Category of Boolean subalgebras


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Category of Boolean subalgebras

Definition 1.11   [2] [3] The category $ \mathcal{W}$ of Boolean subalgebras of the lattice $ P(\mathcal{H})$ has:
  • as objects, the individual Boolean subalgebras, i.e.elements $ W\in\mathcal{W}$ which represent spectral algebras associated with different operators.
  • as morphisms, the arrows between objects of $ \mathcal{W}$, such that a morphism $ i_{W_1W_2}:W_1\rightarrow W_2$ exists iff $ W_1\subseteq W_2$.

From the definition of morphisms it follows that there is, at most, one morphisms between any two elements of $ \mathcal{W}$, therefore $ \mathcal{W}$ forms a poset under subalgebras inclusion $ W_1\subseteq W_2$. To show that $ \mathcal{W}$, as defined above, is indeed a category, we need to define the identity arrow and the composite arrow. The identity arrow in $ \mathcal{W}$ is defined as $ id_W:W\rightarrow W$, which corresponds to $ W\subseteq W$, whereas, given two $ \mathcal{W}$-arrows $ i_{W_1W_2}:W_1\rightarrow W_2$ $ (W_1\subseteq W_2)$ and $ i_{W_2W_3}:W_2\rightarrow W_3$ $ (W_2\subseteq W_3)$ the composite $ i_{W_2W_3}o i_{W_1W_2}$ corresponds to $ W_1\subseteq W_3$.

Example 1.2   An example of the category $ \mathcal{W}$ can be formed in the following way: consider a category formed by four objects: $ \hat{A}$,$ \hat{B}$,$ \hat{C}$,$ \hat{1}$ such that the spectral decomposition is the following:

$\displaystyle \hat{A}$ $\displaystyle =a_1\hat{P}_1+a_2\hat{P}_2+a_3\hat{P}_3$    
$\displaystyle \hat{B}$ $\displaystyle =b_1(\hat{P}_1\vee\hat{P}_2)+b_2\hat{P}_3$    
$\displaystyle \hat{C}$ $\displaystyle =C_1(\hat{P}_1\vee\hat{P}_3)+c_2\hat{P}_2$    

then, the spectral algebras are the following:

$\displaystyle W_A$ $\displaystyle =\{\hat{0},\hat{P}_1,\hat{P}_2,\hat{P}_3,\hat{P}_1\vee\hat{P}_3,\hat{P}_1\vee\hat{P}_2,\hat{P}_3\vee\hat{P}_2,\hat{1}\}$    
$\displaystyle W_B$ $\displaystyle =\{\hat{0},\hat{P}_3,\hat{P}_1\vee\hat{P}_2,\hat{1}\}$    
$\displaystyle W_C$ $\displaystyle =\{\hat{0},\hat{P}_2,\hat{P}_1\vee\hat{P}_3\hat{1}\}$    
$\displaystyle W_1$ $\displaystyle =\{\hat{1}\}$    

The relation between the spectral algebras is given by the following diagram:

$\displaystyle \xymatrix{ &&W_B\ar[rrd]&&\\ W_1\ar[rru]\ar[rrd]&&&&W_A\\ &&W_C\ar[rru]&& }$

where the arrows are subset inclusions.

Relation between categories
The Categories, as defined above, can be related to one and another through the spectral algebra functor.

Definition 1.12   The spectral algebra functor is a contravariant functor $ W:\mathcal{O}\rightarrow\mathcal{W}$ such that:
  • each object $ \hat{A}\in\mathcal{O}$ is mapped to the object $ W_A\in\mathcal{W}$, where $ W_A$ is the spectral algebra of $ \hat{A}$
  • given an $ \mathcal{O}$-arrow $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$ then, the corresponding $ \mathcal{W}$-arrow is $ i_{W_AW_B}:W_A\rightarrow W_B$ which is defined as subset inclusion.

The above definition of morphisms in W as subset inclusions is motivated by the following reasoning: let us consider an object $ \hat{A}\in\mathcal{O}$, whose spectral algebra is $ W_A\in\mathcal{W}$. If there exists a map $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$, such that $ \hat{B}=f(\hat{A})$, then, from the Spectral Theorem it follows that the spectral algebra $ W_B$ of $ \hat{B}$ is a subalgebra of $ W_A$, i.e. $ W_B\subseteq W_A$. Therefore, to each map $ f_{\mathcal{O}}:\hat{B}\rightarrow\hat{A}$, there corresponds a unique map $ i_{W_BW_A}:W_B\rightarrow W_A$, which represents subset inclusion.

Next: Category Hilb (Hilbert spaces) Up: Categories in Quantum Mechanics Previous: The Category of bounded
Cecilia Flori 2006-11-20