Jordan-Banach and Jordan-Lie algebras

0.0.1 Definitions of Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras

Firstly, a specific algebraMathworldPlanetmathPlanetmathPlanetmath consists of a vector spaceMathworldPlanetmath E over a ground field (typically or ) equipped with a bilinearPlanetmathPlanetmath and distributive multiplication  . Note that E is not necessarily commutativePlanetmathPlanetmathPlanetmath or associative.

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


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}=12{STW+WTS} .

A Jordan Lie algebraMathworldPlanetmath is a real vector space 𝔄 together with a Jordan product and Poisson bracket

{,}, satisfying :

  • 1.

    for all S,T𝔄, ST=TS{S,T}=-{T,S}

  • 2.

    the Leibniz rulePlanetmathPlanetmath holds

    {S,TW}={S,T}W+T{S,W} for all S,T,W𝔄, along with

  • 3.


  • 4.

    for some 2, there is the associatorMathworldPlanetmath identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath  :


0.0.2 Poisson algebra

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

Consider the classical configuration space Q=3 of a moving particle whose phase space is the cotangent bundle T*36, and for which the space of (classical) observables is taken to be the real vector space of smooth functionsMathworldPlanetmath


. The usual pointwise multiplication of functions fg defines a bilinear map on 𝔄0, 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 involutionPlanetmathPlanetmathPlanetmath on a complex algebra 𝔄 is a real–linear map TT* such that for all

S,T𝔄 and λ, we have T**=T,(ST)*=T*S*,(λT)*=λ¯T*.

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 spaceMathworldPlanetmath 𝔄, satisfying for all S,T𝔄 :


We can easily see that A*=A . By the above axioms a C*–algebra is a special case of a Banach algebraMathworldPlanetmath where the latter requires the above norm property but not the involution (*) property. Given Banach spaces E,F the space (E,F) of (bounded) linear operators from E to F forms a Banach space, where for E=F, the space (E)=(E,E) is a Banach algebra with respect to the norm


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

T:=sup{(Tu,Tu):uH,(u,u)=1}, and Tu2=(Tu,Tu)=(u,T*Tu)T*Tu2.

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


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

For a C*–algebra 𝔄, we say that T𝔄 is self–adjoint if T=T* . Accordingly, the self–adjoint part 𝔄sa of 𝔄 is a real vector space since we can decompose T𝔄sa 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 functionsMathworldPlanetmathPlanetmath 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,T𝔄, we have


A JLB–algebra is a JB–algebra 𝔄 together with a Poisson bracket for which it becomes a Jordan–Lie algebra for some 20 . 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 𝔄sa, JLB and Poisson algebras.

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

A JB–algebra which is monotoneMathworldPlanetmath completePlanetmathPlanetmathPlanetmathPlanetmath 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 projectionsPlanetmathPlanetmath on (H) on which to study quantum logicPlanetmathPlanetmath (see later). BW-algebras have the following property: whereas 𝔄sa is a J(L)B–algebra, the self adjoint part of a von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath is a JBW–algebra.

A JC–algebra is a norm closed real linear subspace of (H)sa which is closed under the bilinear productMathworldPlanetmathPlanetmath ST=12(ST+TS) (non–commutative and nonassociative). Since any norm closed Jordan subalgebra of (H)sa is a JB–algebra, it is natural to specify the exact relationship between JB and JC–algebras, at least in finite dimensionsPlanetmathPlanetmathPlanetmathPlanetmath. In order to do this, one introduces the ‘exceptional’ algebra H3(𝕆), the algebra of 3×3 Hermitian matricesMathworldPlanetmath 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 Landsman (1998).


  • 1 Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston-Basel-Berlin.(2003).
Title Jordan-Banach and Jordan-Lie algebras
Canonical name JordanBanachAndJordanLieAlgebras
Date of creation 2013-03-22 18:14:05
Last modified on 2013-03-22 18:14:05
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 32
Author bci1 (20947)
Entry type Topic
Classification msc 08A99
Classification msc 08A05
Classification msc 08A70
Synonym quantum operator algebrasPlanetmathPlanetmath
Related topic Algebras2
Related topic CAlgebra3
Related topic AlgebraicCategoryOfLMnLogicAlgebras
Related topic NonAbelianStructures
Related topic AbelianCategory
Related topic AxiomsForAnAbelianCategory
Related topic GeneralizedVanKampenTheoremsHigherDimensional
Related topic AxiomaticTheoryOfSupercategories
Related topic AlgebraicCategoryOfLMnLogicAlgebras
Related topic Categorical
Defines Jordan algebra
Defines Jordan-Banach algebra
Defines Jordan-Lie algebra