Coarse Graining Presheaf G


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Coarse graining presheaf G

Definition 2.4   A coarse graining presheaf on the category $ \mathcal{O}$ of bounded self adjoints operators (see Category of self-acjoint operators. ) is the contravariant functor 3.2 $ G:\mathcal{O}\rightarrow Set$ such that
  • each object $ \hat{A}\in\mathcal{O}$ gets mapped to its spectral algebra $ W_A$ i.e. $ G(\hat{A})=W_A$


  • given an $ \mathcal{O}$-map $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$ the corresponding function in G is $ G(f_{\mathcal{O}}):W_A\rightarrow W_B$ such that

    $\displaystyle G(f_{\mathcal{O}})(\hat{E}[A\in\Delta])=\hat{E}[f(A)\in f(\Delta)]$    

    or alternatively

    $\displaystyle G(f_{\mathcal{O}})(\hat{P}_{A\in\Delta})=\hat{P}_{f(A)\in f(\Delta)}$    

    where $ \hat{P}_{A\in\Delta}\in W_A$ and $ \hat{P}_{f(A)\in f(\Delta)}\in W_B$

In order to show that G, as defined above, is indeed a presheaf (Definition 3.6), we need to show that

$\displaystyle G(f_{\mathcal{O}}o g_{\mathcal{O}})=G(g_{\mathcal{O}}) o G(f_{\mathcal{O}})$    

for $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$ and $ g_{\mathcal{O}}:\hat{C}\rightarrow\hat{B}$

Proof.

$\displaystyle G(f_{\mathcal{O}}o g_{\mathcal{O}})(\hat{P}_{A\in\Delta})$ $\displaystyle =\hat{P}_{f(g(A))\in f(g(\Delta))}=$    
$\displaystyle \hat{P}_{(f o g)(A))\in(f o g)(\Delta))}$ $\displaystyle =G(g_{\mathcal{O}})(\hat{P}_{f(A)\in f(\Delta))}=$    
$\displaystyle G(g_{\mathcal{O}})[G(f_{\mathcal{O}}(\hat{P}_{A\in\Delta})]$ $\displaystyle =[G(g_{\mathcal{O}})o G(f_{\mathcal{O}})](\hat{P}_{A\in\Delta})$    

$ \qedsymbol$

Since this result is independent of $ \hat{P}_{A\in\Delta}$ then $ G(f_{\mathcal{O}}o g_{\mathcal{O}})=G(f_{\mathcal{O}})o G(g_{\mathcal{O}})$ follows. From the above definition it is clear that the map $ G(f_{\mathcal{O}})$ is an inclusion map. In fact, given any two spectral algebras $ W_A$ and $ W_B$ such that $ W_B\subseteq W_A$ ( $ \hat{B}=f(\hat{A}$)), then $ G(f_{\mathcal{O}}):W_B\rightarrow W_A$ represents the embedding of the spectral algebra $ W_{f(A)}=W_B$ into the spectral algebra $ W_A$. In terms of the lattice of operators, the action of $ G(f_{\mathcal{O}})$ is to move from one operator to the one immediately above it, i.e. to the operator that covers it, where the latter operator represents a weaker statement than the former operator.

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Cecilia Flori 2007-01-02