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 [[../NormInducedByInnerProduct/|vector space]] $E$ over a ground [[../CosmologicalConstant/|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 [[../CoIntersections/|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 [[../TopologicalOrder2/|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~:

 \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 [[../Cod/|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 [[../PoissonRing/|Poisson algebra]] we mean a Jordan algebra in which $\circ$ is associative. The usual algebraic [[../Bijective/|types]] of [[../TrivialGroupoid/|morphisms]] automorphism, [[../IsomorphicObjectsUnderAnIsomorphism/|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/|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) [[../QuantumSpinNetworkFunctor2/|observables]] is taken to be the real vector space of smooth [[../Bijective/|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 [[../BilinearMap/|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 [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] which the [[../Parameter/|parameter]] $k^2$ suggests.

C*--algebras (C*--A), JLB and JBW Algebras

An involution on a complex algebra $𝔄$ is a real--linear map $T\mapsto T^{*},$ such that for all $S,T\in 𝔄$ and Failed to parse (unknown function "\bC"): {\displaystyle \lambda \in \bC} , we have also $T^{**}=T,(ST{)}^{*}=T^{*}{S}^{*},(\lambda T{)}^{*}=\overline{\lambda }{T}^{*}.$

A *--algebra is said to be a complex associative algebra together with an involution $*$~.

A C*--algebra is a simultaneously a *--algebra and a [[../NormInducedByInnerProduct/|Banach space]] $𝔄$, satisfying for all $S,T\in 𝔄$~: \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 [[../NormInducedByInnerProduct/|norm]] property but not the involution (*) property. Given Banach spaces $E,F$ the space $ℒ(E,F)$ of (bounded) [[../Commutator/|linear operators]] from $E$ to $F$ forms a Banach space, where for $E=F$, the space $ℒ(E)=ℒ(E,E)$ is a Banach algebra with respect to the norm $\Vert T\Vert :=\sup \{\Vert Tu\Vert :u\in E,\Vert u\Vert =1\}.$

In [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theory]] one may start with a [[../NormInducedByInnerProduct/|Hilbert space]] $H$, and consider the Banach algebra of bounded linear operators $ℒ(H)$ which given to be closed under the usual algebraic [[../Cod/|operations]] and taking adjoints, forms a $*$--algebra of bounded [[../QuantumOperatorAlgebra4/|operators]], where the adjoint operation functions as the involution, and for $T\in ℒ(H)$ 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~.$ByamorphismbetweenC*--algebras<math>𝔄,𝔅$ we mean a linear map Failed to parse (unknown function "\lra"): {\displaystyle \phi : \mathfrak A \lra \mathfrak B} , such that for all $S,T\in 𝔄$, the following hold~:

$\varphi (ST)=\varphi (S)\varphi (T),\varphi (T^{*})=\varphi (T{)}^{*},$

where a [[../Bijective/|bijective]] morphism is said to be an isomorphism (in which case it is then an isometry). A fundamental [[../Bijective/|relation]] is that any norm-closed $*$--algebra $𝒜$ in $ℒ(H)$ is a C*--algebra, and conversely, any C*--algebra is isomorphic to a norm--closed $*$--algebra in $ℒ(H)$ for some Hilbert space $H$~.

For a C*--algebra $𝔄$, we say that $T\in 𝔄$ is self--adjoint if $T=T^{*}$~. Accordingly, the self--adjoint part ${𝔄}^{sa}$ of $𝔄$ is a real vector space since we can decompose $T\in {𝔄}^{sa}$ as ~:

$T=T^{\prime }+T^{{\text{'}}^{\prime }}:=\frac{1}{2}(T+T^{*})+\iota (\frac{-\iota }{2})(T-T^{*}).$

A commutative C*--algebra is one for which the associative multiplication is commutative. Given a commutative C*--algebra $𝔄$, we have $𝔄\cong C(Y)$, 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 ${\hslash }^2\ge 0$~. Such JLB--algebras often constitute the real part of several widely studied complex associative algebras.

For the purpose of [[../MoyalDeformation/|quantization]], there are fundamental relations between ${𝔄}^{sa}$, JLB and Poisson algebras.

Conversely, given a JLB--algebra Failed to parse (unknown function "\bR"): {\displaystyle \mathfrak A_{\bR}} with $k^2\ge 0$, its complexification $𝔄$ 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 [[../LocallyCompactQuantumGroup/|JBW-algebra]]. These appeared in the [[../Work/|work]] of von Neumann who developed a (orthomodular) lattice theory of projections on $ℒ(H)$ on which to study [[../LQG2/|quantum logic]]. BW-algebras have the following property: whereas ${𝔄}^{sa}$ is a J(L)B--algebra, the self adjoint part of a [[../CoordinateSpace/|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(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).