Jordan-Banach and Jordan-Lie algebras

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).



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.