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 (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 \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 (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 \mathfrak A_{\bR}} together with a Jordan product $\circ$ and Poisson bracket

$\{~,~\}$ , satisfying~:

 \item[1.] for all Failed to parse (unknown function "\bR"): {\displaystyle S, T \in \mathfrak A_{\bR}}
,  </math> S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} Failed to parse (unknown function "\item"): {\displaystyle   \item[2.] the ''Leibniz rule''  holds  <math> \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\},}
 for all Failed to parse (unknown function "\bR"): {\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 Failed to parse (unknown function "\bR"): {\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 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 (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 \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 (unknown function "\bR"): {\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 (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 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 ^[1].