordan unit since it is annihilated by the Lie product. Finally, because the Lie product is continuous, we have that the normalizer N J is closed.
It is now possible to consider those derivations associated with J which preserve J , i.e., J = N J ∩ J is the subset of J such that: Proof: The subspace J = N J ∩ J is closed because both N J and J are closed sets. Because J is a Jordan ideal, given a ∈ N J , we will have a • N J ∩ J ⊂ J and because N J is a Jordan subalgebra, we will get a • N J ∩ J ⊂ N J , hence a • J ⊂ J for all a ∈ N J . Similarly, it is shown that J is a Lie algebra ideal of N J .
The closed Lie-Jordan ideal J of the LJB-algebra N J induces a canonical Lie-Jordan algebra structure in the quotient: We will denote elements of L by a.
Observe that all the reduction procedure works also for the case in which L is a (pre-) Lie-Jordan algebra (see section 2) since we did not make use of the compatibility condition (6) and the reduced algebra L will be a (pre-) Lie-Jordan algebra as well.
The quotient Lie-Jordan algebra L carries the quotient norm, where a ∈ N J is an element of the equivalence class [a] of N J with respect to the ideal J = N J ∩ J . The quotient norm provides a LJB-algebra structure to L.
Hence the reduction of the Lie-Jordan algebra L with respect to the closed Jordan ideal J is given by the short exact sequence (20) 0 → J → N J → L → 0 and L is the reduction algebra of L, that will be denoted as L//J too.

Reduction of Lie-Jordan algebras by Lie-Jordan subalgebras
The reduction procedure of the previous section is given by means of a Jordan ideal J of the given Lie-Jordan algebra L. Actually the assumption for J being an ideal is not necessary to obtain a reduced Lie-Jordan algebra. We will prove in this section that it is still possible to obtain a reduction of L by relaxing the ideal property of J and requiring it to be a Lie-Jordan subalgebra of L. In fact by using the Leibniz property (5): The subspace N J • J is also a Jordan ideal of N J , in fact by using the compatibility condition (6): and since J ⊂ N J we obtain Note that N J • J also satisfies the compatibility conditions (5) and (6) and hence it is a Lie-Jordan subalgebra.
Because of the previous Theorem, it is canonically defined a Lie-Jordan algebra structure in the quotient space and it is an unital algebra. As in the previous section if L is a LJB-algebra and J is a closed subspace, then L inherits a LJB-algebra algebra structure with respect to the quotient norm. Observe that if J is a unital Lie-Jordan subalgebra, then N J •J would be a trivial ideal of N J since they would coincide and the reduction would have no sense.

Reduction of Lie-Jordan algebas and constraints in
C * -algebras Following [12], [13] we briefly recall how to deal with quantum constraints in a C * -algebra setting. The aim of this section is to prove that the reduction procedure of C * -algebras used to analize quantum constraints, also called T-reduction, can be equivalently described by using the theory of reduction of LJB-algebras discussed above.
A quantum system with constraints is a pair (F , C) where the field algebra F is a unital C * -algebra containing the self-adjoint constraint set C, i.e. C = C * ∀C ∈ C. The constraints select the physical state space, also called Dirac states where the notation [ · ] denotes the closed linear space generated by its argument. Then we have: For any set Ω ∈ F , define as before its normalized or "weak commutant" as (21) Ω W = {F ∈ F | [F, H] ∈ Ω ∀H ∈ Ω} which corresponds roughly to Dirac's concept of "first class variables" [22]. Consider now the multiplier algebra of Ω as i.e. the largest set for which Ω is a bilateral ideal. M(Ω) is clearly an unital C * -algebra and we obtain: That is, the weak commutant of D is also the largest set for which D is a bilateral ideal and it will be denoted by O. It follows that the maximal (and unital) C * -algebra of physical observables determined by the constraints C is given by: We will now show that this procedure is equivalent to a LJB-algebra reduction by a Lie-Jordan subalgebra as discussed in section 5).
Define L and L such that F = L ⊕ iL and F = L ⊕ i L, i.e. they are the self-adjoint part of F and F respectively. From Corollary 1 it follows that L and L are unital LJB-algebras. Similarly define the LJB-algebras N J and J as the self-adjoint parts of O and D respec- Proof: Using the expressions provided by eqs. (14) and (15), the statement becomes an easy computation.
Proof: J is a LJB-subalgebra of L and hence from the previous Lemma 7 it is a Lie-Jordan ideal of N J . From the Lemma 6, it also follows that N J is its normalizer. Hence the reduced space N J /J is a LJB-algebra whose complexification is F since It follows form Cor. 1 that L = N J /J .
Thus it is evident that the C * -algebra reduction gives rise to a reduction of the LJB-algebra by a Lie-Jordan subalgebra J and, conversely, a reduction of a LJB-algebra with respect to a Lie-Jordan subalgebra J always gives rise to a C * -algebra reduction with respect to D. The two approaches are therefore completely equivalent in this sense, as the following "functorial" diagram shows: s Let us now consider the reduction of a LJB-algebra by a Jordan ideal first and then pass to the complexified C * -algebra. We will obtain in this way an "extension" of the C * -algebra reduction procedure. Let L be the initial LJB-algebra with Jordan ideal J , whose normalizer N J enables us to obtain the reduced LJB-algebra L = N J N J ∩ J . Define now O = N J ⊕ iN J and D = J ⊕ iJ . From Lemma 6, O is the normalizer of D and from Lemma 7,D ∩O is an associative bilateral ideal of O.
We obtain immediately the following: Theorem 7. With the notations above, given an ideal J of the LJB-algebra L, the reduced LJB-algebra L = L//J defines a C * -algebra F that is obtained from the field algebra F = L ⊕ iL as: Again, the following diagramme summarizes the discussion above: Note that if in addition J is a Lie-subalgebra then D ⊂ O and D would be a bilateral C * -ideal, recovering the usual reduction procedure of C * -algebras.

