# 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 Failed to parse (unknown function "\bR"): {\displaystyle \bR} or Failed to parse (unknown function "\bC"): {\displaystyle \bC} ) equipped with a bilinear and distributive multiplication $\circ$~. Note that $E$ 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 (unknown function "\bR"): {\displaystyle \bR} for which:
</math> S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 $,forallelements<math>S,T$ 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 Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}}
together with a Jordan product $\circ$ and Poisson bracket
$\{~,~\}$, satisfying~:

#### 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 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 $fg$ 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 $k^{2}$ 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 Failed to parse (unknown function "\bC"): {\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
</math> \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. $Onecaneasilyseethat<math>\Vert A^{*}\Vert =\Vert A\Vert$~. 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 </math> \Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$ByamorphismbetweenC*--algebras<math>{\mathfrak {A}},{\mathfrak {B}}$ we mean a linear map Failed to parse (unknown function "\lra"): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\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 $\hslash ^{2}\geq 0$~. 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 Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} with $k^{2}\geq 0$, its complexification ${\mathfrak {A}}$ is a $C^{*}$-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 ${\mathcal {L}}(H)$ 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 </math>\mathcal L(H)^{sa}$whichisclosedunderthebilinearproduct$S \circ T = \frac{1}{2}(ST + TS)$(non--commutativeandnonassociative).SinceanynormclosedJordansubalgebraof$\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})$,thealgebraof$3 \times 3$Hermitianmatriceswithvaluesintheoctonians$\mathbb O$~.ThenafinitedimensionalJB--algebraisaJC--algebraifandonlyifitdoesnotcontain<math>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).