# 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 ${\displaystyle E}$ over a ground field (typically $\displaystyle \bR$ or $\displaystyle \bC$ ) equipped with a bilinear and distributive multiplication ${\displaystyle \circ }$~. Note that ${\displaystyle E}$ is not necessarily commutative or associative.

A Jordan algebra (over $\displaystyle \bR$ ), is an algebra over $\displaystyle \bR$ for which:

[/itex] S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 ${\displaystyle ,forallelements$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 ${\displaystyle \{STW\}}$ as defined by: ${\displaystyle \{STW\}=(S\circ T)\circ W+(T\circ W)\circ S-(S\circ W)\circ T~,}$ which is linear in each factor and for which ${\displaystyle \{STW\}=\{WTS\}}$~. Certain examples entail setting ${\displaystyle \{STW\}={\frac {1}{2}}\{STW+WTS\}}$~. A Jordan Lie Algebra is a real vector space $\displaystyle \mathfrak A_{\bR}$ together with a Jordan product ${\displaystyle \circ }$ and Poisson bracket ${\displaystyle \{~,~\}}$, satisfying~:  \item[1.] for all $\displaystyle S, T \in \mathfrak A_{\bR}$ ,$ S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} $\displaystyle \item[2.] the ''Leibniz rule'' holds $\{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\},$ for all $\displaystyle S, T, W \in \mathfrak A_{\bR}$ , along with \item[3.] the Jacobi identity~: ${\displaystyle \{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}}$ \item[4.] for some $\displaystyle \hslash^2 \in \bR$ , there is the associator identity ~: ${\displaystyle (S\circ T)\circ W-S\circ (T\circ W)={\frac {1}{4}}\hslash ^{2}\{\{S,W\},T\}~.}$  #### Poisson algebra By a Poisson algebra we mean a Jordan algebra in which ${\displaystyle \circ }$ 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 $\displaystyle Q = \bR^3$ of a moving particle whose phase space is the cotangent bundle $\displaystyle T^* \bR^3 \cong \bR^6$ , and for which the space of (classical) observables is taken to be the real vector space of smooth functions $\displaystyle \mathfrak A^0_{\bR} = C^{\infty}(T^* R^3, \bR)$ ~. The usual pointwise multiplication of functions ${\displaystyle fg}$ defines a bilinear map on $\displaystyle \mathfrak A^0_{\bR}$ , which is seen to be commutative and associative. Further, the Poisson bracket on functions $\displaystyle \{f, g \} := \frac{\del f}{\del p^i} \frac{\del g}{\del q_i} - \frac{\del f}{\del q_i} \frac{\del g}{\del p^i} ~,$ 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 ${\displaystyle k^{2}}$ suggests. #### C*--algebras (C*--A), JLB and JBW Algebras An involution on a complex algebra ${\displaystyle {\mathfrak {A}}}$ is a real--linear map ${\displaystyle T\mapsto T^{*},}$ such that for all ${\displaystyle S,T\in {\mathfrak {A}}}$ and $\displaystyle \lambda \in \bC$ , we have also ${\displaystyle T^{**}=T~,~(ST)^{*}=T^{*}S^{*}~,~(\lambda T)^{*}={\bar {\lambda }}T^{*}~.}$ A *--algebra is said to be a complex associative algebra together with an involution ${\displaystyle *}$~. A C*--algebra is a simultaneously a *--algebra and a Banach space ${\displaystyle {\mathfrak {A}}}$, satisfying for all ${\displaystyle S,T\in {\mathfrak {A}}}$~: \bigbreak$ \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. ${\displaystyle Onecaneasilyseethat$\Vert A^{*}\Vert =\Vert A\Vert }$~. 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 ${\displaystyle E,F}$ the space ${\displaystyle {\mathcal {L}}(E,F)}$ of (bounded) linear operators from ${\displaystyle E}$ to ${\displaystyle F}$ forms a Banach space, where for ${\displaystyle E=F}$, the space ${\displaystyle {\mathcal {L}}(E)={\mathcal {L}}(E,E)}$ is a Banach algebra with respect to the norm ${\displaystyle \Vert T\Vert :=\sup\{\Vert Tu\Vert :u\in E~,~\Vert u\Vert =1\}~.