0.0.1 Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras: Definitions and Relationships to Poisson and C*-algebras

Firstly, a specific algebra consists of a vector space E over a ground field (typically ℝ or ℂ) equipped with a bilinear and distributive multiplication ∘ . Note that E is not necessarily commutative or associative.

A Jordan algebra (over ℝ), is an algebra over ℝ for which:

S ∘ T = T ∘ S , S ∘ (T ∘ S²) = (S ∘ T) ∘ S²,

for all elements S,T of the algebra.

It is worthwhile noting now that in the algebraic theory of Jordan algebras, an important role is played by the Jordan triple product {STW} as defined by:

which is linear in each factor and for which {STW} = {WTS} . Certain examples entail setting {STW} = {STW + WTS} .

A Jordan Lie algebra is a real vector space 𝔄_ℝ together with a Jordan product ∘ and Poisson bracket

{ , }, satisfying :

• for all S,T ∈ 𝔄_ℝ,

  S ∘ T = T ∘ S {S,T} = −{T,S}

• the Leibniz rule holds
  

  for all S,T,W ∈ 𝔄_ℝ, along with

• the Jacobi identity :

  

• for some ℏ² ∈ ℝ, there is the associator identity  :

  

0.0.2 Poisson algebra

By a Poisson algebra we mean a Jordan algebra in which ∘ is associative. The usual algebraic types of morphisms automorphism, isomorphism, etc.) apply to Jordan-Lie (Poisson) algebras (see Landsman, 2003).

Consider the classical configuration space Q = ℝ³ of a moving particle whose phase space is the cotangent bundle T^∗ℝ³ℝ⁶, and for which the space of (classical) observables is taken to be the real vector space of smooth functions

 . The usual pointwise multiplication of functions fg defines a bilinear map on 𝔄_ℝ⁰, which is seen to be commutative and associative. Further, the Poisson bracket on functions

which can be easily seen to satisfy the Liebniz rule above. The axioms above then set the stage of passage to quantum mechanical systems which the parameter k² suggests.

0.0.3 C*–algebras (C*–A), JLB and JBW Algebras

An involution on a complex algebra 𝔄 is a real–linear map TT^∗, such that for all S,T ∈ 𝔄 and λ ∈ ℂ, we have also

A *–algebra is said to be a complex associative algebra together with an involution ∗ .

A C*–algebra is a simultaneously a *–algebra and a Banach space 𝔄, satisfying for all S,T ∈ 𝔄 :

∥S ∘ T∥ ≤∥S∥ ∥T∥ , ∥T^∗T∥² = ∥T∥² .

One can easily see that ∥A^∗∥ = ∥A∥ . By the above axioms a C*–algebra is a special case of a Banach algebra where the latter requires the above norm property but not the involution (*) property. Given Banach spaces E,F the space ℒ(E,F) of (bounded) linear operators from E to F forms a Banach space, where for E = F, the space ℒ(E) = ℒ(E,E) is a Banach algebra with respect to the norm ∥T∥ := sup{∥Tu∥ : u ∈ E , ∥u∥ = 1} .

In quantum field theory one may start with a Hilbert space H, and consider the Banach algebra of bounded linear operators ℒ(H) which given to be closed under the usual algebraic operations and taking adjoints, forms a ∗–algebra of bounded operators, where the adjoint operation functions as the involution, and for T ∈ℒ(H) we have :

∥T∥ := sup{(Tu,Tu) : u ∈ H , (u,u) = 1} , and ∥Tu∥² = (Tu,Tu) = (u,T^∗Tu) ≤∥T^∗T∥ ∥u∥² .

By a morphism between C*–algebras 𝔄,𝔅 we mean a linear map ϕ : 𝔄→𝔅, such that for all S,T ∈ 𝔄, the following hold :

where a bijective morphism is said to be an isomorphism (in which case it is then an isometry). A fundamental relation is that any norm-closed ∗–algebra 𝒜 in ℒ(H) is a C*–algebra, and conversely, any C*–algebra is isomorphic to a norm–closed ∗–algebra in ℒ(H) for some Hilbert space H .

For a C*–algebra 𝔄, we say that T ∈ 𝔄 is self–adjoint if T = T^∗ . Accordingly, the self–adjoint part 𝔄^sa of 𝔄 is a real vector space since we can decompose T ∈ 𝔄^sa as  :

A commutative C*–algebra is one for which the associative multiplication is commutative. Given a commutative C*–algebra 𝔄, we have 𝔄C(Y ), the algebra of continuous functions on a compact Hausdorff space Y  .

A Jordan–Banach algebra (a JB–algebra for short) is both a real Jordan algebra and a Banach space, where for all S,T ∈ 𝔄_ℝ, we have

∥S ∘ T∥ ≤∥S∥ ∥T∥ , ∥T∥² ≤∥S² + T²∥ .

A JLB–algebra is a JB–algebra 𝔄_ℝ together with a Poisson bracket for which it becomes a Jordan–Lie algebra for some ℏ² ≥ 0 . Such JLB–algebras often constitute the real part of several widely studied complex associative algebras.

For the purpose of quantization, there are fundamental relations between 𝔄^sa, JLB and Poisson algebras.

Conversely, given a JLB–algebra 𝔄_ℝ with k² ≥ 0, its complexification 𝔄 is a C^∗-algebra under the operations :

ST := S ∘ T −k ×_k , (S + ιT)^∗ := S − ιT.

For further details see Landsman (2003) (Thm. 1.1.9).

A JB–algebra which is monotone complete and admits a separating set of normal sets is called a JBW-algebra. These appeared in the work of von Neumann who developed a (orthomodular) lattice theory of projections on ℒ(H) on which to study quantum logic. BW-algebras have the following property: whereas 𝔄^sa is a J(L)B–algebra, the self adjoint part of a von Neumann algebra is a JBW–algebra.

A JC–algebra is a norm closed real linear subspace of ℒ(H)^sa which is closed under the bilinear product S ∘ T = (ST + TS) (non–commutative and nonassociative). Since any norm closed Jordan subalgebra of ℒ(H)^sa is a JB–algebra, it is natural to specify the exact relationship between JB and JC–algebras, at least in finite dimensions. In order to do this, one introduces the ‘exceptional’ algebra H₃(𝕆), the algebra of 3 × 3 Hermitian matrices with values in the octonians 𝕆 . Then a finite dimensional JB–algebra is a JC–algebra if and only if it does not contain H₃(𝕆) as a (direct) summand [1].

The above definitions and constructions follow the approach of Alfsen and Schultz (2003), and also reported earlier by Landsman (1998).

References