PlanetPhysics/Jordan Banach and Jordan Lie Algebras

\subsubsection{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 over a ground field (typically Failed to parse (unknown function "\bR"): {\displaystyle \bR} or Failed to parse (unknown function "\bC"): {\displaystyle \bC} ) equipped with a bilinear and distributive multiplication ~. Note that is not necessarily commutative or associative.

A Jordan algebra (over Failed to parse (unknown function "\bR"): {\displaystyle \bR} ), is an algebra over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \bR} for which:

</math> S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 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 as defined by:

which is linear in each factor and for which ~. Certain examples entail setting ~.

A Jordan Lie Algebra is a real vector space Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} together with a Jordan product and Poisson bracket

, satisfying~:

 \item[1.] for all Failed to parse (unknown function "\bR"): {\displaystyle S, T \in \mathfrak A_{\bR}}
,  </math> S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} Failed to parse (unknown function "\item"): {\displaystyle   \item[2.] the ''Leibniz rule''  holds  <math> \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\},}
 for all Failed to parse (unknown function "\bR"): {\displaystyle S, T, W \in \mathfrak A_{\bR}}
, along with  \item[3.]  the Jacobi identity~:    \item[4.]  for some Failed to parse (unknown function "\bR"): {\displaystyle \hslash^2 \in \bR}
, there is the associator identity  ~: 

Poisson algebraEdit

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 Failed to parse (unknown function "\bR"): {\displaystyle Q = \bR^3} of a moving particle whose phase space is the cotangent bundle Failed to parse (unknown function "\bR"): {\displaystyle T^* \bR^3 \cong \bR^6} , and for which the space of (classical) observables is taken to be the real vector space of smooth functions Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A^0_{\bR} = C^{\infty}(T^* R^3, \bR)} ~. The usual pointwise multiplication of functions defines a bilinear map on Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A^0_{\bR}} , which is seen to be commutative and associative. Further, the Poisson bracket on functions

Failed to parse (unknown function "\del"): {\displaystyle \{f, g \} := \frac{\del f}{\del p^i} \frac{\del g}{\del q_i} - \frac{\del f}{\del q_i} \frac{\del g}{\del p^i} ~,}

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 suggests.

C*--algebras (C*--A), JLB and JBW AlgebrasEdit

An involution on a complex algebra is a real--linear map such that for all and Failed to parse (unknown function "\bC"): {\displaystyle \lambda \in \bC} , 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 , satisfying for all ~: \bigbreak

</math> \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. ~. 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 the space of (bounded) linear operators from to forms a Banach space, where for , the space is a Banach algebra with respect to the norm

In quantum field theory one may start with a Hilbert space , and consider the Banach algebra of bounded linear operators 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 we have~:

and </math> \Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~. we mean a linear map Failed to parse (unknown function "\lra"): {\displaystyle \phi : \mathfrak A \lra \mathfrak B} , such that for all , 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 in is a C*--algebra, and conversely, any C*--algebra is isomorphic to a norm--closed --algebra in for some Hilbert space ~.

For a C*--algebra , we say that is self--adjoint if ~. Accordingly, the self--adjoint part of is a real vector space since we can decompose as ~:

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

A Jordan--Banach algebra (a JB--algebra for short) is both a real Jordan algebra and a Banach space, where for all Failed to parse (unknown function "\bR"): {\displaystyle S, T \in \mathfrak A_{\bR}} , we have

Failed to parse (syntax error): {\displaystyle \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. }

A JLB--algebra is a JB--algebra Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} together with a Poisson bracket for which it becomes a Jordan--Lie algebra for some ~. 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 , JLB and Poisson algebras.

Conversely, given a JLB--algebra Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} with , its complexification is a -algebra under the operations~:

Failed to parse (syntax error): {\displaystyle S T &:= S \circ T - \frac{\iota}{2} k \times{\left\{S,T\right\}}_k ~, {(S + \iota T)}^* &:= S-\iota T . }

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 on which to study quantum logic. BW-algebras have the following property: whereas 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 </math>\mathcal L(H)^{sa}S \circ T = \frac{1}{2}(ST + TS)\mathcal L(H)^{sa}Failed to parse (syntax error): {\displaystyle 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 } H_3({\mathbb O})3 \times 3\mathbb 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).

All SourcesEdit



  1. 1.0 1.1 Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkh\"auser, Boston-Basel-Berlin.(2003).