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# Jordan-Banach and Jordan-Lie algebras

# 0.0.1 Definitions of Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras

Firstly, a specific *algebra* consists of a vector space $E$ over a ground field (typically $\mathbb{R}$ or $\mathbb{C}$)
equipped with a bilinear and distributive multiplication $\circ$ . Note that $E$ is not
necessarily commutative or associative.

A *Jordan algebra* (over $\mathbb{R}$), is an algebra over $\mathbb{R}$ for which:

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

for all elements $S,T$ 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 $\mathfrak{A}_{{\mathbb{R}}}$
together with a *Jordan product* $\circ$ and *Poisson bracket*

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

- 1.
for all $S,T\in\mathfrak{A}_{{\mathbb{R}}}$, $\begin{aligned}\displaystyle S\circ T&\displaystyle=T\circ S\\ \displaystyle\{S,T\}&\displaystyle=-\{T,S\}\end{aligned}$

- 2.
the

*Leibniz rule*holds$\{S,T\circ W\}=\{S,T\}\circ W+T\circ\{S,W\}$ for all $S,T,W\in\mathfrak{A}_{{\mathbb{R}}}$, along with

- 3.
the

*Jacobi identity*:$\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$

- 4.
for some $\hslash^{2}\in\mathbb{R}$, there is the

*associator identity*:$(S\circ T)\circ W-S\circ(T\circ W)=\frac{1}{4}\hslash^{2}\{\{S,W\},T\}~{}.$

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

$\mathfrak{A}^{0}_{{\mathbb{R}}}=C^{{\infty}}(T^{*}R^{3},\mathbb{R})$ |

. The usual pointwise multiplication of functions $fg$ defines a bilinear map on $\mathfrak{A}^{0}_{{\mathbb{R}}}$, which is seen to be commutative and associative. Further, the Poisson bracket on functions

$\{f,g\}:=\frac{\partial f}{\partial p^{i}}\frac{\partial g}{\partial q_{i}}-% \frac{\partial f}{\partial q_{i}}\frac{\partial g}{\partial p^{i}}~{},$ |

# 0.0.3 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 $\lambda\in\mathbb{C}$, we have $T^{{**}}=T~{},~{}(ST)^{*}=T^{*}S^{*}~{},~{}(\lambda T)^{*}=\bar{\lambda}T^{*}~% {}.$

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 $S,T\in\mathfrak{A}$ :

$\begin{aligned}\displaystyle\|S\circ T\|&\displaystyle\leq\|S\|~{}\|T\|~{},\\ \displaystyle\|T^{*}T\|^{2}&\displaystyle=\|T\|^{2}~{}.\end{aligned}$

We can easily see that $\|A^{*}\|=\|A\|$ . 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

$\|T\|:=\sup\{\|Tu\|:u\in E~{},~{}\|u\|=1\}~{}.$

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 :

$\|T\|:=\sup\{(Tu,Tu):u\in H~{},~{}(u,u)=1\}~{},$ and $\|Tu\|^{2}=(Tu,Tu)=(u,T^{*}Tu)\leq\|T^{*}T\|~{}\|u\|^{2}~{}.$

By a morphism between C*–algebras $\mathfrak{A},\mathfrak{B}$ we mean a linear map $\phi:\mathfrak{A}{\longrightarrow}\mathfrak{B}$, 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 :

$T=T^{{\prime}}+T^{{{}^{{\prime\prime}}}}:=\frac{1}{2}(T+T^{*})+\iota(\frac{-% \iota}{2})(T-T^{*})~{}.$

A *commutative* C*–algebra is one for which the associative multiplication is commutative. Given a commutative C*–algebra $\mathfrak{A}$, 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 $S,T\in\mathfrak{A}_{{\mathbb{R}}}$, we have

$\begin{aligned}\displaystyle\|S\circ T\|&\displaystyle\leq\|S\|~{}\|T\|~{},\\ \displaystyle\|T\|^{2}&\displaystyle\leq\|S^{2}+T^{2}\|~{}.\end{aligned}$

A *JLB–algebra* is a JB–algebra $\mathfrak{A}_{{\mathbb{R}}}$ 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.

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 (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.

A *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 [1].

The above definitions and constructions follow the approach of Alfsen and Schultz (2003) and Landsman (1998).

# References

- 1 Alfsen, E.M. and F. W. Schultz: Geometry of State Spaces of Operator Algebras, Birkhäuser, Boston-Basel-Berlin.(2003).

## Mathematics Subject Classification

08A99*no label found*08A05

*no label found*08A70

*no label found*

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