PlanetMath: $C^*$-Clifford algebra

Given a general Hilbert space $\mathcal{H}$ , one can define an associated

-Clifford algebra, ${\rm Cl}[\mathcal{H}]$ , which admits a canonical representation on $\mathcal L(\mathbb{F}(\mathcal{H}))$ the bounded linear operators on the Fock space $\mathbb{F}(\mathcal{H})$ of $\mathcal{H}$ , (as in Plymen and Robinson, 1994), and hence one has a natural sequence of maps $\mathcal{H} {\longrightarrow}{\rm Cl}[\mathcal{H}] {\longrightarrow}\mathcal L(\mathbb{F}(\mathcal{H}))~.$

The details and notation related to the definition of a

-Clifford algebra, are presented in the following brief paragraph and diagram.

The Non-Commutative Quantum Observable Algebra is a Clifford Algebra.

Definition 0.1 Let us briefly define the notion of a Clifford algebra with the above notations and auxiliary concepts. Consider first a pair

, where

denotes a real vector space and

is a quadratic form on

. Then, the Clifford algebra associated to

, denoted here as ${\rm Cl}(V) = {\rm Cl}(V, Q)$ , is the algebra over $\mathbb{R}$ generated by

, where for all $v, w \in V$ , the relations: $v \cdot w + w \cdot v = -2 Q(v,w)~,$ are satisfied; in particular,

If is an algebra and $c : V {\longrightarrow}W$ is a linear map satisfying then there exists a unique algebra homomorphism $\phi : {\rm Cl}(V) {\longrightarrow}W$ such that the diagram

$\xymatrix{&&\hspace*{-1mm}{\rm Cl}(V)\ar[ddrr]^{\phi}&&\\ &&&&\\ \hspace{1mm} V \ar[uurr]^{{\rm Cl}} \ar[rrrr]_{c}&&&& W\hspace{1mm}}$

commutes. (It is in this sense that ${\rm Cl}(V)$ is considered to be `universal').

Then, with the above notation, one has the precise definition of the -Clifford algebra as ${\rm Cl}[\mathcal{H}]$ when $\mathcal{H}$ is selected as . To simplify notation we shall then replace ${\rm Cl}[\mathcal{H}]$ by .