C*-Clifford algebra

Preliminary data for the definition of a C*-Clifford algebra

Given a general Hilbert space $ \mathcal{H}$ , one can define an associated $ C^*$ -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 $ C^*$ -Clifford algebra, are presented in the following brief paragraph and diagram.

A non-commutative quantum observable algebra (QOA) is a Clifford algebra.

Definition 0.1   Let us briefly recall the notion of a Clifford algebra with the above notations and auxiliary concepts. Consider first a pair $ (V, Q)$ , where $ V$ denotes a real vector space and $ Q$ is a quadratic form on $ V$  . Then, the Clifford algebra associated to $ V$ , denoted here as $ {\rm Cl}(V) = {\rm Cl}(V, Q)$ , is the algebra over $ \mathbb{R}$ generated by $ V$ , where for all $ v, w \in V$ , the relations: $ v \cdot w + w \cdot v = -2 Q(v,w)~,$ are satisfied; in particular, $ v^2 = -2Q(v,v)$  .

If $ W$ is an algebra and $ c : V {\longrightarrow}W$ is a linear map satisfying $ c(w) c(v) + c(v) c(w) = - 2Q (v, w)~, $ 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 $ C^*$ -Clifford algebra as $ {\rm Cl}[\mathcal{H}]$ when

$\displaystyle \mathcal{H} = V, $

where $ V$ is a real vector space, as specified above.

Also note that the Clifford algebra is sometimes denoted as $ Cliff(Q,V)$ .



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