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 $\bR$ or $\bC$ ) equipped with a bilinear and distributive multiplication $\circ$ . Note that $E$ is not necessarily commutative or associative. A Jordan algebra (over $\bR$ ), is an algebra over $\bR$ for which:

,

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 $\mathfrak A_{\bR}$ together with a Jordan product $\circ$ and Poisson bracket $\{~,~\}$ , satisfying :

1.

for all $S, T \in \mathfrak A_{\bR}$ ,

2.

the Leibniz rule holds $ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$ for all $S, T, W \in \mathfrak A_{\bR}$ , along with

3.

the Jacobi identity :

$ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$

4.

for some $\hslash^2 \in \bR$ , 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 $Q = \bR^3$ of a moving particle whose phase space is the cotangent bundle $T^* \bR^3 \cong \bR^6$ , and for which the space of (classical) observables is taken to be the real vector space of smooth functions

$$\mathfrak A^0_{\bR} = C^{\infty}(T^* R^3, \bR)$$ . The usual pointwise multiplication of functions $fg$ defines a bilinear map on $\mathfrak A^0_{\bR}$ , which is seen to be commutative and associative. Further, the Poisson bracket on functions

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

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 $\lambda \in \bC$ , 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 $S, T \in \mathfrak A$ :

We 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 $\phi : %%@ \mathfrak A \lra \mathfrak B$ , 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 $S, T \in \mathfrak A_{\bR}$ , we have A JLB-algebra is a JB-algebra $\mathfrak A_{\bR}$ 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.

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).

Bibliography

1

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