}$ In quantum field theory one may start with a Hilbert space ${\displaystyle H}$, and consider the Banach algebra of bounded linear operators ${\displaystyle {\mathcal {L}}(H)}$ which given to be closed under the usual algebraic operations and taking adjoints, forms a ${\displaystyle *}$--algebra of bounded operators, where the adjoint operation functions as the involution, and for ${\displaystyle T\in {\mathcal {L}}(H)}$ we have~: ${\displaystyle \Vert T\Vert :=\sup\{(Tu,Tu):u\in H~,~(u,u)=1\}~,}$ and$ \Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.${\displaystyle ByamorphismbetweenC*--algebras${\mathfrak {A}},{\mathfrak {B}}}$ we mean a linear map $\displaystyle \phi : \mathfrak A \lra \mathfrak B$ , such that for all ${\displaystyle S,T\in {\mathfrak {A}}}$, the following hold~: ${\displaystyle \phi (ST)=\phi (S)\phi (T)~,~\phi (T^{*})=\phi (T)^{*}~,}$ 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 ${\displaystyle *}$--algebra ${\displaystyle {\mathcal {A}}}$ in ${\displaystyle {\mathcal {L}}(H)}$ is a C*--algebra, and conversely, any C*--algebra is isomorphic to a norm--closed ${\displaystyle *}$--algebra in ${\displaystyle {\mathcal {L}}(H)}$ for some Hilbert space ${\displaystyle H}$~. For a C*--algebra ${\displaystyle {\mathfrak {A}}}$, we say that ${\displaystyle T\in {\mathfrak {A}}}$ is self--adjoint if ${\displaystyle T=T^{*}}$~. Accordingly, the self--adjoint part ${\displaystyle {\mathfrak {A}}^{sa}}$ of ${\displaystyle {\mathfrak {A}}}$ is a real vector space since we can decompose ${\displaystyle T\in {\mathfrak {A}}^{sa}}$ as ~: ${\displaystyle T=T'+T^{''}:={\frac {1}{2}}(T+T^{*})+\iota ({\frac {-\iota }{2}})(T-T^{*})~.}$ A commutative C*--algebra is one for which the associative multiplication is commutative. Given a commutative C*--algebra ${\displaystyle {\mathfrak {A}}}$, we have ${\displaystyle {\mathfrak {A}}\cong C(Y)}$, the algebra of continuous functions on a compact Hausdorff space ${\displaystyle Y~}$. A Jordan--Banach algebra (a JB--algebra for short) is both a real Jordan algebra and a Banach space, where for all $\displaystyle S, T \in \mathfrak A_{\bR}$ , we have $\displaystyle \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~.$ A JLB--algebra is a JB--algebra $\displaystyle \mathfrak A_{\bR}$ together with a Poisson bracket for which it becomes a Jordan--Lie algebra for some ${\displaystyle \hslash ^{2}\geq 0}$~. 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 ${\displaystyle {\mathfrak {A}}^{sa}}$, JLB and Poisson algebras. Conversely, given a JLB--algebra $\displaystyle \mathfrak A_{\bR}$ with ${\displaystyle k^{2}\geq 0}$, its complexification ${\displaystyle {\mathfrak {A}}}$ is a ${\displaystyle C^{*}}$-algebra under the operations~: $\displaystyle S T &:= S \circ T - \frac{\iota}{2} k \times{\left\{S,T\right\}}_k ~, {(S + \iota T)}^* &:= S-\iota T .$ 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 ${\displaystyle {\mathcal {L}}(H)}$ on which to study quantum logic. BW-algebras have the following property: whereas ${\displaystyle {\mathfrak {A}}^{sa}}$ 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$\mathcal L(H)^{sa}${\displaystyle whichisclosedunderthebilinearproduct}$S \circ T = \frac{1}{2}(ST + TS)${\displaystyle (non--commutativeandnonassociative).SinceanynormclosedJordansubalgebraof}$\mathcal L(H)^{sa}$\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})${\displaystyle ,thealgebraof}$3 \times 3${\displaystyle Hermitianmatriceswithvaluesintheoctonians}$\mathbb O${\displaystyle ~.ThenafinitedimensionalJB--algebraisaJC--algebraifandonlyifitdoesnotcontain[itex]H_{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).

## References

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