08A99-JordanBanachAndJordanLieAlgebras.tex

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\pmtitle{Jordan-Banach and Jordan-Lie algebras}
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%\pmkeywords{Jordan-Lie algebras}
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%\pmkeywords{Jordan algebras in quantum operator theory and quantum algebraic topology}
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\begin{document}
\subsubsection{Definitions of Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras}

Firstly, a specific \emph{algebra} consists of a vector space $E$ over a ground field (typically $\bR$ or $\bC$) 
equipped with a bilinear and distributive multiplication $\circ$~. Note that $E$ is not 
necessarily commutative or associative.
\med
A \emph{Jordan algebra} (over $\bR$), is an algebra over $\bR$ for which:
\med 

$ \begin{aligned} S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 
\end{aligned}$, 


\med
for all elements $S, T$ of the algebra.
\med
It is worthwhile noting now that in the algebraic 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~, $
\med
which is linear in each factor and for which $\{STW\} = \{WTS\}$~. Certain examples entail 
setting $\{STW\} = \frac{1}{2}\{STW + WTS\}$~.

\med

 A \emph{Jordan Lie algebra} is a real vector space $\mathfrak A_{\bR}$
 together with a \emph{Jordan product} $\circ$ and \emph{Poisson bracket}
\bigbreak
 $\{~,~\}$, satisfying~:
 \begin{itemize}
\item[1.] for all $S, T \in \mathfrak A_{\bR}$,
$\begin{aligned} S \circ T &= T \circ S \\ \{S, T \} &= - \{T,
S\} \end{aligned}$
\med

\item[2.] the \emph{Leibniz rule} holds
\bigbreak
$ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$ 
for all $S, T, W \in \mathfrak A_{\bR}$, along with



\med

\item[3.]

the \emph{Jacobi identity}~:\\

 $ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$



\med

\item[4.]

for some $\hslash^2 \in \bR$, there is the \emph{associator identity} ~:
\bigbreak 
$(S \circ T) \circ W - S \circ (T \circ W) = \frac{1}{4} \hslash^2 \{\{S, W \}, T \}~.$



\end{itemize}



\subsubsection{Poisson algebra}


  By a \emph{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 $Q = \bR^3$ of a moving particle whose phase space 
is the cotangent bundle $T^* \bR^3 \cong \bR^6$, and for which the space of (classical) 
observables  is taken to be the real vector space of smooth functions

$$\mathfrak A^0_{\bR} = C^{\infty}(T^* R^3, \bR)$$~. The usual pointwise multiplication of 
functions $fg$ defines a bilinear map 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 systems which the parameter $k^2$ 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 
\bigbreak
$S, T \in \mathfrak A$ and $\lambda \in \bC$, we have $ T^{**} = T~,~ (ST)^* = T^* S^*~,~ %%@
(\lambda T)^* = \bar{\lambda} T^*~. $ 
\bigbreak
A \emph{*--algebra} is said to be a complex associative algebra together with an involution %%@
$*$~.


\med

A \emph{C*--algebra} is a simultaneously a *--algebra and a Banach space $\mathfrak A$, %%@
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}$
\bigbreak
We 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 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 \bigbreak
$\Vert T \Vert := \sup\{ \Vert Tu \Vert : u \in E~,~ \Vert u \Vert= 1 \}~. $
\bigbreak
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~: 
\bigbreak

$ \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~.$
\bigbreak


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~: 
\bigbreak
$\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, $ 
\bigbreak
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 $H$~.

\med

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 ~:
\bigbreak
$ T = T' + T^{''} := \frac{1}{2} (T + T^*) + \iota (\frac{-\iota}{2})(T - T^*)~.$
\bigbreak
 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 $Y~$.
 \med
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 
\bigbreak
$ \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}$
\bigbreak
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.
\bigbreak
For the purpose of quantization, there are fundamental relations between $\mathfrak A^{sa}$, %%@
JLB and Poisson algebras. 
\med
%%In fact, if $\mathfrak A$ is a C*--algebra 
%%and $\hslash \in \bR/{0}$, then $\mathfrak A^{sa}$ is a JLB--algebra when it takes 
%%its norm from $\mathfrak A$, with $k= \hslash$, and is equipped with the operations~:
%%\bigbreak
%%$\begin{aligned} S \circ T &:= \frac{1}{2}{(ST + TS)} ~,{\left\{S,T\right\}}_k &:=\frac{\iota}_k \times [S,T]} ~.$ 
%%\end{aligned}$
%%\med
%%Conversely, given a JLB--algebra $\mathfrak A_{\bR}$ with $k^2 \geq 0$, its 
%%complexification $\mathfrak A$ is a $C^*$-algebra under the operations~: 
%%\bigbreak
%%$\begin{aligned} S T &:= S \circ T - \frac{\iota}{2} k \times \left\{S,T\right\}}_ ~, {(S + \iota T)}^* &:=S-\iota T %%%~. \end{aligned}$
\bigbreak
For further details see Landsman (2003) (Thm. 1.1.9).

\med
\bigbreak
A JB--algebra which is monotone complete and admits a separating set of normal sets is 
called a \emph{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 (see later). 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.

\bigbreak
\med

	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 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}

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\end{document}