The space of states of a reduced LJB-algebra
The purpose of the remaining two sections is to discuss the structure of the space of states and the GNS construction of reduced states for reduced LJB-algebras with respect to the space of states of the unreduced LJB-algebra.
As it was discussed in the previous section, let A be a C * -algebra, L = A sa its real part and J a Jordan ideal of L. Let N J be the normalizer of J with respect to the Lie algebra structure of the Lie-Jordan algebra (L, •, [·, ·] L ). Then we will denote as before by L the reduced Lie-Jordan Banach algebra N J /(N J ∩ J ) and its elements by a.
Let S = S( L) be the state space of the reduced LJB-algebra L, i.e. ω ∈ S means that ω( a 2 ) ≥ 0 ∀ a ∈ L, and ω is normalized. Notice that if L is unital, then 1 ∈ N J and 1 + N J ∩ J is the unit element of L. We will denote it by 1. We have the following: Thus if ω ′ is positive on N J , ω ′ (a 2 ) ≥ 0, hence: and if ω ′ vanishes on the closed ideal N J ∩ J , then ω ′ induces a positive linear functional on L. Clearly ω ′ is normalized then the induced functional is normalized too because 1 = 1 + N J ∩ J . Conversely, if ω : L −→ R is positive and we define then ω ′ is well-defined, positive, normalized and ω ′ | N J ∩J = 0. Now we will relate positive linear functionals on N J vanishing on N J ∩ J with a class of linear functionals on L. For this purpose, we will first notice that given a positive linear functional on N J there exists an extension of it to L which is positive too. Proof: Since L is a JB-algebra, it is also a Banach space. Due to the Hanh-Banach extension theorem, there exists a continous extension ω of ω ′ , i.e. ω(A) = ω ′ (A) ∀A ∈ N J , and moreover ω = ω ′ .
From the equality of norms and the fact that ω ′ is positive we have ω = ω ′ (1), but ω is an extension of ω ′ then ω = ω(1), which implies that ω is a positive functional and satisfies all the requirements stated in the lemma.
We will denote by S((N J ∩J ) 0 ) the set of states vanishing on N J ∩J . We will consider the equivalence classes of states on S((N J ∩ J ) 0 ) (which is again a convex set) with respect to the equivalence relation induced by N 0 J . This is, the canonical projection π : (N J ∩ J ) 0 −→ (N J ∩ J ) 0 N 0 J induces an equivalence relation on S((N J ∩ J ) 0 ) whose equivalence classes are π −1 ([ω]) ∩ S(L) with [ω] ∈ (N J ∩ J ) 0 /N 0 J . We will call this equivalence relation the N 0 J -equivalence relation on S((N J ∩ J ) 0 ). Hence we have proved the following: Theorem 8. There is a one-to-one correspondence between the states ω of the reduced Lie-Jordan Banach algebra L and the N 0 J -equivalence classes of states ω ∈ S((N J ∩ J ) 0 ) ⊂ S(L).
Proof: The proof is almost trivial using the results above. Thus if ω is an state on S((N J ∩ J ) 0 ), then its restriction to N J vanishes on N J ∩ J and because of Lemma 8 it induces an state on L.
Conversely, states on L are in one-to-one correspondence with states on N J vanishing on N J ∩ L, hence because of Lemma 9 there exists an extension of any such state to a state on L. However two such extension define the same state on N J if they differ on an element of N 0 J , hence they are N 0 J -equivalent.

The GNS representation of reduced states
Finally, we will try to describe the GNS representation of a reduced state in terms of data from the unreduced LJB-algebra. Let L be, as before, the reduced LJB-algebra of L with respect to the Jordan ideal J . Denote by A = L ⊕ i L the corresponding C * -algebra and by S its state space. Let ω ∈ S be a normalized state on A. The GNS representation of A associated to the state ω, denoted by and J ω is the Gelfand ideal of ω. Let ω be a state on A that extends the state ω ′ on N C J induced by ω according to Lemmas 8 and 9. Notice that ω| N J vanishes on N J ∩ J , thus the Gelfand ideal J ω of ω contains N J ∩ J . We will have then: Theorem 9. There is a unitary equivalence between H ω and the completion of the pre-Hilbert space: