# 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 $E$ over a ground field (typically $\displaystyle \bR$ or $\displaystyle \bC$ ) equipped with a bilinear and distributive multiplication $\circ$ ~. Note that $E$ 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 $,forallelements$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 $\{STW\}$ as defined by: $\{STW\}=(S\circ T)\circ W+(T\circ W)\circ S-(S\circ W)\circ T~,$ which is linear in each factor and for which $\{STW\}=\{WTS\}$ ~. Certain examples entail setting $\{STW\}={\frac {1}{2}}\{STW+WTS\}$ ~. A Jordan Lie Algebra is a real vector space $\displaystyle \mathfrak A_{\bR}$ together with a Jordan product $\circ$ and Poisson bracket $\{~,~\}$ , 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~: $\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$ \item[4.] for some $\displaystyle \hslash^2 \in \bR$ , there is the associator identity ~: $(S\circ T)\circ W-S\circ (T\circ W)={\frac {1}{4}}\hslash ^{2}\{\{S,W\},T\}~.$ #### Poisson algebra By a Poisson algebra we mean a Jordan algebra in which $\circ$ 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 $fg$ 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 $k^{2}$ suggests. #### C*--algebras (C*--A), JLB and JBW Algebras An involution on a complex algebra ${\mathfrak {A}}$ is a real--linear map $T\mapsto T^{*},$ such that for all $S,T\in {\mathfrak {A}}$ and $\displaystyle \lambda \in \bC$ , we have also $T^{**}=T~,~(ST)^{*}=T^{*}S^{*}~,~(\lambda T)^{*}={\bar {\lambda }}T^{*}~.$ 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 space ${\mathfrak {A}}$ , satisfying for all $S,T\in {\mathfrak {A}}$ ~: \bigbreak$ \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. $Onecaneasilyseethat$\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 $E,F$ the space ${\mathcal {L}}(E,F)$ of (bounded) linear operators from $E$ to $F$ forms a Banach space, where for $E=F$ , the space ${\mathcal {L}}(E)={\mathcal {L}}(E,E)$ is a Banach algebra with respect to the norm $\Vert T\Vert :=\sup\{\Vert Tu\Vert :u\in E~,~\Vert u\Vert =1\}~.$ In quantum field theory one may start with a Hilbert space $H$ , and consider the Banach algebra of bounded linear operators ${\mathcal {L}}(H)$ 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 $T\in {\mathcal {L}}(H)$ we have~: $\Vert T\Vert :=\sup\{(Tu,Tu):u\in H~,~(u,u)=1\}~,$ and$ \Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$ByamorphismbetweenC*--algebras${\mathfrak {A}},{\mathfrak {B}}$ we mean a linear map $\displaystyle \phi : \mathfrak A \lra \mathfrak B$ , such that for all $S,T\in {\mathfrak {A}}$ , the following hold~: $\phi (ST)=\phi (S)\phi (T)~,~\phi (T^{*})=\phi (T)^{*}~,$ 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 $*$ --algebra ${\mathcal {A}}$ in ${\mathcal {L}}(H)$ is a C*--algebra, and conversely, any C*--algebra is isomorphic to a norm--closed $*$ --algebra in ${\mathcal {L}}(H)$ for some Hilbert space $H$ ~. For a C*--algebra ${\mathfrak {A}}$ , we say that $T\in {\mathfrak {A}}$ is self--adjoint if $T=T^{*}$ ~. Accordingly, the self--adjoint part ${\mathfrak {A}}^{sa}$ of ${\mathfrak {A}}$ is a real vector space since we can decompose $T\in {\mathfrak {A}}^{sa}$ as ~: $T=T'+T^{''}:={\frac {1}{2}}(T+T^{*})+\iota ({\frac {-\iota }{2}})(T-T^{*})~.$ A commutative C*--algebra is one for which the associative multiplication is commutative. Given a commutative C*--algebra ${\mathfrak {A}}$ , we have ${\mathfrak {A}}\cong C(Y)$ , 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 $\displaystyle S, T \in \mathfrak A_{\bR}$ , 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 $\hslash ^{2}\geq 0$ ~. 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 ${\mathfrak {A}}^{sa}$ , JLB and Poisson algebras. Conversely, given a JLB--algebra $\displaystyle \mathfrak A_{\bR}$ with $k^{2}\geq 0$ , its complexification ${\mathfrak {A}}$ is a $C^{*}$ -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 ${\mathcal {L}}(H)$ on which to study quantum logic. BW-algebras have the following property: whereas ${\mathfrak {A}}^{sa}$ 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$\mathcal L(H)^{sa}$whichisclosedunderthebilinearproduct$ S \circ T = \frac{1}{2}(ST + TS)$(non--commutativeandnonassociative).SinceanynormclosedJordansubalgebraof$ \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})$,thealgebraof$ 3 \times 3$Hermitianmatriceswithvaluesintheoctonians$ \mathbb O$~.ThenafinitedimensionalJB--algebraisaJC--algebraifandonlyifitdoesnotcontain[itex]H_{3}({\mathbb {O} })$ as a (direct) summand .

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