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Jordan-Banach and Jordan-Lie algebras (Topic)

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 $\mathbb{R}$ 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:

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

for all elements $S, T$ of the algebra.

It is worthwhile noting now that in the algebraic theory of Jordan algebras, an important role is played by the Jordan triple product $\{STW\}$ as defined by:

$\displaystyle \{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{equation*}\begin{aligned}S \circ T &= T \circ S \\ \{S, T \} &= - \{T, S\} \end{aligned}\end{equation*}
2.  the Leibniz rule holds

$\displaystyle \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\},$
for all $S, T, W \in \mathfrak{A}_{\mathbb{R}}$, along with
3.  

the Jacobi identity :

$\displaystyle \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$
4.  

for some $\hslash^2 \in \mathbb{R}$, there is the associator identity :

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

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 $Q = \mathbb{R}^3$ 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

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

$\displaystyle \{f, g \} := \frac{\partial f}{\partial p^i} \frac{\partial g}{\partial q_i} - \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p^i} ~,$

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 $\lambda \in \mathbb{C}$, we have also

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

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

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 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 $\Vert Tu \Vert^2 = (Tu, % Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$

By a morphism between C*–algebras $\mathfrak{A},\mathfrak{B}$ we mean a linear map $\phi : \mathfrak{A} {\longrightarrow}\mathfrak{B}$, such that for all $S, T \in \mathfrak{A}$, the following hold :

$\displaystyle \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 :

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

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 $S, T \in \mathfrak{A}_{\mathbb{R}}$, 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 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.

Conversely, given a JLB–algebra $\mathfrak{A}_{\mathbb{R}}$ with $k^2 \geq 0$, its complexification $\mathfrak{A}$ is a $C^*$-algebra under the operations :

\begin{equation*}\begin{aligned}S T &:= S \circ T - \frac{\iota}{2} k \times{\le... ...S,T\right\}}_k ~, {(S + \iota T)}^* &:= S-\iota T . \end{aligned}\end{equation*}

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 $\mathcal L(H)^{sa}$ 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 also reported earlier by Landsman (1998).

Bibliography

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



"Jordan-Banach and Jordan-Lie algebras" is owned by bci1.

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Also defines:  Jordan-Banach algebra, JB-algebra, JL-algebra, Jordan-Lie algebra, exceptional Lie algebra, Hermitian matrices, bilinear product, JC--algebra, Jordan triple product, JLB algebra, JBW algebra, complex associative algebra, commutative C*--algebra, orthomodular lattice theory
Keywords:  Jordan-Banach algebra, JB-algebra, JL-algebra, Jordan-Lie algebra, exceptional Lie algebra, Hermitian matrices, bilinear product, JC--algebra, Jordan triple product

Cross-references: von Neumann algebra, quantum logic, work, JBW-algebra, quantization, relation, bijective, operators, operations, Hilbert space, quantum field theory, linear operators, norm, Banach space, parameter, systems, bilinear map, functions, observables, isomorphism, morphisms, types, Poisson algebra, identity, Lie Algebra, algebraic, field, vector space
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This is version 5 of Jordan-Banach and Jordan-Lie algebras, born on 2009-05-03, modified 2009-05-03.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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