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%%% Filename: JordanBanachAndJordanLieAlgebras.tex
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\begin{document}
\subsubsection{Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras:
Definitions and Relationships to Poisson and C*-algebras}
Firstly, a specific \emph{algebra} consists of a \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $E$ over a ground \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (typically $\bR$ or $\bC$) equipped with a bilinear and distributive multiplication $\circ$~. Note that $E$ is not necessarily commutative or associative.
A \emph{Jordan algebra} (over $\bR$), is an algebra over $\bR$ for which:
$ \begin{aligned} S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2
\end{aligned}$,
for all elements $S, T$ of the algebra.
It is worthwhile noting now that in the \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} theory of Jordan algebras, an important role is played by the \emph{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 \emph{Jordan \htmladdnormallink{Lie Algebra}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html}} is a real vector space $\mathfrak A_{\bR}$
together with a \emph{Jordan product} $\circ$ and \emph{Poisson bracket}
$\{~,~\}$, satisfying~:
\begin{itemize}
\item[1.] for all $S, T \in \mathfrak A_{\bR}$,
$\begin{aligned} S \circ T &= T \circ S \\ \{S, T \} &= - \{T,
S\} \end{aligned}$
\item[2.] the \emph{Leibniz rule} holds
$$ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\},$$
for all $S, T, W \in \mathfrak A_{\bR}$, along with
\item[3.]
the \emph{Jacobi \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html}}~:
$$ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$$
\item[4.]
for some $\hslash^2 \in \bR$, there is the \emph{associator identity} ~:
$$(S \circ T) \circ W - S \circ (T \circ W) = \frac{1}{4} \hslash^2 \{\{S, W \}, T \}~.$$
\end{itemize}
\subsubsection{Poisson algebra}
By a \emph{\htmladdnormallink{Poisson algebra}{http://planetphysics.us/encyclopedia/PoissonRing.html}} we mean a Jordan algebra in which $\circ$ is associative. The
usual algebraic \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} automorphism, \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}, etc.) apply to Jordan-Lie (Poisson) algebras (see Landsman, 2003).
Consider the classical configuration space $Q = \bR^3$ of a moving \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} whose phase space is the cotangent bundle $T^* \bR^3 \cong \bR^6$, and for which the space of (classical) \htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} is taken to be the real vector space of smooth \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} $$\mathfrak A^0_{\bR} = C^{\infty}(T^* R^3, \bR)$$~. The usual pointwise multiplication of functions $fg$ defines a \htmladdnormallink{bilinear map}{http://planetphysics.us/encyclopedia/BilinearMap.html} on $\mathfrak A^0_{\bR}$, which is seen to be commutative and associative. Further, the Poisson bracket on functions
$$\{f, g \} := \frac{\del f}{\del p^i} \frac{\del g}{\del q_i} - \frac{\del
f}{\del q_i} \frac{\del g}{\del p^i} ~,$$
which can be easily seen to satisfy the Liebniz rule above. The axioms above then set the stage of passage to quantum mechanical \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} which the \htmladdnormallink{parameter}{http://planetphysics.us/encyclopedia/Parameter.html} $k^2$ suggests.
\subsubsection{C*--algebras (C*--A), JLB and JBW Algebras}
An \emph{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
$\lambda \in \bC$, we have also
$$ T^{**} = T~,~ (ST)^* = T^* S^*~,~ (\lambda T)^* = \bar{\lambda} T^*~. $$
A \emph{*--algebra} is said to be a complex associative algebra together with an involution $*$~.
A \emph{C*--algebra} is a simultaneously a *--algebra and a \htmladdnormallink{Banach space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $\mathfrak A$, satisfying for all $S, T \in \mathfrak A$~:
\bigbreak
$ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T
\Vert^2 & = \Vert T\Vert^2 ~. \end{aligned}$
One can easily see that $\Vert A^* \Vert = \Vert A \Vert$~. By the above axioms a C*--algebra is a special case of a Banach algebra where the latter requires the above \htmladdnormallink{norm}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} property but not the involution (*) property. Given Banach spaces $E, F$ the space $\mathcal L(E, F)$ of (bounded) \htmladdnormallink{linear operators}{http://planetphysics.us/encyclopedia/Commutator.html} 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 \htmladdnormallink{quantum field theory}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} one may start with a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $H$, and consider the Banach algebra of bounded linear operators $\mathcal L(H)$ which given to be closed under the usual algebraic \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} and taking adjoints, forms a
$*$--algebra of bounded \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}, 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~.$
By a morphism between C*--algebras $\mathfrak A,\mathfrak B$ we mean a linear map $\phi : \mathfrak A \lra \mathfrak B$, such that for all $S, T \in \mathfrak A$, the following hold~:
$$\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, $$
where a \htmladdnormallink{bijective}{http://planetphysics.us/encyclopedia/Bijective.html} morphism is said to be an isomorphism (in which case it is then an isometry). A fundamental \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \emph{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 \emph{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 \emph{Jordan--Banach algebra} (a JB--algebra for short) is both a real Jordan algebra and a Banach space, where for all $S, T \in \mathfrak A_{\bR}$, we have
$ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. \end{aligned}$
A \emph{JLB--algebra} is a JB--algebra $\mathfrak A_{\bR}$ 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 \htmladdnormallink{quantization}{http://planetphysics.us/encyclopedia/MoyalDeformation.html}, there are fundamental relations between $\mathfrak A^{sa}$, JLB and Poisson algebras.
Conversely, given a JLB--algebra $\mathfrak A_{\bR}$ with $k^2 \geq 0$, its
complexification $\mathfrak A$ is a $C^*$-algebra under the operations~:
$\begin{aligned} S T &:= S \circ T - \frac{\iota}{2} k \times{\left\{S,T\right\}}_k ~, {(S + \iota T)}^* &:= S-\iota T . \end{aligned}$
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 \emph{\htmladdnormallink{JBW-algebra}{http://planetphysics.us/encyclopedia/LocallyCompactQuantumGroup.html}}. These appeared in the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} of von Neumann who developed a (orthomodular) lattice theory of projections on $\mathcal L(H)$ on which to study \htmladdnormallink{quantum logic}{http://planetphysics.us/encyclopedia/LQG2.html}. BW-algebras have the following property: whereas $\mathfrak A^{sa}$ is a J(L)B--algebra, the self adjoint part of a \htmladdnormallink{von Neumann algebra}{http://planetphysics.us/encyclopedia/CoordinateSpace.html} is a JBW--algebra.
A \emph{JC--algebra} is a norm closed real linear subspace of $\mathcal
L(H)^{sa}$ which is closed under the bilinear product $S \circ T = \frac{1}{2}(ST + TS)$ (non--commutative and nonassociative). Since any norm closed Jordan subalgebra of $\mathcal L(H)^{sa}$ 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 $H_3({\mathbb O})$, the algebra of $3 \times 3$ Hermitian matrices with values in the octonians $\mathbb O$~. Then a finite dimensional JB--algebra is a
JC--algebra if and only if it does not contain $H_3({\mathbb O})$ as a (direct) summand \cite{AS}.
The above definitions and constructions follow the approach of Alfsen and Schultz (2003), and also reported earlier by Landsman (1998).
\begin{thebibliography}{9}
\bibitem{AS}
Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkh\"auser, Boston-Basel-Berlin.(2003).
\end{thebibliography}
\end{document}
